Thermodynamic concentric multi-layered shell model and simple symmetry diagram method

Abstract

The present invention provides a thermodynamic concentric multi-layered polyhedral shell model and a simplified symmetry diagram method. The present invention uses geometry to reveal the existence of symmetry in thermodynamics and uses intuitive symmetry to summarize and interpret a large number of thermodynamic relationships, thereby making thermodynamics simple and easy. According to physical meaning, the present invention evenly and reasonably arranges 44 thermodynamic variables of four categories of unit single-phase systems on the vertices of a concentric multi-layered polyhedral shell to form a self-consistent and complete thermodynamic model; according to the principle of symmetry equivalence, by superimposing twelve specially created rigid active patterns on a fixed two-dimensional {1,0,0} projection diagram and performing symmetry transformation, twelve categories of more than 300 thermodynamic relationships can be described one by one; using the various created patterns and the obtained C P As a result of class variables, parametric expressions for any desired partial derivatives can be derived simply and reliably.

Description

Technical Field

[0001] The present invention relates to the technical field of thermodynamics, in particular to a thermodynamic concentric multi-layered polyhedral shell model and a simple symmetric diagram method. Background Art

[0002] H.Callen proposed a theoretical explanation of thermodynamics. He believed that thermodynamics is a symmetrical science. [1,2] However, this view has not been fully verified. If one were to do so, one would first have to build a self-consistent and complete framework based on familiar thermodynamic relations, select as many thermodynamic variables as possible as elements, and then use specific and precise symmetries to summarize and interpret a large number of thermodynamic relations to verify this view.

[0003] On the other hand, many scholars have revealed the symmetry of thermodynamics in their papers. For example, FO Koenig classified many important thermodynamic relations into many different families according to whether they have the same standard form. [3,4] Other scholars use geometric figures (squares [5] ,cuboctahedron [6] , multi-layer circle [7] ,cube [8] , and Venn diagram [9] ) describes thermodynamic relationships.

[0004] From a mathematical perspective, thermodynamic quantities behave like multivariate functions. They can be manipulated using calculus. Many thermodynamic relations with similar functional forms can be grouped together. These self-similar relations can be precisely defined as symmetric functions. Through symmetry transformations, they preserve not only the functional form but also the types and relationships of the variables. Summary of the Invention

[0005] It is well known that the simplest and most intuitive symmetry transformations can be observed in an object with geometric symmetry. Therefore, we can use geometry to reveal the symmetry of thermodynamics, and then combine this symmetry to build a self-consistent and complete structural framework of thermodynamic variables, and then use symmetry to make thermodynamics simpler and easier.

[0006] In view of the above situation, the present invention provides a thermodynamic concentric multi-layer polyhedral shell model and a simple symmetry diagram method to solve the technical problem of how to use symmetry to make thermodynamics simple and easy.

[0007] To achieve the above-mentioned purpose, the technical solution adopted by the present invention is to provide a thermodynamic concentric multi-layer polyhedral shell model and a simple symmetry diagram method, wherein the method is based on the principle of symmetry equivalence, by superimposing various specially created rigid movable patterns on a fixed two-dimensional {1,0,0} projection diagram, performing symmetry transformation, and describing more than 300 thermodynamic relationships of twelve categories in a unit single-phase system one by one.

[0008] A further improvement of the thermodynamic concentric multi-layer polyhedral shell model and the simplified symmetric diagram method of the present invention is that the two-dimensional {1,0,0} projection diagram is obtained by dissecting a thermodynamic concentric multi-layer polyhedral shell model and projecting it outward from the central plane along six different <1,0,0> directions.

[0009] A further improvement of the thermodynamic concentric multi-layer polyhedral shell model and the simplified symmetric diagram method of the present invention is that the thermodynamic concentric multi-layer polyhedral shell model is composed of a cubic shell sandwiched between two octahedral shells and an outer twenty-six-sided shell, with a total of four layers; according to physical significance, forty-four thermodynamic variables of four categories in the unit single-phase system are evenly and reasonably arranged on the forty-four vertices of the model, including three pairs of conjugate independent variables, eight complete thermodynamic potentials, six first-order partial derivatives of the thermodynamic potentials, and twenty-four complete second-order partial derivatives of the thermodynamic potentials.

[0010] A further improvement of the thermodynamic concentric multi-layered polyhedral shell model and the simplified symmetric diagram method of the present invention is that, among the twenty-four secondary partial derivatives of the thermodynamic potential, in addition to the isobaric heat capacity (C P ) and isochoric heat capacity (C V ), the remaining twenty-two C P Class variables are invented according to the principle of symmetric equivalence. They are O PN ,O VN ,J TN ,J SN ,R TN ,R SN ,C Pμ ,C Vμ ,O Pμ ,O Vμ ,J Tμ ,J Sμ ,R Tμ ,R Sμ ,Λ PT ,Λ VT ,Γ PT ,Γ VT ,Λ PS ,Λ VS ,Γ PS , and Γ VS .

[0011] A further improvement of the thermodynamic concentric multi-layer polyhedral shell model and simplified symmetry diagram method of the present invention is that, after verification, it has been found that the thermodynamic symmetry exhibited by the concentric multi-layer polyhedral shell model carrying numerous thermodynamic variables is three-fold rotational axial symmetry (C3) with 'U~Φ' as the axis, and has mirror symmetry (σ) and four-fold rotational axial symmetry (C4) on three squares containing internal energy.

[0012] A further improvement of the thermodynamic concentric multi-layered polyhedral shell model and the simplified symmetric diagram method of the present invention is that, based on the principle of symmetry equivalence, a unified four-step diagram method for describing various thermodynamic relationships using specially created patterns is designed and created.

[0013] A further improvement of the thermodynamic concentric multi-layer polyhedral shell model and the simplified symmetric diagram method of the present invention is that twelve special drawings are developed specifically for describing twelve different types of thermodynamic relationships, each of which is arranged in the order of writing and is a mixture of mathematical symbols and variable selection symbols.

[0014] The further improvement of the thermodynamic concentric multi-layered polyhedral shell model and the simplified symmetric diagram method of the present invention is that these developed diagrams can not only describe and distinguish some similar but rather confusing relationships, but also create new C P class variables, derived new thermodynamic potential dependencies, and discovered three types of C P New relations between class variables.

[0015] The further improvement of the thermodynamic concentric multi-layered multi-faceted shell model and the simplified symmetric diagram method of the present invention is to utilize the newly discovered C P The relationship between class variables is derived from the complete twenty-four C P Argument expressions for class variables.

[0016] The further improvement of the thermodynamic concentric multi-layered multi-faceted shell model and the simplified symmetric diagram method of the present invention is that the symmetric diagram method utilizes the specially created various patterns and the obtained C P As a result of class variables, parametric expressions for any desired partial derivatives can be derived simply and reliably.

[0017] A further improvement of the thermodynamic concentric multi-layer polyhedral shell model and the simplified symmetric diagram method of the present invention is that, based on the research result that the sum of the thermodynamic potentials at the two ends of the diagonal of the cubic shell is always equal to the internal energy, that is, □+□*=TS-PV+μN=U(S,V,N), this is used as the criterion for the conjugate thermodynamic potential, and three thermodynamic potentials that have not yet been formally named, Φ(T,P,μ), ψ(S,V,μ), and χ(S,P,μ), are respectively given meaningful names: conjugate internal energy, conjugate Gibbs free energy, and conjugate Helmholtz free energy.

[0018] A further improvement of the thermodynamic concentric multi-layered polyhedral shell model and the simplified symmetry diagram method of the present invention is that the diagram method utilizes a self-consistent and complete overall structural model composed of elements containing forty-four thermodynamic variables and the principle of symmetry equivalence to summarize and interpret a large number of thermodynamic relationships through specific and accurate symmetry transformations, fully proving that thermodynamics is a symmetrical science.

[0019] The present invention adopts the above technical solution, so it has the following beneficial effects:

[0020] (1) The forty-four different categories of thermodynamic variables of the unit single-phase system are reasonably distributed in a three-dimensional model according to their physical meanings;

[0021] (2) It can use specially created different rigid activity patterns, superimposed on a fixed two-dimensional projection diagram, and through symmetry transformation, simply and reliably describe more than 300 thermodynamic relationships of twelve categories;

[0022] (3) Use different patterns to distinguish some similar but quite confusing partial derivatives;

[0023] (4) Ability to derive parametric expressions for any required partial derivatives;

[0024] (5) Determine the specific thermodynamic symmetry. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 It is a schematic diagram of the thermodynamic concentric multi-layer polyhedral shell model of the present invention.

[0026] Figure 2 .1 to Figure 2 .6 is the two-dimensional {1,0,0} projection diagram used in the method of the present invention, where Figure 2 .1 is the most important.

[0027] Figure 3 is a schematic diagram of a pattern used in the present invention to describe Legendre transformation, wherein: Figure 3 include Figure 2 .1((0,0,-1) projection), Figure 3 .1 (Use Figure 1 to describe Equations (1-1) and (1-2)), Figure 3 .2 (Use Figure 1 to describe Equations (1-3) and (1-4)).

[0028] Figure 4 FIG2 is a schematic diagram of the second figure used in the present invention to describe the thermodynamic identity equation, wherein: Figure 4 include Figure 2 .1((0,0,-1) projection), Figure 4 .1 (Use Figure 2 to describe formula (2-1)), Figure 4.2 (Use Figure 2 to describe formula (2-2)).

[0029] Figure 5 is a schematic diagram of the third figure used in the present invention to describe Maxwell's equations, wherein: Figure 5 include Figure 2 .1((0,0,-1) projection), Figure 5 .1 (using Figure 3 to describe formula (3-1)), Figure 5 .2 (Use Figure 3 to describe formula (3-2)).

[0030] Figure 6 is a fourth schematic diagram of the present invention for describing the second type of Maxwell equations, wherein, Figure 6 include Figure 2 .1((0,0,-1) projection), Figure 6 .1 (using Figure 4 to describe formula (4-1)), Figure 6 .2 (Use Figure 4 to describe formula (4-2)).

[0031] Figure 7 is a schematic diagram of the fifth figure used in the present invention to describe the total differential of the thermodynamic potential, wherein: Figure 7 include Figure 2 .1((0,0,-1) projection), Figure 7 .1 (Use Figure 5 to describe formula (5-1)), Figure 7 .2 (Use Figure 5 to describe formula (5-2)).

[0032] Figure 8 is a schematic diagram of the sixth figure used in the present invention to describe the Gibbs-Helmholtz equation, wherein: Figure 8 include Figure 2 .1((0,0,-1) projection), Figure 8 .1 (using Figure 6 to describe formula (6-1)), Figure 8 .2 (Use Figure 6 to describe Equation (6-2)).

[0033] Figure 9 The present invention is used to describe the isobaric heat capacity (C PN ) Schematic diagram of the class variable, where Figure 9 include Figure 2 .1((0,0,-1) projection), Figure 9 .1 (Use Figure 7 to describe Equations (7-1) and (7-2)), Figure 9 .2 (Use Figure 7 to describe Equations (7-3) and (7-4)).

