Research: Science and Education
A Vector Representation for Thermodynamic Relationships Lionello Pogliani Dipartimento di Chimica, Università della Calabria, 87030 Rende (CS), Italy;
[email protected] A deeper understanding of a subject is always advantageously promoted by alternative approaches, which not only have the advantage of strengthening curiosity, but which can also become a subject worth study on their own. The learning process is not a simple linear procedure, that is, a series of “forward-step” concepts, but it includes a good deal of reshaping and revisualizing the acquired material. Thermodynamics is surely the branch of science where students and teachers encounter great difficulty. Thermodynamics is useful but, at the same time, many of its concepts are highly abstract, and its mathematical formalism is feared by many students, especially if they already fear mathematics on its own. The present article, based on row vectors, is an extension of the diagrammatic method for thermodynamic relationships for simple one-component PVT systems, suggested first by Max Born and worked out, recently, by other authors (1–6). The diagrammatic method is a “quasi-geometrical” approach that allows the derivation of a huge series of thermodynamic relations in a straightforward way. The method is easily tractable and does not require the use of higher mathematics. The present vector formalism method for thermodynamic relationships maintains tractability and uses accessible mathematics, which can be seen as a diverting and entertaining step into the mathematical formalism of thermodynamics and as an elementary application of matrix algebra. The method is based on ideas and operations apt to improve the skills of teachers and the level of inquisitiveness and curiosity of students.
for the natural variables:
Ordered Sets of Thermodynamic Objects: The Potentials and their Natural Variables
≡
{ V , T, P, S }
( 1, −1 )
=
( 0, 1 )
+ ( 1, 0 )
(3)
= ( 1, 0 ) + ( 0, −1 )
(4)
( −1,, −1 )
=
( 0, −1 )
+ ( −1, 0 )
(5)
( −1, 1 )
=
( −1, 0 )
+ ( 0, 1 )
(6)
These sums can be read as encoding the functional dependence of the potentials, A = A(V, T), G = G(T, P), H = H(P, S), and U = U(S, V). Further, (ii) the set of potentials can be subdivided into subsets of adjacent potentials, and this subdivision uncovers that the potentials are orthogonal to each other. Orthogonality means that the result of multiplying a vector with components values of ±1 by the transpose, t (column vector), of an adjacent vector with components values of ±1, is zero1:
{ {A, G}, {G, H}, {H, U}, {U, A} }
(7)
{ { ( 1, 1 ), ( 1, −1 ) }, { ( 1, −1 ), ( −1, −1 ) }, { ( −1, −1 ), ( −1,, 1 ) }, { ( −1, 1 ) , ( 1, 1 ) } }
(8)
( 1, 1 ) ( 1, −1 )t = ( −1, −1 ) ( −1, 1 )t
t
0; ( 1, −1 ) ( −1, −1 ) = 0 t
= 0; ( −1, 1 ) ( 1, 1 ) = 0
(9)
{ ( 1, 1 ), ( 1, −1 ) , ( −1, −1 ) , ( −1, 1 ) }
(1)
The same holds for the transposed multiplications, that is, (1, ᎑1)(1, 1)t = 0, and so on. The result of multiplying any adjacent potentials is, thus, always zero. Further, multiplications between nonadjacent vectors with components values of ±1, that is, (1, 1)(᎑1, ᎑1)t = ᎑2, gives rise to a scalar different from 0 and 1, which shows that the two thermodynamic potentials are related by two transformations. Geometrically, such thermodynamic potentials lie at opposite corners of the E-diagram for thermodynamic relationships. The third rule says that (iii) the set of natural variables can be divided into subsets of adjacent natural variables that are orthogonal to each other:
{ ( 0, 1 ), ( 1, 0 ), ( 0, −1 ) , ( −1, 0 ) }
(2)
{ { V , T }, { T, P }, { P, S }, { S, V } }
Two sets of ordered objects are defined, the first is the set of four energy parameters, {A, G, H, U}, and the second is the set of four natural variables, {P, S, T, V}. These two sets of objects can be seen as a set of points in a bi-dimensional Cartesian coordinate system, where each point can be characterized by a two component (x, y) row vector. Thus, the set of energy parameters and the set of natural variables can be rewritten as two sets of (x, y) vectors:
{ A, G, H, U }
( 1, 1 )
≡
These vectors could be used to draw the energy E-diagram for thermodynamic relationships in the Cartesian plane. It is evident that potentials are encoded by vectors with components values of ±1, while natural variables are encoded by vectors with component values of ±1 and 0. This vector representation allows the detection of some interesting properties: (i) the vector representing a thermodynamic potential is the sum of the two adjacent vectors (complementary vectors)
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(10)
{ { ( 0, 1 ), ( 1, 0 ) }, { ( 1, 0 ), ( 0, −1 ) }, { ( 0, −1 ), ( −1, 0 ) }, { ( −1, 0 ), ( 0, 1 ) } } ( 0, 1 ) ( 1, 0 )t = ( 0, −1 ) ( −1, 0 )t
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(11)
t
0; ( 1, 0 ) ( 0, −1 ) = 0 t
= 0; ( −1, 0 ) ( 0, 1 ) = 0
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(12)
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Now, (iv) the subset of natural variables nonadjacent to each other are collinear to each other, as can be seen from the corresponding row vector operations:
{ { P, V }, { S, T } } ≡ { { ( 0, −1 ) , ( 0, 1 ) } , { ( −1, 0 ) , ( 1, 0 ) } } ( 0, −1 ) ( 0, 1 )t
t
≠ 0 ; ( −1, 0 ) ( 1, 0 ) ≠ 0
(13) (14)
This collinearity will be interpreted so that the only allowed multiplications between the natural variables are the ones that give rise to an energy-dimensioned term: PV or ST, that is, (0, ᎑1)(0, 1), and (᎑1, 0)(1, 0). Clearly, these two multiplications cannot be further simplified.
Relations between Neighboring Potentials The relation that relates together the two orthogonal (᎑1, ᎑1) and (᎑1, 1) vectors, where the y component undergoes the change ᎑1 → 1, is obtained by adding to (᎑1, 1) the collinear energy-dimensioned term, where the y component also changes as ᎑1 → 1. To avoid further simplification, that is, multiplication and addition, the second vector of this last term should not be transposed. This change in the positive direction requires a positive sign, that is,
( −1, −11 )
=
( −1, 1 )
+ ( 0, −1 ) ( 0, 1 )
H = U + PV
(15)
The vector format, which cannot be further simplified, and the “translation” into the regular thermodynamic equation are listed together. The relation between (1, ᎑1) and (᎑1, ᎑1), where the x component changes from 1 to ᎑1, is obtained by adding, with negative sign, the collinear energy-dimensioned term, where x also changes as: 1 → ᎑1. This change in the negative direction requires a negative sign,
( 1, −1 )
=
( −1, −1 )
− ( 1, 0 ) ( −1, 0 )
G = H − TS
(16)
The relation that ties together (᎑1, 1) and (1, 1), where the x component changes from ᎑1 to 1, requires adding to (1, 1) the corresponding collinear energy-dimensioned term with this same change for the x component. This change in the positive direction requires positive sign,