[0034] Figure 10 This invention is used to describe the third kind of Maxwell partial derivative and C PSchematic diagram of the relationship between class variables, where Figure 10 include Figure 2 .1((0,0,-1) projection), Figure 10 .1 (Use Figure 8 to describe Equations (8-1) and (8-2)), Figure 10 .2 (Use Figure 8 to describe Equations (8-3) and (8-4)).

[0035] Figure 11 The present invention is used to describe the two nearest neighbors C P Figure 9 shows the relationship between class variables, where Figure 11 include Figure 2 .1((0,0,-1) projection), Figure 11 .1 (using Figure 9 to describe formula (9-1)), Figure 11 .2 (Use Figure 9 to describe Equation (9-3)).

[0036] Figure 12 This invention is used to describe the parallel C P A diagram showing the relationship between class variables, where Figure 12 include Figure 2 .1((0,0,-1) projection), Figure 12 .1 (Use Figure 10 to describe Equations (10-1) and (10.2)), Figure 12 .2 (Use Figure 10 to describe Equations (10-3) and (10-4)).

[0037] Figure 13 This invention is used to describe the cross C P Figure 11 shows the relationship between class variables, where: Figure 13 include Figure 2 .1((0,0,-1) projection), Figure 13 .1 (using Figure 11 to describe formula (11-1)), Figure 13 .2 (Use Figure 11 to describe Equation (11-2)).

[0038] Figure 14 is a schematic diagram of the twelve figures used in the present invention to describe the Jacobian equation, wherein: Figure 14 include Figure 2 .1((0,0,-1) projection), Figure 14 .1 (using Figure 12 to describe formula (12-1)), Figure 14 .2 (Use Figure 12 to describe formula (12-2)).

[0039] Figure 15 It is one of the pattern collections of the method of the present invention, including patterns 1 to 6 and 12.

[0040] Figure 16This is the second of the diagrams of the method of the present invention, including diagrams 7 to 11.

[0041] Figure 17 It is a schematic diagram of the thermodynamic symmetry disclosed by the present invention.

[0042] Figure 18 This is a thermodynamic model diagram hand-made by the inventor.

[0043] Figure 19 This is a thermodynamic concentric multi-layered multi-faceted shell model diagram produced by the present invention using a three-dimensional printer.

[0044] Figure 20 This is a self-consistent and complete thermodynamic model diagram produced by the present invention using a three-dimensional printer.

[0045] Figure 21 This is a diagram of the thermodynamic symmetry model disclosed by the present invention produced using a 3D printer.

[0046] Figure 22 It is a two-dimensional (1,-1,1) projection diagram of the method of the present invention, which shows that the overall thermodynamic model has a three-dimensional rotational axis symmetry (C3) with 'U~Φ' as the axis. DETAILED DESCRIPTION

[0047] To facilitate understanding of the present invention, the following description is given with reference to the accompanying drawings and embodiments.

[0048] The following details how to complete the following questions:

[0049] (1) How to reasonably distribute the forty-four different categories of thermodynamic variables of a unit single-phase system in a three-dimensional model according to their physical meanings?

[0050] (2) How to use specially created different rigid activity patterns, superimpose them on a fixed two-dimensional projection diagram, and use symmetry transformations to simply and reliably describe more than 300 thermodynamic relationships of twelve categories?

[0051] (3) How to use different graphs to distinguish some similar but quite confusing partial derivatives?

[0052] (4) How to derive the parametric expression for any required partial derivative?

[0053] (5) How to determine the specific and exact thermodynamic symmetry?

[0054] 1. Thermodynamic concentric multi-layered shell model

[0055] According to the physical meaning, the 44 thermodynamic variables of four different categories in the unit single-phase system are used as elements to evenly and reasonably form a concentric four-layer polyhedral shell model ( Figure 1 ).

[0056] 1. First level: Place three pairs of conjugated (intensity-extension) independent variables: temperature (T)-entropy (S), pressure (P)-volume (V), and chemical potential (μ)-number of moles (N) at the six vertices of a small octahedral shell. Their coordinates are: T[1,0,0]-S[-1,0,0], P[0,-1,0]-V[0,1,0], and μ[0,0,1]-N[0,0,-1].

[0057] 2. The second layer: In order to reflect the close relationship between each thermodynamic potential and its three related independent variables, the four complete pairs of conjugate thermodynamic potentials, internal energy U(S,V,N)~conjugate internal energy Φ(T,P,μ), thermal enthalpy H(S,P,N)~conjugate thermal enthalpy (giant potential) Ω(T,V,μ), Gibbs free energy G(T,P,N)~conjugate Gibbs free energy ψ(S,V,μ), and Helmholtz free energy A(T,V,N)~conjugate Helmholtz free energy χ(S,P,μ), are placed on the eight vertices of a cubic shell close to their three independent variables. Their coordinates are: U[-1,1,-1]~Φ[1,-1,1], H[-1,-1,-1]~Ω[1,1,1], G[1,-1,-1]~ψ[-1,1,1] and A[1,1,-1]~χ[-1,-1,1].

[0058] 3. Third level: Similar to the first level, place the first partial derivatives of the three pairs of conjugated thermodynamic potentials (T to -S, -P to V, and μ to -N) at the six vertices of a large octahedral shell. Their coordinates are: T[3,0,0] to -S[-3,0,0], -P[0,-3,0] to V[0,3,0], and μ[0,0,3] to -N[0,0,-3]. The signs of the six conjugated first partial derivatives of the thermodynamic potentials are exactly the same as the signs of the six conjugated independent variables (T, S, P, V, μ, and N), except that three of them are negative.

[0059] The presence or absence of a negative sign before the symbol of a thermodynamic variable has a special physical meaning: in spontaneous changes and equilibrium, these negative variables (-S, -P, and -N) will reach their maximum, not their minimum. This is the opposite of other variables that do not have a negative sign.

[0060] 4. Fourth level: Usually, the second-order partial derivatives of the thermodynamic potential describe the properties of the thermodynamic system. For example, isobaric and isochoric heat capacity (C P and C V ), isobaric thermal expansion coefficient (α), isothermal compressibility coefficient (κ T or β). According to the isobaric heat capacity (C P) and combined with the principle of symmetry equivalence, through symmetry transformation, another twenty-two new C P Class variables. The second-order partial derivatives of the complete twenty-four thermodynamic potentials (C PN ,C VN ,O PN ,O VN ,J TN ,J SN ,R TN ,R SN ,C Pμ ,C Vμ ,O Pμ ,O Vμ ,J Tμ ,J Sμ ,R Tμ ,R Sμ ,Λ PT ,Λ VT ,Γ PT ,Γ VT ,Λ PS ,Λ VS ,Γ PS , and Γ VS ), evenly and rationally placed at the vertices of a rhombicuboctahedron, and close to the thermodynamic potential and independent variables associated with them. Their coordinates are the full arrangement of <±h, ±h, ±k>, where h is equal to one and a half units (h = 1.50) and k is greater than h. times (k=3.615).

[0061] This approach, in which different categories of thermodynamic variables are treated as elements and placed on different shells, forms a comprehensive structural framework (the coordinates of all thermodynamic variables are summarized in Appendix 1). This approach is meaningful for a holistic understanding of thermodynamics. This conception aligns with Ehrenfest's classification of thermodynamic phase transitions and is helpful in understanding the criteria for their classification.

[0062] 5. Three-Dimensional Model Simplification: In this concentric multi-layered polyhedral shell model (a cubic shell sandwiched between two octahedral shells, which are then enclosed by a 26-sided shell), the variables on the two octahedral shells of different sizes are identical, except for the three variables (-S, -P, and -N) on the larger octahedral shell, which have different negative signs. Therefore, if the negative sign problem can be handled using one of the methods described below, the two octahedral shells can be simplified into a single large octahedral shell.

[0063] 6. Two-dimensional projection diagram: In a three-dimensional model, describing symmetry transformation is quite complex and difficult. However, in a two-dimensional diagram, it becomes simple and easy. Therefore, this diagram method uses a two-dimensional diagram. The simplified three-dimensional model is dissected to obtain six two-dimensional {1,0,0} projection diagrams ( Figure 2 .1 to Figure 2 .6).

[0064] When actually making a two-dimensional graph, all the variables on the simplified three-dimensional model cut in half are projected outwards from the central plane along six different <1,0,0> directions in parallel onto the corresponding six {1,0,0} planes ( Figure 2 .1~ Figure 2 .6). In order to [7] Consistent, without affecting the graphical effect, the outermost four C P Class variables are omitted. For example, in Figure 2 .1,Γ PT ,Γ VT ,Γ PS and Γ VS Omitted.

[0065] In theory, any variable in a three-dimensional model can be expressed with the help of matrix methods as long as it has its three-dimensional coordinates [x, y, z].

[10] Project it onto any desired (hkl) plane. The position of the variable on the two-dimensional projection surface can be determined by the corresponding two-dimensional vector,V,.

[0066] The two-dimensional vector, V, is expressed as:

[0067]

[0068] in, and are two mutually orthogonal unit vectors on the projection plane (hkl), which correspond to the normals of two mutually orthogonal planes, (h1 k1 l1) and (h2 k2 l2).

[0069] σ ij is the element of the projection transformation matrix, that is

[0070] σ 11 =d' 11 h1,σ 12 =d' 11 k1,σ 13 =d' 11 l1,

[0071] σ 21 =d' 22 h2,σ 22 =d'22 k2,σ 23 =d' 22 l2,

[0072] Among them, d' ii Determined by the following formula:

[0073]

[0074] Figure 2 The six {1,0,0} projection images shown are centered on -N. Figure 2 .1, centered on μ Figure 2 .2, centered around -P Figure 2 .3, V-centered Figure 2 .4, T-centered Figure 2 .5, and centered on -S Figure 2 .6. Each diagram has two concentric squares and an octagon, all showing four-fold rotational symmetry and mirror symmetry (C4 and σ). Figure 2 .1 is used as an example because it contains many commonly used thermodynamic variables and the most familiar thermodynamic relationships can be described on it.

[0075] 2. Special symbols

[0076] Most of the mathematical operations in thermodynamics are algebraic and calculus, with very little geometry. Therefore, before introducing this diagrammatic method, it is necessary to introduce some special symbols.

[0077] 1. Symbols used to select variables: large and small circles ('○' and ) selects the variables located on the large square (T, -S, -P, V, μ, and -N). The difference between the large and small circles is significant only for the three variables with negative signs (-S, -P, and -N). For example, if -S is selected by the large circle, it retains the negative sign, representing -S with a negative value (-). However, if -S is selected by the small circle, it removes the negative sign, representing S with a positive value (+). Use the square symbol ('□') to select the variables located on the small square (U, H, G, and A). Use the special pattern symbol Select C on the octagon P Class variables (C PN ,C VN ,O PN ,O VN ,J TN ,J SN ,R TN , and R SN ).

[0078] 2. Symbols used to represent mathematical operations: Use a line segment ('-----') connecting two variables, such as '○-----○' or Indicates the product of two selected variables ('●'). Use a slash (' / ') between two selected variables, such as ' / ○', represents the quotient or ratio of two selected variables. The addition ('+') and subtraction ('-') symbols are omitted. Other mathematical operation symbols, such as =, d, and J, still retain their original meanings, representing equality, differential, first-order partial differential, second-order partial differential and Jacobian notation respectively.

[0079] 3. The meaning of the arrow: The arrow (→) can indicate the direction of change between variables, the order in which mathematical expressions are written, and the order in which variables are selected.

[0080] 4. Graphical method of describing partial derivatives: The first-order partial derivative of a multivariate function, f = f(x, y, z), is expressed as It means that under the condition that two independent variables, y and z, remain unchanged, the change of the multivariate function, f = f(x, y, z), is caused by the change of the other independent variable, x. This mathematical expression, It can be divided into two parts. One part is arranged in the order of writing, with mixed mathematical symbols and variable selection symbols ( or ). The other part is a list of different variables (f, x, y and z). For example, when you want to describe When you can make a special pattern, Overlapping Figure 2 .1, select the variables involved (G, T, P and N) in the order of the arrows, and then merge the two parts to become That is,

[0081] 5. Symmetry Symbols: This thermodynamic concentric multi-layered polyhedral shell model exhibits geometric symmetries. These symmetries include mirror symmetry (σ), as well as cubic and quartic rotational symmetries (C3 and C4). Symmetry plays a crucial role in this diagrammatic representation.

[0082] 3. Steps to describe thermodynamic relationships using unified diagrams

[0083] According to the symmetric equivalence principle

[11] If we know a certain relationship in a certain class of thermodynamic relationships, then through symmetry transformation, we can know all the relationships in this class. The specific unification steps are as follows:

[0084] Step 1: Use the two-dimensional (0,0,-1) projection map ( Figure 2 .1) as the basis. On it, the sixteen most commonly used thermodynamic variables are distributed on the vertices of two squares and one octagon.

[0085] Step 2: Select a familiar relationship from a class of thermodynamic relationships and use it as a model. Figure 2 .1 to describe it. The diagram is a mixture of mathematical symbols and variable selection symbols arranged in the order of writing.

[0086] Step 3: Overlay this movable rigid special pattern on the fixed (0,0,-1) projection map, and transform it by symmetry (σ,C4 1 ,C4 2 and C4 3 ), describe other such relationships one by one.

[0087] Step 4: Then use other two-dimensional {1,0,0} projections ( Figure 2 .2~ Figure 2 .6) Replace one by one Figure 2 .1, repeat the third step to describe all such relationships.

[0088] Using this unified approach, we can test whether thermodynamics is a symmetrical science.

[0089] 4. Introduction to various types of drawings

[0090] For twelve different types of thermodynamic relationships, the following diagram ( Figures 3 to 14 ), briefly introduces and explains how to create special diagrams to describe them, and how to use these diagrams to describe a large variety of thermodynamic relationships.

[0091] 1. Used to describe Legendre transformation

[12] Pattern 1 ( Figure 3 )

[0092] Legendre transformation template: U=H-PV (1-1)

[0093] Or H=U+PV

[0094] Analysis: U=H-PV=H+V●(-P)

[0095] Or H=U+PV=U+P●(V)

[0096] from Figure 2.1 It can be seen that the above two relations are a pair of reversible linear transformations between the two nearest-neighbor thermodynamic potentials, U and H, located on the small square. The second term is the product of two variables parallel to U and H, located at opposite diagonals of the large square. The sign of this product term is determined by the sign of the variable closest to the thermodynamic potential being transformed (rather than being transformed).

[0097]

[0098]

[0099] The first circle must be small to remove the negative sign before the selected variable, while the second circle must be large to retain the negative sign before the selected variable.

[0100] Then add an arrow (→) between the two squares to indicate the direction of the transformation. Add a line segment (---) between the two circles to indicate the product of the two selected variables (●). Finally, the pattern (becomes

[0101]

[0102] This diagram, specifically used to describe the Legendre transformation, can be described as:

[0103] Two line segments, □→□ parallel to each other

[0104] exist Figure 3 As can be seen in .1, a pair of figures describes the equation (1-1), U=H+V●(-P)=HP●V, and the equation (1-2), A=G+V●(-P)=GP●V. Both show mirror symmetry (σ) with the parallel diagonal of the large square, V~-P, as the mirror plane. Figure 3 In .2, another pair of mirror-symmetric patterns describes equation (1-3), U = A + S ● (T) = A + T ● S, and equation (1-4), H = G + S ● (T) = G + T ● S. Figure 3 .1 and Figure 3 The difference between the two is that the mirror-symmetrical pattern 1 is rotated 90 degrees counterclockwise around the center (-N) (C4 1 ).

[0105] Then, following steps 3 and 4, all these relationships can be described one by one. Since the cubic shell has twelve sides, and there are two reversible transformations on each side, there are twenty-four total relationships.

[0106] 2. Diagram 2 used to describe the thermodynamic identity equation ( Figure 4 )

[0107] Thermodynamic identity equation template:

[0108] Analysis: Figure 2 .1 As can be seen, the left side of the equation is the partial derivative of the Helmholtz free energy, A, with respect to the temperature variable, T, holding its two dependent independent variables, V and N, constant. The right side of the equation is a first-order partial derivative variable, -S, located on the diagonal of the large square and conjugated to the temperature. In fact, this relationship is the definition of the first-order partial derivative variable, -S, of the thermodynamic potential.

[0109] Thus, the second diagram specifically used to describe the thermodynamic identity equation is expressed as:

[0110]

[0111] The last circle must be large to preserve the negative sign before the chosen first partial derivative.

[0112] Overlay the active pattern on the fixed two-dimensional {1,0,0} projection ( Figure 2 .1 to Figure 2 .6), through the symmetry transformation (σ, C4 1 ,C4 2 and C4 3 ), all relationships can be described one by one. For example, equations (2-1) and (2-2) are respectively Figure 4 .1 and Figure 4 .2. There are twenty-four such relationships in total, since there are eight complete thermodynamic potentials, each with three associated independent variables.

[0113] 3. Diagram 3 used to describe Maxwell's equations ( Figure 5 )

[0114] Maxwell's equations template:

[0115] Analysis: Figure 2 .1 and the rewritten formula (3-1), As can be seen, both sides of the equation are standard Maxwell partial derivatives. Their first three variables are located at the vertices of the large square, and the last variable is located at the center of the diagram. The two opposing paths that choose the first three variables both go around the edges of the large square and have mirror symmetry (σ) about the diagonal of the small square. Using these analysis results, we created Figure 3 specifically describing Maxwell's equations:

[0116] Two opposite mirror-symmetric paths, Equal to each other

[0117] The first circle must be large in order to preserve the negative sign of the first variable.

[0118] Overlay the active pattern on the fixed two-dimensional {1,0,0} projection ( Figure 2 .1 to Figure 2 .6), through the symmetry transformation (σ, C4 1 ,C4 2 and C4 3 ), all Maxwell equations can be described one by one. For example, equations (3-1) and (3-2) are respectively Figure 5 .1 and Figure 5 .2. There are twenty-four such relationships that can be deduced.

[0119] 4. Diagram 4 for describing Maxwell's equations of the second kind ( Figure 6 )

[0120] Maxwell's equations of the second kind example:

[0121] Analysis: This relationship is actually the inverted Maxwell equation. Figure 2 .1 and the rewritten formula (4-1), As you can see, both sides of the equation are inverted Maxwell partial derivatives, or Maxwell-II partial derivatives of the second kind. Their first three variables and their last variable are also located at the corners of the large square and the center of the diagram, respectively. The only difference is the path (order) of the first three variables. They first go around the edge of the large square and then pass through the center of the diagram. The two paths form a closed figure eight (8 or ∞). Using this analytical result, Figure 4 was created specifically to describe Maxwell's equations of the second kind:

[0122] Two closed figure eight paths, Equal to each other

[0123] The first circle must be large in order to preserve the negative sign of the first variable.

[0124] Overlay the active pattern on the fixed two-dimensional {1,0,0} projection ( Figure 2 .1 to Figure 2 .6), through the symmetry transformation (σ, C4 1 ,C4 2 and C4 3 ), all the second-class Maxwell equations can be described one by one. For example, equations (4-1) and (4-2) are respectively Figure 6 .1 and Figure 6 .2. There are twenty-four such relationships that can be deduced.

[0125] 5. Diagram 5 for describing the total differential of thermodynamic potential ( Figure 7 )

[0126] Thermodynamic potential total differential template: dU=T dS-P dV(5-1)

[0127] Analysis: According to the rewritten formula (5-1), dU=T·dS+(-P)·dV, and Figure 2 .1 As can be seen, the left side of the equation is the internal energy at the corner of the small square, U = U(S, V, N), and its total differential, dU, under the condition that N remains constant. The right side of the equation is the sum of the differentials (dS and dV) of two pairs of first-nearest internal energy variables (-S and V) on the large square and the product of their conjugate second-nearest internal energy variables (T and -P) (T ● dS and (-P) ● dV).

[0128]

[0129] Among them, the square (□) is used to select the thermodynamic potential, U, and the small circle The first-nearest variables of the thermodynamic potential, -S and V, must be selected to eliminate the negative sign. The large circle (○) must be used to select the second-nearest variables of the thermodynamic potential, T and -P, to retain the negative sign. Finally, the symbols for product and differential (---- and d) are added. Thus, a diagram specifically for describing the total differential of the thermodynamic potential becomes:

[0130]

[0131] Overlay the active pattern 5 on the fixed two-dimensional {1,0,0} projection ( Figure 2 .1 to Figure 2 .6), through the symmetry transformation (σ, C4 1 ,C4 2 and C4 3 ), all thermodynamic potential total differentials can be described one by one. For example, Equation (5-1) and Equation (5-2) are described in Figure 7 .1 and Figure 7 .2. There are twenty-four such relationships that can be deduced. Among them, dΦ = (-S) ● dT + (V) ● dP = 0 is the Gibbs-Duhem equation.

[0132] From the introduction to the five diagrams above, we can see that symmetry does exist in thermodynamics, because the fundamental thermodynamic relationships are described by this symmetric diagramming method. Below, we will continue to use this method to invent new variables, explore and verify new relationships, and establish and improve the relationships between new variables.

[0133] 6. Diagram 6 used to describe the Gibbs-Helmholtz equation ( Figure 8 )

[0134] When discussing the dependence of Gibbs free energy on temperature, the Gibbs-Helmholtz equation holds true.