( −1, 1 )
= ( 1, 1 ) + ( −1, 0 ) ( 1, 0 )
U = A + ST
(17)
The relation that holds between (1, ᎑1) and (1, 1), where the y component changes from ᎑1 to 1 (positive change), requires the addition of (1, 1) to the corresponding energy-dimensioned term with this same change, that is,
( 1, −1 )
= ( 1, 1 ) + ( 0, −1 ) ( 0, 1 )
G = A + PV
156
= ( 1, −1 ) − ( 0, 1 ) ( 0, −1 )
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Relations for the Differential Forms of the Potentials To derive an expression for d(᎑1, 1) differentiating eq 6 we obtain d(᎑1, 1) = d(᎑1, 0) + d(0, 1), where the two sides do not have the same dimensions. To obtain an energy-dimensioned term also on the right, the two differentials on this side are multiplied by the collinear companion, that is, (1, 0) and (0, ᎑1). The sign for the whole multiplicative term is given by the sign of this companion, d ( −1, 1 ) = ( 1, 0 ) d ( −1, 0 ) − ( 0, −1 ) d ( 0, 1 ) d U = T d S − P dV
(20)
Let us see how this formalism is able to derive dA = ᎑PdV − SdT and dG = ᎑SdT + VdP, that is, eqs 21, and 22 d ( 1, 1 ) = − ( 0, −1 ) d ( 0, 1 ) − ( −1, 0 ) d ( 1, 0 ) (21) d ( 1, −1 ) = − ( −1, 0 ) d ( 1, 0 ) + ( 0, 1 ) d ( 0, −1 ) (22)
For eq 21 use is made of eq 3, of the corresponding collinear vector (see eq 14), and the signs are given by the collinear companions, both negative. For eq 22 use is made of eq 4, of the corresponding collinear companion, of the negative sign of (᎑1, 0), and of the positive sign of (0, 1). To generate d(᎑1, ᎑1), eq 5 is differentiated, and the collinear companion is multiplied, with its sign, d ( −1, −1 ) =
( 0, 1 ) d ( 0, −1 )
+ ( 1, 0 ) d ( −1, 0 )
d H = V d P + Td S
(23)
Relations between Potentials and Natural Variables Let us take eq 20 and consider the partial derivative where S is held constant, that is, the variable (᎑1, 0) is held constant. After some algebra we obtain the following relation, ∂ ( −1, 1 ) = − ( 0, −1 ) : ∂ ( 0, 1 ) (−1, 0)
∂ U ∂ V
S
= −P (24)
That is, the partial differentiation of a vector with component values of ±1 by a vector with component values of ±1 and 0, with its complementary vector constant (see eqs 3– 6), equals the collinear vector of the denominator with its sign. Let us check this formalism with (1, 1)兾(1, 0), with (1, ᎑1)兾(1, 0), and with (᎑1, 1)兾(᎑1, 0) ratios of vectors
(18)
∂ (1, 1) = − ( −1, 0 ) : ∂ (1, 0) (0, 1)
(19)
∂ (1, −1) ∂ (1, 0 )
From G = A + PV we obtain, A = G − VP, that is,
( 1, 1 )
Here y changes from 1 to ᎑1 (negative change); thus the term to be added must have this same change, and it must be negative. In this way one can derive every expression that holds between thermodynamic potentials; it suffices to know the component whose sign changes.
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= − (−1, 0 ) : ( 0,−1)
∂ A ∂ T ∂ G ∂ T
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= −S
(25)
= −S
(26)
V
P
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∂ ( −11, 1) ∂ ( −1, 0 )
=
∂ U ∂ S
(1, 0 ) :
( 0,1)
= T V
(27)
It is not possible to differentiate (1, 1) with respect to either (᎑1, 0) or (0, ᎑1) using the present simple rules, as these last vectors have no complement vector that sums to (1, 1). Note that subtracting the denominator from the numerator vector gives the constant vector.