[0135]

[0136] Analysis: According to formula (6-1) and Figure 2 .1 As can be seen, the left side of the equation is a complex expression of first-order partial derivatives. The right side, however, is simply a thermodynamic potential (enthalpy, H) located at the corner of a small square. The variables involved are (G / T), (1 / T), P, N, and H. Using the above analysis results and the experience gained in creating diagrams, the mathematical symbols and symbols for the selected variables are mixed and arranged in the order in which they are written, forming Diagram 6 specifically for describing the Gibbs-Helmholtz equation:

[0137]

[0138] A '1' must be added to the pattern. Figure 2 .1, it should be on the extension line of the diagonal line of the small square (H~A). Figure 8 As can be seen in .1, Figure 6 describes formula (6-1),

[0139] If we change formula (6-1), Consider as a model of this type of relationship, that is, this type of thermodynamic potential has a certain

[0140] If we can find a model of the dependence of the independent variables, then we can infer and predict other new relationships of this type of thermodynamic potential dependence based on the principle of symmetry equivalence. Figure 8 In .1, taking the diagonal of the small square (H~A) as the mirror plane, and performing a mirror symmetry transformation (σ) on pattern 6, we can describe the formula (6-1'), That is, the dependence of internal energy (U) on its independent variable volume (V). Similarly, Figure 8 .1 The movable figure 6 on the left is rotated 90 degrees clockwise with the center of the figure as the axis (C4 1 ), another new pair of dependencies, Just Figure 8 .2 is described.

[0141] By symmetry transformation of pattern six (σ and C4 1 ), the three new relations inferred above can be proved to be correct in theory.

[0142] Proof 1: Dependence of internal energy (U) on volume (V), formula (6-1')

[0143] U=H+V·(-P)=H-PV (using pattern 1)

[0144]

[0145] rewrite

[0146]

[0147] rewrite Establishment

[0148] Proof 2: Dependence of enthalpy (H) on pressure (P), formula (6-2')

[0149] H=U+P·(V)=U+PV (using Figure 1)

[0150]

[0151] rewrite

[0152]

[0153] rewrite Establishment

[0154] Proof 3: Dependence of Helmholtz free energy (A) on temperature (T), Equation (6-2)

[0155] A=U+T·(-S)=U-TS (using pattern 1)

[0156]

[0157] rewrite

[0158]

[0159] rewrite Establishment

[0160] Then, following the third and fourth steps of the diagrammatic method, the remaining twenty thermodynamic potential dependencies can be described one by one. Furthermore, they can be theoretically verified to be correct. After verification, it was found that, since Φ(T, P, μ) = 0, three of the twenty-four dependencies are unreasonable. For example,

[0161] 7. Used to describe isobaric heat capacity (C P ) Class variable pattern seven ( Figure 9 )

[0162] Isobaric heat capacity (C P ) and isochoric heat capacity (C V ) are very important properties in thermodynamics. They are the second-order partial derivatives of the thermodynamic potential, Gibbs free energy (G), and Helmholtz free energy (A), respectively.

[0163]

[0164] and

[0165] So, according to C PN and C VN exist Figure 2 .1 are located closest to their variables (G, T, P and A, T, V) on the outer octagon and their definitions, equations (7-1) and (7-2), a diagram specifically used to describe isobaric heat capacity (C P ) class variable is created as:

[0166]

[0167] exist Figure 9 .1, a pair of C P Class C PN and C VN , that is, equations (7-1) and (7-2), can be described by a pair of mirror-symmetrical patterns seven with the diagonal lines (-S to T) of the large square as the mirror plane. If this pair of patterns seven is placed in Figure 9 .1 Take the center of the figure (-N) as the axis and rotate 90 degrees clockwise (C4 1 ), another pair of C P Class variables (R TN and R SN ), that is, formula (7-3) and formula (7-4), can be Figure 9 .2 was created. Further, if this pair of patterns is placed on Figure 9 .1 Take the center of the figure (-N) as the axis and rotate 180 degrees clockwise (C4 2 ) and 270 degrees (C4 3 ), the other four C P Class variables ( PN ,O VN ,J TN , and J SN ), namely, equations (7-5) to (7-8), were created.

[0168]

[0169]

[0170]

[0171]

[0172] The other sixteen Cs P Class variables (C Pμ ,C Vμ ,O Pμ ,O Vμ ,J Tμ ,J Sμ ,R Tμ ,R Sμ ,Λ PT ,Λ VT ,Γ PT ,Γ VT ,Λ PS ,Λ VS ,Γ PS , and Γ VS ) is to similarly overlay pattern seven on other two-dimensional {1,0,0} projection images ( Figure 2 .2 to Figure 2 .6) and invented by the same method as above. Among them, there are three C P Class variables ( Pμ ,J Tμ ,Γ PT ) is zero value, the other three C P Class variables (C Pμ ,R Tμ ,Λ PT ) is an infinite value.

[0173] 8. Used to describe the third kind of Maxwell partial derivatives and C P The relationship between class variables Figure 10 )

[0174] C P Class variables are second-order partial derivatives of the thermodynamic potential and are related to the so-called Maxwell-III partial derivatives. For example, and

[0175] In other words, the third kind of Maxwell partial derivative and C P There are relationships between class variables.

[0176] A sample of this type of relationship:

[0177] Analysis: Comparison (8-1) and Figure 2.1, we can see that the left side of the equation is a so-called third-order Maxwell partial derivative. Its first three variables are also located at the vertices of the large square, but the path of selecting these variables is different. The path first passes through the center of the graph and then goes around the edge of the large square, like a "hook". The right side of the equation is the second-order partial derivative variable (C P Class variables, C PN ) and the quotient (or ratio) of the first-order partial derivative variable (temperature, T) of the nearest neighbor. Based on the above analysis results, a method is designed to describe the third kind of Maxwell partial derivative and C P Figure 8 of the relationship between class variables:

[0178] Hook Path

[0179] The last circle must be large in order to retain the negative sign of the last variable.

[0180] Depend on Figure 10 It can be seen that the two pairs of relationships, equations (8-1) to (8-4), are described by two pairs of mirror-symmetrical figures eight, respectively. Figure 10 .1 and Figure 10 .2. And it can be seen that the difference between the two pairs of mirror-symmetrical patterns 8 is that the two pairs of mirror-symmetrical patterns 8 are Figure 2 .1 is rotated 90 degrees clockwise around the center of the figure (-N) (C4 1 ).

[0181] Similarly, the movable pattern is superimposed on the fixed two-dimensional {1,0,0} projection image ( Figure 2 .1 to Figure 2 .6), through the symmetry transformation (σ, C4 1 ,C4 2 and C4 3 ), all twenty-four of these categories of relationships can be described one by one.

[0182] 9. Used to describe things like C P and C V Similarly, the two nearest neighbors C P Figure 9: Relationship between class variables Figure 11 )

[0183] Everyone knows C P and C V There is an important relationship between:

[0184] or

[0185] Among them, α and κ T They are defined as:

[0186] Isobaric thermal expansion coefficient:

[0187] Isothermal compressibility:

[0188] In order to find C P General relationships between class variables,The above relationships are best expressed in terms of the independent thermodynamic variables (T, S, P, V, μ, and N).

[0189] Under the condition that the number of moles (N) remains unchanged, make the total differential of S=S(V,T)

[0190]

[0191] Under the condition of constant pressure (P), taking the partial derivative of the above equation with respect to temperature (T), we can get

[0192]

[0193] Rewrite into

[0194] On the other hand, using the above derivation results and C P and C V Definition of , and their relationship to the third kind of Maxwell partial derivatives:

[0195]

[0196] Rewrite the above formula into formula (9-1) and formula (9-2) under the condition that the number of moles (N) remains unchanged

[0197]

[0198]

[0199] From formula (9-1) and formula (9-2), we can see that C VN and C PN The product term in the two reversible transformation relations between consists of three parts: two standard Maxwell partial derivatives and a key independent variable (temperature, T) in the middle; all variables in the product term are in Figure 2 .1 are all close to C VN and C PN The sign of the product term is determined by the sign of the numerator (first) variable of the second Maxwell partial derivative. As for which variable is selected as the first (numerator) variable of the second Maxwell partial derivative, it is determined by the transformation direction. For example, when Equation (9-1) represents the product term C VN Transformed into C PN When Figure 2 .1 near C PNThe variable (-P) is chosen as the numerator variable of the second Maxwell partial derivative, When formula (9-2) is expressed by C PN Transformed into C VN When Figure 2 .1 near C VN The variable (V) is chosen as the first variable of the second Maxwell partial derivative,

[0200] In order to verify whether such a pair of reversible transformation relations also exists symmetrically in another pair of C P Class variables (R T and R S ), under the condition that the number of moles (N) remains unchanged, make the total differential of V=V(T,P):

[0201]

[0202] Under the condition that entropy (S) remains unchanged, taking the partial derivative of the above formula with respect to pressure (P), we can obtain:

[0203]

[0204] Rewrite into

[0205] On the other hand, using the above derivation results and R T and R S Definition of , and their relationship to the third kind of Maxwell partial derivatives:

[0206]

[0207] Rewrite the above formula into formula (9-3) and formula (9-4) under the condition that the number of moles (N) remains unchanged

[0208]

[0209]

[0210] From equations (9-3) and (9-4), we can see that the two reversible transformation relationships of this type are completely similar to equations (9-1) and (9-2). Figure 2 .1 are close to R TN and R SN The sign of the product term is also determined by the sign of the numerator (first) variable of the second Maxwell partial derivative. As for which variable is selected as the first (numerator) variable of the second Maxwell partial derivative, it is determined by the transformation direction. When Equation (9-3) is represented by R TN Transformed into R SN When Figure 2 .1 near R SN The variable (-S) is chosen as the first variable of the second Maxwell partial derivative, When formula (9-4) is expressed by R SN Transformed into R TN When Figure 2 .1 near R TN The variable (T) is chosen as the numerator variable of the second Maxwell partial derivative, When the transformation direction (arrow direction) changes, the selected variable that determines the sign of the product term changes from -S to T in a mirror-symmetrical (σ) manner.

[0211] Using the above analysis results, Figure 2 .1 Based on formula (9-1) as a template, a rather complex diagram is designed to describe the relationship between two nearest neighbors C P Figure 9: Relationship between class variables

[0212]

[0213] Among them, the two paths for selecting the first three variables of Maxwell partial derivatives are exactly opposite, with mirror symmetry (σ) with the diagonal of the large square as the mirror plane, close to the transformation variable (the second ), rather than being close to the transformed variable (the first ) The symbol for the numerator (first) variable of the second Maxwell partial derivative must be a large circle to preserve the sign of the selected variable.

[0214] All twenty-four pairs (forty-eight) of this type of relations can be transformed by symmetry (σ, C4) by superimposing the movable pattern nine on the fixed two-dimensional {1,0,0} projection. 1 ,C4 2 and C4 3 ), are described one by one. For example, Figure 11 .1 and Figure 11 .2 describes equations (9-1) and (9-3).

[0215] 10. Used to describe parallel C P The relationship between class variables Figure 12 )

[0216] It can be found that the following relationship holds:

[0217] C VN ●O VN =T●(-S)=-TS (10-1)

[0218] C PN ●O PN=T●(-S)=-TS (10-2)

[0219] J TN ●R TN =V●(-P)=-PV (10-3)

[0220] J SN ●R SN =V●(-P)=-PV (10-4)

[0221] For example, formula (10-1) can be easily proved as follows:

[0222]

[0223] exist Figure 2 As can be seen from the figure 10, Equation (10-1) can be described by a fairly concise parallel line diagram as follows:

[0224] The products of the variables at both ends of parallel line segments are equal to each other.

[0225] Here, both circles must be large to preserve the negative sign of the chosen variable.

[0226] All 24 relations of this type can be solved by superimposing the movable pattern on the fixed two-dimensional {1,0,0} projection and performing symmetric transformations (σ, C4 1 ,C4 2 and C4 3 ), are described one by one. For example, Figure 12 .1 and Figure 12 .2 describes equations (10-1) to (10-4).

[0227] 11. Used to describe cross C P The relationship between class variables Figure 13 )

[0228] It can also be proved that the following relationship holds:

[0229] J TN ·C PN =J SN ·C VN (11-1)

[0230] C VN ·R TN =C PN ·R SN (11-2)

[0231] R TN ·O PN =R SN ·O VN(11-3)

[0232] O PN ·J SN =O VN ·J TN (11-4)

[0233] For example, using Maxwell's equations (Figure 3), Maxwell's equations of the second kind (Figure 4), and Maxwell's equations of the third kind, partial derivatives and C P The relationship between class variables (Figure 8) can prove that formula (11-1) is valid.

[0234]

[0235] exist Figure 2 As can be seen from the figure 11-1, equation (11-1) can be described by another rather concise cross-line diagram 11:

[0236] The products of the variables at both ends of the intersecting line segment are equal to each other.

[0237] Similarly, all 24 relations of this type can be solved by superimposing the movable pattern 11 on the fixed two-dimensional {1,0,0} projection and performing symmetric transformations (σ, C4 1 ,C4 2 and C4 3 ), are described one by one. For example, Figure 13 .1 and Figure 13 .2 describes equations (11-1) and (11-2).

[0238] From the above introduction, it can be seen that the symbols and positions of all variables in this method, all the regulations and various diagrams describing the steps of thermodynamic relations are consistent and completely self-consistent. For example, the complete C P The class variables show a symmetrical relationship between them, which is the most prominent example of self-consistency. At the same time, it proves that symmetry runs through the entire process of describing thermodynamic relationships in various diagrams and plays an extremely important role.

[0239] 12. Diagram 12 used to describe the Jacobian equation ( Figure 14 )

[0240] The Jacobian method is simple, reliable, and very useful [13,14] This diagram method will be more practical if it is combined with other methods.

[0241] The Jacobian equation can be derived from the total differential of the thermodynamic potential. For example, under the condition that the number of moles (N) remains unchanged, the total differential of the internal energy, U = U(S, V) is:

[0242] dU=TdS-PdV=(T)·dS+(-P)·dV (5-1)

[0243] Choose x and y as any two variables. When y is constant, take the partial derivative of the total differential with respect to x and we get

[0244]

[0245] According to the Jacobian symbol, J(,),

[0246]

[0247] The above formula becomes

[0248]

[0249] Multiplying by J(X,Y) gives the Jacobian equation for internal energy

[0250] J(U,Y)=(T)·J(S,Y)+(-P)·J(V,Y) (12-1)

[0251] So, a similar The figure 12 specifically used to describe the Jacobian equation becomes

[0252]

[0253] The difference between Figure 12 and Figure 5 is only the replacement of the symbols. J(□,Y) and The pattern 12 has become more complex, and its usefulness has also increased. This will be seen in later examples.

[0254] Overlay the movable pattern twelve on the fixed two-dimensional {1,0,0} projection map, and transform it by symmetry (σ, C4 1 ,C4 2 and C4 3 ), all Jacobian equations can be described. For example, Figure 14 .1 and Figure 14 .2 describes equations (12-1) and (12-2) respectively.

[0255] For the convenience of readers to use this diagram, the above twelve pictures are collected in Figure 15 and Figure 16 The twelve categories and more than three hundred thermodynamic relations involved are collected in Appendix 2 for easy reference.

[0256] 5. Twenty-four Cs P Expressions for class variables

[0257] If we want to know the total differential of a thermodynamic quantity, we must know its partial derivatives. In order to solve numerical problems, it is often found that there is no convenient experimental method to estimate the required partial derivatives. In this case, we must calculate the partial derivatives and relate them to some known quantities. In this case, the partial derivatives are calculated using six independent variables (T, S, P, V, μ and N) and several other parameters (C P ,α,κ T and ω), where ω is the Moore's potential.

[0258] Using C P The relationship between class variables can be derived from the twenty-four C P The parameter expression of the class variable is:

[0259] 1.

[0260] 2.

[0261] 3.

[0262] 4.

[0263] 5.

[0264] 6.

[0265] 7.

[0266] 8.

[0267] 9.

[0268] 10.

[0269] 11.

[0270] 12.

[0271] 13.

[0272] 14.

[0273] 15. 16. 17.

[0274] 18.

[0275] 19.

[0276] 20.

[0277] twenty one.

[0278] twenty two.

[0279]

[0280] twenty three.

[0281] twenty four.

[0282] The above twenty-four C P Parameter expressions of class variables are very useful for obtaining the results of other partial derivatives. Some of them have special values ​​(zero and infinity) C P Class variables ( Pμ ,J Tμ ,Γ PT and C Pμ ,R Tμ ,Λ PT ) in the model is also helpful in determining the exact thermodynamic symmetry.

[0283] 6. Symmetry in Thermodynamics

[0284] FO Koenig has conducted long-term research on thermodynamics and revealed the symmetry of thermodynamics. [3,4] He classified many important thermodynamic relations into families based on whether they had the same standard form. He listed the standard forms of each relationship and counted the number of members in each family: 48, 24, 12, 8, 6, 4, 3, and 1.

[0285] In the twelve diagrams that describe the twelve types of thermodynamic relationships, the relationships of the largest number (48, 24, 12, and 8) are confirmed by the geometric symmetry of this model. However, because the special thermodynamic potential at one vertex of the cubic shell is equal to zero (Φ = 0), it destroys the perfect geometric symmetry of the cubic shell, resulting in an imperfect thermodynamic symmetry. In order to accurately determine the thermodynamic symmetry, the model is placed in a specific orientation ( Figure 17), providing a geometric explanation of the relationships between classes of small numbers (6, 4, 3, and 1).

[0286] exist Figure 17 In the figure, eight complete thermodynamic potentials are distributed on the eight vertices of the cubic shell. Their parametric expressions (Euler equations) are listed on the right side of the figure. If classified by the number of products of conjugate variables in their expressions, they can be divided into four categories: the zero-term class (Φ = 0) located on the bottom (1, -1, 1) base plane, the monoterm class (G, Ω, and χ) located on the first (1, -1, 1) plane, the two-term class (H, A, and ψ) located on the second (1, -1, 1) plane, and the three-term class (U = TS-PV + μN) located on the top internal energy (1, -1, 1) plane. Figure 17 The positions in the and their Euler equations are tested and explained for small numbers as follows:

[0287] 1. Six-member class: This standard form, U-A+GH=0, Figure 17 In the equation, the sum of the thermodynamic potentials at the two diagonals of any square on the surface of a cubic shell is equal, that is, (U + G) = (A + H). Since a cubic shell has six surfaces, this relationship has six members.

[0288] 2. Four-member class: This standard form, U-Φ=TS-PV+μN=U(S,V,N), Figure 17 In the equation, the difference between the two thermodynamic potentials at the ends of any diagonal in a cubic shell is equal to the internal energy (U). After verification, it was found that this relationship only holds for one special diagonal (U-Φ), but not for the other three diagonals. For example, H-Ω=(TS+μN)-(-PV)=TS+PV+μN≠TS-PV+μN. Therefore, the standard form of the four-member class should be changed to: U+Φ=TS-PV+μN=U(S,V,N). It is in Figure 17 In the literal translation, it is stated that the sum of the two thermodynamic potentials at the ends of any diagonal in a cubic shell, rather than their difference, is always equal to the internal energy (U). This important relationship can be used to define the criterion for conjugate pairing of thermodynamic potentials. Since a cubic shell has four diagonals, this conjugate pairing class has four members.

[0289] 3. Three-member class: This standard form, U+A+G+H-χ-Φ-Ω-ψ=4μN, Figure 17In this particular orientation, the difference between the sums of the four thermodynamic potentials on the parallel upper and lower surfaces (squares) of the cubic shell is four times greater than the product of the two conjugate variables on their normals. Since this particular orientation of the cubic shell has only three pairs of parallel upper and lower squares, this class of relations has only three members. Furthermore, the single-term thermodynamic potentials (G, Ω, and χ) and the double-term thermodynamic potentials (H, A, and ψ) on the first and second (1, -1, 1) faces also belong to the three-member class.

[0290] 4. Single member class: This type of standard form, U-A+G-H+χ-Φ+Ω-ψ=0, Figure 17 It states that for a special pair of conjugate thermodynamic potentials (U-Φ), the sum of the internal energy (U) and its three second-nearest-neighbor thermodynamic potentials (U+G+χ+Ω) equals the sum of its conjugate internal energy (Φ) and its three second-nearest-neighbor thermodynamic potentials (Φ+A+H+ψ). This relationship has been verified to hold not only for the U-Φ conjugate pair but also for the other three pairs of conjugate thermodynamic potentials. This is because the sum of any thermodynamic potential and its three second-nearest-neighbor thermodynamic potentials equals twice the internal energy (2U). Therefore, this relationship is not suitable for the standard form of a single-member class. The standard form of a single-member class can be replaced by the previously mentioned relationship: U-Φ = TS-PV + μN = U(S,V,N), or the internal energy (U = TS-PV + μN), or the conjugate internal energy (Φ = 0).

[0291] Please refer to Figures 18 to 21 ,in, Figure 18 Showing the thermodynamic model diagram hand-made by the inventor; Figure 19 A diagram showing a thermodynamic concentric multi-layered multi-faceted shell model produced by a 3D printer according to the present invention; Figure 20 A diagram showing a self-consistent and complete thermodynamic model produced by a 3D printer according to the present invention; Figure 21 A diagram showing the thermodynamic symmetry model disclosed by the present invention produced using a 3D printer ( Figure 17 ).

[0292] Through the above geometric interpretation and verification, it can be considered that the thermodynamic symmetry exhibited by the concentric multi-layered polyhedral shell model carrying many thermodynamic variables is three-fold rotational symmetry (C3) with 'U~Φ' as the axis, and has mirror symmetry (σ) and four-fold rotational symmetry (C4) on the three squares containing internal energy. That is, in the six two-dimensional {1,0,0} projection diagrams ( Figure 2 ), there are only three two-dimensional {1,0,0} graphs centered at -N, V, and -S respectively ( Figure 2 .1, Figure 2 .4 and Figure 2 .6) has perfect symmetry, while the other three two-dimensional {1,0,0} images are not so perfect. This conclusion is also confirmed by the two-dimensional (1,-1,1) projection image ( Figure 22 ) three zero values ​​C P Class variables ( Pμ ,J Tμ ,Γ PT ) or three infinite C P Class variables (C Pμ ,R Tμ ,Λ PT ) are confirmed by the relationship that they are one hundred and twenty degrees apart. Figure 22 It is a (1,-1,1) projection diagram that brings together six first-order and twenty-four second-order partial derivatives of the thermodynamic potential.

[0293] 7. Derive parametric expressions for any desired partial derivatives

[0294] This symmetry diagram method can use the various patterns introduced and the obtained C P Class variable results, easily and reliably derive any required partial derivatives, Here are two examples:

[0295] 1.Example 1

[0296]

[0297]

[0298] 2.Example 2

[0299]

[0300] in,

[0301] and,

[0302]

[0303] Finally, substitute the results of J(U,T) and J(U,P) into the following formula to get

[0304]

[0305] (Note: Example 2 is a Bridgman thermodynamic equation

[15] . )

[0306] 8. Conclusion

[0307] 1. In physical terms, the 44 thermodynamic variables of the four categories of the unit single-phase system are evenly and rationally arranged at the vertices of a concentric multi-layered polyhedral shell, forming a complete, self-consistent, and symmetrical thermodynamic model. To some extent, it is very similar to the Bohr model of the atom and the periodic table of chemical elements.

[0308] 2. The symmetry of thermodynamics is not as perfect as the symmetry of geometric models. It exhibits three-fold rotational symmetry (C3) about the special conjugated U-Φ pair, as well as mirror symmetry (σ) and four-fold rotational symmetry (C4) about the three squares containing the internal energy (U).

[0309] 3. This graphical method utilizes a complete self-consistent symmetric model composed of numerous thermodynamic variables as elements. By creating special movable patterns, it performs specific and accurate symmetry transformations, summarizes and interprets them simply and reliably, and describes a large number of thermodynamic relationships, fully proving that thermodynamics is a symmetrical science.

[0310] IX. References

[0311] 1. Herbert Callen, 'Thermodynamics as a Science of Symmetry', Foundations of Physics, Vol. 4, No. 4, pp. 423~443 (1974).

[0312] 2. Herbert B. Callen, Thermodynamics and An Introduction to Thermostatistics', 2nd Edition, 131, 458 (1985).

[0313] 3. FO Koenig, 'Families of Thermodynamic Equations. I-The Method of Transformations by the Characteristic Group', J. Chem. Phys., 3, 29 (1935).

[0314] 4. FO Koenig, 'Families of Thermodynamic Equations. II-The Case of Eight Characteristic Functions', J. Chem. Phys., 56, 4556 (1972).

[0315] 5. JAPrins, 'On the Thermodynamic Substitution Group and ItsRepresentation by the Rotation of a Square', J. Chem. Phys., 16, 65 (1948).

[0316] 6. RFFox, 'The Thermodynamic Cuboctahedron', J. Chem. Edu., 53, 441 (1976).

[0317] 7. Li Zhenchuan, "Graphical Study of Thermodynamic State Function Relationship", Chemical Bulletin, No. 1, 1982, pp. 48-55. Chemical Abstract, 96, 488.96: 188-159t (1982).

[0318] 8. SFPate, 'The thermodynamic cube: A mnemonic and learning device for students of classic thermodynamics', Am.J.Phys., 67(12), 1111(1999).

[0319] 9.WCKerr and JCMacosko, 'Thermodynamic Venn diagram: Sorting outforce, fluxes, and Legendre transforms', Am.J.Phys., 79(9), 950~953, (2011).

[0320] 10.ZCLi(Li Zhenchuan)and SHWhang,'Planar defects in{113}planes of L1 o type TiAl-Their structures and energies', Phil.Mag.,A,1993,Vol.68,No.1,169-182.

[0321] 11. Joe Rosen, Symmetry in Science, 97 (1995).

[0322] 12.Robert A.Alberty,'Use of Legendre Transforms in ChemicalThermodynamics',Pure Appl.Chem.,73(8),1350(2001)

[0323] 13. FHCrawford, 'Jacobian Methods in Thermodynamics', Am.J.Phys., 17(1), 1(1949).

[0324] 14. Charles E. Reid, Principles of Chemical Thermodynamics, 36&249, Reinhold, New York (1960).

[0325] 15.PW Bridgman,Phys.Rev.,2 nd series,3,273(1914).

[0326] 10. Appendix

[0327] Appendix 1 Coordinates of the 44 Thermodynamic Variables in the Three-Dimensional Model

[0328] 1. First layer: three pairs of conjugate (intensity-extension) independent variables located at the vertices of the small octahedral shell

[0329] T[1,0,0]~S[-1,0,0]; V[0,1,0]~P[0,-1,0]; μ[0,0,1]~N[0,0,-1].

[0330] 2. Second layer: four pairs of complete conjugate thermodynamic potentials located at the vertices of the square shell

[0331] U[-1,1,-1]~Φ[1,-1,1]; H[-1,-1,-1]~Ω[1,1,1];

[0332] G[1,-1,-1]~ψ[-1,1,1]; A[1,1,-1]~χ[-1,-1,1].

[0333] 3. The third layer: the first partial derivatives of the three pairs of conjugated thermodynamic potentials at the vertices of the large octahedral shell

[0334] T[3,0,0]~-S[-3,0,0]; V[0,3,0]~-P[0,-3,0]; μ[0,0,3]~-N[0,0,-3].

[0335] 4. The fourth layer: the second-order partial derivatives of the twenty-four complete thermodynamic potentials at the vertices of the twenty-six-sided shell (C P class variables)

[0336]

[0337] Appendix 2: All twelve categories of over 300 thermodynamic relationships described in this simplified symmetry diagram

[0338] 1. Legendre Transformation

[0339] (1) Under the condition that the number of moles (N) remains unchanged ( Figure 2 .1),

[0340] U(S, V, N) = H + V·(-P) = H - P·V (1-1)

[0341] A(T, V, N) = G + V·(-P) = G - P·V (1-2)

[0342] U(S, V, N) = A + S·(T) = A + T·S (1-3)

[0343] H(S, P, N) = G + S·(T) = G + T·S (1-4)

[0344] G(T, P, N) = A + P·(V) = A + P·V (1-5)

[0345] H(S, P, N) = U + P·(V) = U + P·V (1-6)

[0346] A(T, V, N) = U + T·(-S) = U - T·S (1-7)

[0347] G(T, P, N) = H + T·(-S) = H - T·S (1-8)

[0348] (2) Under the condition that the chemical potential (μ) remains unchanged ( Figure 2 .2),

[0349] χ(S, P, μ) = ψ + P·(V) = ψ + P·V (1-9)

[0350] φ(T, P, μ) = Ω + P·(V) = Ω + P·V = 0 (1-10)

[0351] χ(S, P, μ) = φ + S·(T) = φ + T·S = 0 + T·S = T·S (1-11)

[0352] ψ(S, V, μ) = Ω + S·(T) = Ω + T·S (1-12)

[0353] Ω(T, V, μ) = φ + V·(-P) = 0 - P·V = -P·V (1-13)<​​​​​

[0356] Ω(T, V, μ) = ψ + T·(-S) = ψ - T·S (1-16)

[0357] (3) When the pressure (P) is kept constant ( Figure 2 .3),

[0358] H(S, P, N) = χ + N·(μ) = χ + μ·N (1-17)

[0359] G(T, P, N) = φ + N·(μ) = 0 + N·μ = μ·N (1-18)

[0360] H(S, P, N) = G + S·(T) = G + T·S (1-19)

[0361] χ(S, P, μ) = φ + S·(T) = φ + T·S = 0 + T·S = T·S (1-20)

[0362] φ(T, P, μ) = G + μ·(-N) = G - μ·N = 0 (1-21)

[0363] χ(S, P, μ) = H + μ·(-N) = H - μ·N (1-22)

[0364] G(T, P, N) = H + T·(-S) = H - T·S (1-23)

[0365] φ(T, P, μ) = χ + T·(-S) = χ - T·S = 0 (1-24)

[0366] (4) When the volume (V) is kept constant ( Figure 2 .4),

[0367] ψ(S, V, μ) = U + μ·(-N) = U - μ·N (1-25)

[0368] Ω(T, V, μ) = A + μ·(-N) = A - μ·N (1-26)

[0369] ψ(S, V, μ) = Ω + S·(T) = Ω + T·S (1-27)

[0370] U(S, V, N) = A + S·(T) = A + T·S (1-28)

[0371] A(T, V, N) = Ω + N·(μ) = Ω + μ·N (1-29)

[0372] U(S, V, N) = ψ + N·(μ) = ψ + μ·N (1-30)

[0373] Ω(T, V, μ) = ψ + T·(-S) = ψ - T·S (1-31)

[0374] A(T, V, N) = U + T·(-S) = U - T·S (1-32)(5) When under the condition that the temperature (T) remains unchanged ( Figure 2 .5),

[0375] A(T, V, N) = G + V·(-P) = G - P·V (1-33)

[0376] Ω(T, V, μ) = φ + V·(-P) = 0 - P·V = -P·V (1-34)

[0377] A(T, V, N) = Ω + N·(μ) = Ω + μ·N (1-35)

[0378] G(T, P, N) = φ + N·(μ) = 0 + N·μ = μ·N (1-36)

[0379] φ(T, P, μ) = Ω + P·(V) = Ω + P·V = 0 (1-37)

[0380] G(T, P, N) = A + P·(V) = A + P·V (1-38)

[0381] Ω(T, V, μ) = A + μ·(-N) = A - μ·N (1-39)

[0382] φ(T, P, μ) = G + μ·(-N) = G - μ·N = 0 (1-40)(6) When under the condition that the entropy (S) remains unchanged ( Figure 2 .6),

[0383] ψ(S, V, μ) = χ + V·(-P) = χ - P·V (1-41)

[0384] U(S, V, N) = H + V·(-P) = H - P·V (1-42)

[0385] ψ(S, V, μ) = U + μ·(-N) = U - μ·N (1-43)

[0386] χ(S, P, μ) = H + μ·(-N) = H - μ·N (1-44)

[0387] H(S, P, N) = U + P·(V) = U + P·V (1-45)

[0388] χ(S, P, μ) = ψ + P·(V) = ψ + P·V (1-46)

[0389] U(S, V, N) = ψ + N·(μ) = ψ + μ·N (1-47)

[0390] H(S,P,N)=χ+N·(μ)=χ+μ·N (1-48)

[0391] (Note: Each transformation is repeated once. For example, Equation (1-1) and Equation (1-42) are the same. Therefore, there are only 24 independent Legendre transformations.)

[0392] 2. Thermodynamic identities (In fact, this type of relationship is the definition of the first partial derivative of the thermodynamic potential.)

[0393] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0394]

[0395]

[0396]

[0397]

[0398] (2) When the chemical potential (μ) remains unchanged ( Figure 2 .2),

[0399]

[0400]

[0401]

[0402]

[0403] (3) When the pressure (P) remains constant ( Figure 2 .3),

[0404]

[0405]

[0406]

[0407]

[0408] (4) When the volume (V) remains constant ( Figure 2 .4),

[0409]

[0410]

[0411]

[0412]

[0413] (5) When the temperature (T) remains constant ( Figure 2 .5),

[0414]

[0415]

[0416]

[0417]

[0418] (6) When entropy (S) remains constant ( Figure 2 .6),

[0419]

[0420]

[0421]

[0422]

[0423] 3. Maxwell's equations

[0424] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0425]

[0426]

[0427]

[0428]

[0429] (2) When the chemical potential (μ) remains unchanged ( Figure 2 .2),

[0430]

[0431]

[0432]

[0433]

[0434] (3) When the pressure (P) remains constant ( Figure 2.3),

[0435]

[0436]

[0437]

[0438]

[0439] (4) When the volume (V) remains constant ( Figure 2 .4),

[0440]

[0441]

[0442]

[0443]

[0444] (5) When the temperature (T) remains constant ( Figure 2 .5),

[0445]

[0446]

[0447]

[0448]

[0449] (6) When entropy (S) remains constant ( Figure 2 .6),

[0450]

[0451]

[0452]

[0453]

[0454] 4. Maxwell's equations of the second kind (Actually, this type of relationship is the inverted Maxwell equation.)

[0455] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0456]

[0457]

[0458]

[0459]

[0460] (2) When the chemical potential (μ) remains unchanged ( Figure 2 .2),

[0461]

[0462]

[0463]

[0464]

[0465] (3) When the pressure (P) remains constant ( Figure 2 .3),

[0466]

[0467]

[0468]

[0469]

[0470] (4) When the volume (V) remains constant ( Figure 2 .4),

[0471]

[0472]

[0473]

[0474]

[0475] (5) When the temperature (T) remains constant ( Figure 2 .5),

[0476]

[0477]

[0478]

[0479]

[0480] (6) When entropy (S) remains constant ( Figure 2 .6),

[0481]

[0482]

[0483]

[0484]

[0485] 5. Thermodynamic potential total differential equation

[0486] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0487] dU(V,S)=(-P)·dV+(T)·dS=-P·dV+T·dS (5-1)

[0488] dH(S,P)=(T)·dS+(V)·dP=T·dS+V·dP (5-2)

[0489] dG(P,T)=(V)·dP+(-S)·dT=V·dP-S·dT (5-3)

[0490] dA(T,V)=(-S)·dT+(-P)·dV=-S·dT-P·dV (5-4)

[0491] (2) When the chemical potential (μ) remains unchanged ( Figure 2 .2),

[0492] dχ(P,S)=(V)·dP+(T)·dS=V·dP+T·dS (5-5)

[0493] dψ(S,V)=(T)·dS+(-P)·dV=T·dS-P·dV (5-6)

[0494] dΩ(V,T)=(-P)·dV+(-S)·dT=-P·dV-S·dT (5-7)

[0495] dφ(T,P)=(-S)·dT+(V)·dP=-S·dT+V·dP=0 (5-8)

[0496] (3) When the pressure (P) remains constant ( Figure 2 .3),

[0497] dH(N,S)=(μ)·dN+(T)·dS=(μ)·dN+T·dS (5-9)

[0498] dχ(S,μ) = (T)·dS + (-N)·dμ = T·dS - N·dμ (5-10)

[0499] dφ(μ,T) = (-N)·dμ + (-S)·dT = -N·dμ - S·dT = 0 (5-11)

[0500] dG(T,N) = (-S)·dT + (μ)·dN = -S·dT + μ·dN (5-12)

[0501] (4) When under the condition that the volume (V) remains unchanged ( Figure 2 .4),

[0502] dψ(μ,S) = (-N)·dμ + (T)·dS = -N·dμ + T·dS (5-13)

[0503] dU(S,N) = (T)·dS + (μ)·dN = T·dS + μ·dN (5-14)

[0504] dA(N,T) = (μ)·dN + (-S)·dT = μ·dN - S·dT (5-15)

[0505] dΩ(T,μ) = (-S)·dT + (-N)·dμ = -S·dT - N·dμ (5-16)

[0506] (5) When under the condition that the temperature (T) remains unchanged ( Figure 2 .5),

[0507] dA(V,N) = (-P)·dV + (μ)·dN = -P·dV + μ·dN (5-17)

[0508] dG(N,P) = (μ)·dN + (V)·dP = μ·dN + V·dP (5-18)

[0509] dφ(P,μ) = (V)·dP + (-N)·dμ = V·dP - N·dμ = 0 (5-19)

[0510] dΩ(μ,V) = (-N)·dμ + (-P)·dV = -N·dμ - P·dV (5-20)

[0511] (6) When under the condition that the entropy (S) remains unchanged ( Figure 2 .6),

[0512] dψ(V,μ) = (-P)·dV + (-N)·dμ = -P·dV - N·dμ (5-21)

[0513] dχ(μ,P)=(-N)·dμ+(V)·dP=-N·dμ+V·dP (5-22)

[0514] dH(P,N)=(V)·dP+(μ)·dN=V·dP+μ·dN (5-23)

[0515] dU(N,V)=(μ)·dN+(-P)·dV=μ·dN-P·dV (5-24)

[0516] 6. Gibbs-Helmholtz equation and this type of thermodynamic relationship

[0517] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0518]

[0519]

[0520]

[0521]

[0522] (2) When the chemical potential (μ) remains unchanged ( Figure 2 .2),

[0523]

[0524]

[0525]

[0526]

[0527] (3) When the pressure (P) remains constant ( Figure 2 .3),

[0528]

[0529]

[0530]

[0531]

[0532] (4) When the volume (V) remains constant ( Figure 2 .4),

[0533]

[0534]

[0535]

[0536]

[0537] (5) When the temperature (T) remains constant ( Figure 2 .5),

[0538]

[0539]

[0540]

[0541]

[0542] (6) When entropy (S) remains constant ( Figure 2 .6),

[0543]

[0544]

[0545]

[0546]

[0547] 7. Isobaric heat capacity (C P ) class variables

[0548] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0549]

[0550]

[0551]

[0552]

[0553]

[0554]

[0555]

[0556]

[0557] (2) When the chemical potential (μ) remains unchanged ( Figure 2 .2),

[0558]

[0559]

[0560]

[0561]

[0562]

[0563]

[0564]

[0565]

[0566] (3) When the pressure (P) remains constant ( Figure 2 .3),

[0567]

[0568]

[0569]

[0570]

[0571]

[0572]

[0573]

[0574]

[0575] (4) When the volume (V) remains constant ( Figure 2 .4),

[0576]

[0577]

[0578]

[0579]

[0580]

[0581]

[0582]

[0583]

[0584] (5) When the temperature (T) remains constant ( Figure 2 .5),

[0585]

[0586]

[0587]

[0588]

[0589]

[0590]

[0591]

[0592]

[0593] (6) When entropy (S) remains constant ( Figure 2 .6),

[0594]

[0595]

[0596]

[0597]

[0598]

[0599]

[0600]

[0601]

[0602] (Note: Independent C P There are only 24 class variables. Each relationship is repeated once. For example, formula (7-1) and formula (7-26) are equal. VN =C NV .)

[0603] 8. Maxwell Partial Derivatives of the Third Kind and C P Relationships between class variables

[0604] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0605]

[0606]

[0607]

[0608]

[0609]

[0610]

[0611]

[0612]

[0613] (2) When the chemical potential (μ) remains unchanged ( Figure 2 .2),

[0614]

[0615]

[0616]

[0617]

[0618]

[0619]

[0620]

[0621]

[0622] (3) When the pressure (P) remains constant ( Figure 2 .3),

[0623]

[0624]

[0625]

[0626]

[0627]

[0628]

[0629]

[0630]

[0631] (4) When the volume (V) remains constant ( Figure 2 .4),

[0632]

[0633]

[0634]

[0635]

[0636]

[0637]

[0638]

[0639]

[0640] (5) When the temperature (T) remains constant ( Figure 2 .5),

[0641]

[0642]

[0643]

[0644]

[0645]

[0646]

[0647]

[0648]

[0649] (6) When entropy (S) remains constant ( Figure 2 .6),

[0650]

[0651]

[0652]

[0653]

[0654]

[0655]

[0656]

[0657]

[0658] (Note: There are only twenty-four independent relationships of this type. Each relationship is repeated once. For example, equation (8-1) and equation (8-26) are the same.)

[0659] 9. Like C P and C V The same two nearest neighbors C P Relationships between class variables

[0660] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0661]

[0662]

[0663]

[0664]

[0665]

[0666]

[0667]

[0668]

[0669] (2) When the chemical potential (μ) remains unchanged ( Figure 2 .2),

[0670]

[0671]

[0672]

[0673]

[0674]

[0675]

[0676]

[0677]

[0678] (3) When the pressure (P) remains constant ( Figure 2 .3),

[0679]

[0680]

[0681]

[0682]

[0683]

[0684]

[0685]

[0686]

[0687] (4) When the volume (V) remains constant ( Figure 2 .4),

[0688]

[0689]

[0690]

[0691]

[0692]

[0693]

[0694]

[0695]

[0696] (5) When the temperature (T) remains constant ( Figure 2 .5),

[0697]

[0698]

[0699]

[0700]

[0701]

[0702]

[0703]

[0704]

[0705] (6) When entropy (S) remains constant ( Figure 2 .6),

[0706]

[0707]

[0708]

[0709]

[0710]

[0711]

[0712]

[0713]

[0714] 10. Parallel C P Relationships between class variables

[0715] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0716] C VN ·O VN =(T)·(-S)=-TS (10-1)

[0717] C PN ·O PN =(T)·(-S)=-TS (10-2)

[0718] J TN ·R TN =(V)·(-P)=-PV (10-3)

[0719] J SN ·R SN =(V)·(-P)=-PV (10-4)

[0720] (2) When the chemical potential (μ) remains unchanged ( Figure 2 .2)

[0721] C Pμ ·O Pμ ≠(T)·(-S)=-TS (10-5)

[0722] C Vμ ·O Vμ =(T)·(-S)=-TS (10-6)

[0723] R Tμ ·J Tμ ≠(-P)·(V)=-PV (10-7)

[0724] R Sμ ·J Sμ =(-P)·(V)=-PV (10-8)

[0725] (3) When the pressure (P) remains constant ( Figure 2 .3),

[0726] C μP ·O μP ≠(T)·(-S)=-TS (10-9)

[0727] C NP ·O NP =(T)·(-S)=-TS (10-10)

[0728] Γ TP ·Λ TP ≠(-N)·(μ)=-μN (10-11)

[0729] Γ SP ·Λ SP =(-N)·(μ)=-μN (10-12)

[0730] (4) When the volume (V) remains constant ( Figure 2 .4),

[0731] C μV ·O μV =(T)·(-S)=-TS (10-13)

[0732] C NV ·O NV =(T)·(-S)=-TS (10-14)

[0733] Λ TV Γ TV=(μ)·(-N)=-μN (10-15)

[0734] Λ SV Γ SV =(μ)·(-N)=-μN (10-16)

[0735] (5) When the temperature (T) remains constant ( Figure 2 .5),

[0736] Λ VT Γ VT =(μ)·(-N)=-μN (10-17)

[0737] Λ PT Γ PT ≠(μ)·(-N)=-μN (10-18)

[0738] J μT ·R μT ≠(V)·(-P)=-PV (10-19)

[0739] J NT ·R NT =(V)·(-P)=-PV (10-20)

[0740] (6) When entropy (S) remains constant ( Figure 2 .6),

[0741] Γ VS ·Λ VS =(-N)·(μ)=-μN (10-21)

[0742] Γ PS ·Λ PS =(-N)·(μ)=-μN (10-22)

[0743] J NS ·R NS =(V)·(-P)=-PV (10-23)

[0744] J μS ·R μS =(V)·(-P)=-PV (10-24)

[0745] (Note: C Pμ =R Tμ =Λ PT =∞ and O Pμ =J Tμ =Γ PT =0)

[0746] 11. Cross C P Relationships between class variables

[0747] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0748] J SN ·C VN =J TN ·C PN (11-1)

[0749] C VN ·R TN =C PN ·R SN (11-2)

[0750] R TN ·O PN =R SN ·O VN (11-3)

[0751] O PN ·J SN =O VN ·J TN (11-4)

[0752] (2) When the chemical potential (μ) remains unchanged ( Figure 2 .2),

[0753] R Sμ ·C Pμ =R Tμ ·C Vμ =∞ (11-5)

[0754] C Pμ ·J Tμ ≠C Vμ ·J Sμ (11-6)

[0755] J Tμ ·O Vμ =J Sμ ·O Pμ =0 (11-7)

[0756] O Vμ ·R Sμ ≠O Pμ ·R Tμ (11-8)

[0757] (3) When the pressure (P) remains constant ( Figure 2 .3),

[0758] Γ SP ·C NP ≠Γ TP ·C μP(11 - 9)

[0759] C NP ·Λ TP = C μP ·Λ SP = ∞ (11 - 10)

[0760] Λ TP ·O μP ≠ Λ SP ·O NP (11 - 11)

[0761] O μP ·Γ SP = O NP ·Γ TP = 0 (11 - 12)

[0762] (4) When under the condition that the volume (V) remains unchanged ( Figure 2 .4),

[0763] Λ SV ·C<00​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​μT Γ PT ≠R NT Γ VT (11-19)

[0771] Γ PT ·J NT =Γ VT ·J μT =0 (11-20)

[0772] (6) When entropy (S) remains constant ( Figure 2 .6),

[0773] J μS Γ VS =J NS Γ PS (11-21)

[0774] Γ VS ·R NS =Γ PS ·R μS (11-22)

[0775] R NS ·Λ PS =R μS ·Λ VS (11-23)

[0776] Λ PS ·J μS =Λ VS ·J NS (11-24)

[0777] (Note: C Pμ =R Tμ =Λ PT =∞ and O Pμ =J Tμ =Γ PT =0.)

[0778] 12. Jacobian equation

[0779] (1) When the number of moles (N) remains unchanged ( Figure 2 .1),

[0780] J(U,Y)=(-P)·J(V,Y)+(T)·J(S,Y)=-P·J(V,Y)+T·J(S,Y) (12-1)

[0781] J(H,Y)=(T)·J(S,Y)+(V)·J(P,Y)=T·J(S,Y)+V·J(P,Y) (12-2)

[0782] J(G,Y) = (V)·J(P,Y) + (-S)·J(T,Y) = V·J(P,Y) - S·J(T,Y) (12 - 3)

[0783] J(A,Y) = (-S)·J(T,Y) + (-P)·J(V,Y) = -S·J(T,Y) - P·J(V,Y) (12 - 4)

[0784] (2) When the chemical potential (μ) is kept constant ( Figure 2 .2),

[0785] J(χ,Y) = (V)·J(P,Y) + (T)·J(S,Y) = V·J(P,Y) + T·J(S,Y) (12 - 5)

[0786] J(ψ,Y) = (T)·J(S,Y) + (-P)·J(V,Y) = T·J(S,Y) - P·J(V,Y) (I2 - 6)

[0787] J(Ω,Y) = (-P)·J(V,Y) + (-S)·J(T,Y) = -P·J(V,Y) - S·J(T,Y) (12 - 7)

[0788] J(φ,Y) = (-S)·J(T,Y) + (V)·J(P,Y) = -S·J(T,Y) + V·J(P,Y) = 0 (12 - 8)

[0789] (3) When the pressure (P) is kept constant ( Figure 2 .3),

[0790] J(H,Y) = (μ)·J(N,Y) + (T)·J(S,Y) = (μ)·J(N,Y) + T·J(S,Y) (12 - 9)

[0791] J(χ,Y) = (T)·J(S,Y) + (-N)·J(μ,Y) = T·J(S,Y) - N·J(μ,Y) (12 - 10)

[0792] J(φ,Y) = (-N)·J(μ,Y) + (-S)·J(T,Y) = -N·J(μ,Y) - S·J(T,Y) = 0 (12 - 11)

[0793] J(G,Y) = (-S)·J(T,Y) + (μ)·J(N,Y) = -S·J(T,Y) + μ·J(N,Y) (1I - 12)

[0794] (4) When the volume (V) is kept constant ( Figure 2 .4),

[0795] It should be noted that there seems to be an error in the original text where "I2 - 6" should probably be "12 - 6" and "1I - 12" should probably be "12 - 12" in the translated content for better accuracy.J(ψ, Y) = (-N)·J(μ, Y) + (T)·J(S, Y) = -N·J(μ, Y) + T·J(S, Y) (12 - 13)

[0796] J(U, Y) = (T)·J(S, Y) + (μ)·J(N, Y) = T·J(S, Y) + μ·J(N, Y) (12 - 14)

[0797] J(A, Y) = (μ)·J(N, Y) + (-S)·J(T, Y) = μ·J(N, Y) - S·J(T, Y) (12 - 15)

[0798] J(Ω, Y) = (-S)·J(T, Y) + (-N)·J(μ, Y) = -S·J(T, Y) - N·J(μ, Y) (12 - 16)

[0799] (5) When the temperature (T) is constant ( Figure 2 .5),

[0800] J(A, Y) = (-P)·J(V, Y) + (μ)·J(N, Y) = -P·J(V, Y) + μ·J(N, Y) (12 - 17)

[0801] J(G, Y) = (μ)·J(N, Y) + (V)·J(P, Y) = μ·J(N, Y) + V·J(P, Y) (12 - 18)

[0802] J(φ, Y) = (V)·J(P, Y) + (-N)·J(μ, Y) = V·J(P, Y) - N·J(μ, Y) = 0 (12 - 19)

[0803] J(Ω, Y) = (-N)·J(μ, Y) + (-P)·J(V, Y) = -N·J(μ, Y) - P·J(V, Y) (12 - 20)

[0804] (6) When the entropy (S) is constant ( Figure 2 .6),

[0805] J(ψ, Y) = (-P)·J(V, Y) + (-N)·J(μ, Y) = -P·J(V, Y) - N·J(μ, Y) (12 - 21)

[0806] J(χ, Y) = (-N)·J(μ, Y) + (V)·J(P, Y) = -N·J(μ, Y) + V·J(P, Y) (12 - 22)

[0807] J(H, Y) = (V)·J(P, Y) + (μ)·J(N, Y) = V·J(P, Y) + μ·J(N, Y) (12 - 23)

[0808] J(U,Y)=(μ)·J(N,Y)+(-P)·J(V,Y)=μ·J(N,Y)-P·J(V,Y) (12-24)

[0809] The present invention has been described in detail above with reference to the accompanying drawings and embodiments. A person skilled in the art can make various modifications to the present invention based on the above description. Therefore, certain details in the embodiments should not be construed as limiting the present invention. The scope of protection of the present invention shall be determined by the scope defined in the appended claims.

Claims

1. A thermodynamic concentric multi-layered polyhedral shell model, characterized in that: The thermodynamic concentric multi-layer polyhedral shell model can be made by hand or by using a 3D printer; forty-four thermodynamic variables of four categories in the unit single-phase system are used as elements and are respectively placed on the vertices of the concentric multi-layer polyhedral shell model according to their physical meanings, thereby forming a symmetrical and complete self-consistent thermodynamic concentric multi-layer polyhedral shell model; The thermodynamic concentric multi-layered polyhedral shell model is composed of a cubic shell sandwiched between two octahedral shells and an outer 26-hedral shell; Forty-four thermodynamic variables of four categories in the unit single-phase system include three pairs of conjugate independent variables, eight complete thermodynamic potentials, six first-order partial derivatives of thermodynamic potentials, and twenty-four second-order partial derivatives of thermodynamic potentials; The three pairs of conjugate independent variables are placed on six vertices of an octahedral shell located inside the cubic shell; The eight complete thermodynamic potentials are placed on the eight vertices of the cubic shell; The first-order partial derivatives of the six thermodynamic potentials are placed on the six vertices of the octahedral shell located outside the cubic shell; The second-order partial derivatives of the twenty-four complete thermodynamic potentials are placed on the vertices of the twenty-six-sided shell.

2. A simple symmetric diagram method for describing thermodynamic relationships, characterized in that: The simple symmetry diagram method comprises the following steps: S11, providing the thermodynamic concentric multi-layered polyhedral shell model according to claim 1; S12, obtaining a two-dimensional {1,0,0} projection diagram according to the thermodynamic concentric multi-layered polyhedral shell model; S13, the twelve specially created rigid movable patterns are superimposed on the fixed two-dimensional {1,0,0} projection diagram, and symmetric transformation is performed, thereby describing the twelve types of more than 300 thermodynamic relationships in the unit single-phase system one by one.

3. The simplified symmetry diagram method according to claim 2, wherein: The step of obtaining a two-dimensional {1,0,0} projection diagram based on the thermodynamic concentric multi-layer polyhedral shell model includes: The thermodynamic concentric multi-layer polyhedral shell model is dissected and projected outward from the central plane along six different <1,0,0> directions, thereby obtaining six two-dimensional {1,0,0} projection images.

4. The simplified symmetry diagram method according to claim 3, wherein: According to the principle of symmetry equivalence, the steps to describe the same type of thermodynamic relationship include: S21 is a step of selecting a basis: selecting a (0,0,-1) projection map from the six two-dimensional {1,0,0} projection maps as a fixed base map; S22: Step of creating a pattern: selecting one of the twelve types of thermodynamic relations as a model, and creating a special pattern to describe the model; S23 Overlap transformation step: Overlap the pattern on the fixed base pattern, select corresponding mathematical symbols and thermodynamic variables for various graphic symbols in the pattern, so as to describe a thermodynamic relationship; then make the pattern undergo a series of symmetric transformations (σ, C4 1 ,C4 2 and C4 3 ), after each symmetry transformation, corresponding mathematical symbols and thermodynamic variables are selected for various graphical symbols in the pattern, thereby realizing the description of other thermodynamic relationships in the same type of thermodynamic relationship; S24 is a replacement basis step: replacing the (0,0,-1) projection map with the other two-dimensional projection maps in the six two-dimensional {1,0,0} projection maps one by one, and repeating the S23 overlapping transformation step, thereby realizing the description of all thermodynamic relationships in the same type of thermodynamic relationships.

5. The simplified symmetry diagram method according to claim 2, wherein: Also includes: For the twelve different types of thermodynamic relationships, twelve rigid movable patterns are created respectively to describe the twelve types of thermodynamic relationships. Each pattern is arranged in the order of writing, and is a mixture of mathematical symbols and variable selection symbols. According to the differences in the appearance, symbol composition and writing order of the created patterns, the differences between different types of thermodynamic relationships can be intuitively distinguished.

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