The Maxwell Relations Maxwell relations are partial differential relations among the natural variables, that is, among the vectors with component values of ±1 and 0, and, more specifically, between the complementary (0, 1)兾(1, 0) vectors of eqs 3–6. Here, the numerator and the constant term are collinear vectors, while the constant term controls the sign of the partial differential. The two sides of the relation are inverted (0, 1) vectors, and if, in such vectors, two ᎑1 elements are inverted they change sign, as in eqs 29 and 31. Some examples are
Ordered Sets of Thermodynamic Objects: The Massieu Functions and their Natural Variables An S-diagram has recently been added to the E-diagram, (7, 8), which shows the relationship of the entropic functions and their natural variables. The same formalism used here, that is, the same row vectors and the same rules can be used to derive all the relationships that hold for the parameters of the set of entropic Massieu functions and for the set of their natural variables:
{M1, M 2 , M 3 , S} V,
≡
1 P , , U ≡ T T
{ (1, −1) , (1, 1) , ( −1, 1) , ( −1, −1) } (32)
{ ( 0, −1) , (1, 0 ) , ( 0, 1) , ( −1, 0 ) }
(33)
By the aid of these row vectors it is possible to draw the Sdiagram in the Cartesian plane. The M’s are the Massieu’s entropic functions, whose expressions can be found using exactly the same formalism used above.2
Relations between Neighboring Potentials ∂ ( 0, −1) ∂ (1, 0 )
= ( 0,1)
∂ P ∂ T
∂ ( −1, 0 ) ∂ ( 0, −1)
V
(1,0 ) T
( -1,0 )
∂ T ∂ P
∂ ( 0, −1) ∂ ( −1, 0 )
=
∂ S ∂ V
= −
∂ S ∂ P
∂ (1, 0 ) − ∂ ( 0, −1)
∂ (−1, 0 ) ∂ ( 0, 1)
S
V
(1, 1) = ( −1, 1) − ( −1, 0 )(1, 0 )
T
M 2 = M1 −
∂ ( 0, 1) ∂ (1, 0 )
∂ V ∂ S
( 0,−1)
∂ T = − ∂ V
( −1, 1) = ( −1, −1) − ( 0, 1)(0, −1)
(29)
(35)
(1, 1) = ( −1, −1) − ( 0, 1)( 0, −1) − (1, 0 )( −1, 0 ) M1 = S −
( 0,−1)
(30) P
( −1,0 )
(31) S
Actually some other regularities can be found in the vector equations. For example, the sum of numerator and denominator vectors on the two sides of a relation are just sign inverted, that is, in eq 28: (1, ᎑1) and (᎑1, 1). The reader should check vector eqs 29–31. Further, the sum of all the vectors on one side always equals the denominator vector on the same side.
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(34)
If this relation is inserted into eq 34, we obtain
P
∂ (1, 0 ) ∂ ( 0, 1)
VP T
The equation relating (᎑1, 1) with (᎑1, ᎑1) is
∂ ( 0, 1) = − ∂ ( −1, 0 ) =
( 0,1)
(1,0 )
(28)
∂ V = − ∂ T
= −
∂ P ∂ S
The following equation relates (1, 1) with (᎑1, 1):
•
PV U − T T
(36)
Note how the double change in sign in going from (1, 1) to (᎑1, ᎑1) should be followed by the same double change in sign in the required two collinear energy-dimensioned terms, which, consequently, should also be negative. In eq 36, S − PV兾T − U兾T = S − (PV + U)兾T = Y, where Y is the wellknown Planck function (1). Conclusion The reader should check the present method for further regularities and relations, especially, those involving the Massieu functions and their natural variables. The reader should also check when the method does not work (see paragraph after eq 27), and if this formalism could be extended to include three-dimensional row vectors for systems with more than one component, completing what has been done by Phillips with a three-dimensional diagrammatic method (3).
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Actually, the problem of finding “shortcuts” for deriving thermodynamic relationships is far from being a minor challenge. Such methods were also main concerns of the young P. W. Bridgman (Nobel prize for his contributions in high-pressure physics in 1945) (9), who in 1914 devised an interesting algebraic scheme of shorthand notation for a wide variety of thermodynamic relationships. This scheme was further developed by Shaw in 1935 (10, 11). Other attempts to outline a matrix representation of thermodynamics for simple and multicomponent systems involve, in one case (12, 13), a more convoluted matrix formalism, which does not parallel the aims of the present vector formalism, and, in the other case, it concerns a detailed experimental application of matrix algebra for multicomponent systems (14). Nevertheless, these three articles are worthy to be perused, especially, by those who are interested in “new dishes with old ingredients”. Acknowledgment I thank the reviewers for their interesting and helpful suggestions. Notes 1. The product of a row and a column vector equals: (a, b, c)(d, e, f )t = ad + be + cf.
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2. The Mi numbering is somewhat different from the numbering introduced in a previous article (6).
Literature Cited 1. Callen, H. B. Thermodynamics and Introduction to Thermostatistics; Wiley: New York, 1985. 2. Dykstra, C. E. Physical Chemistry; Prentice Hall: Upper Saddle River, NJ, 1997. 3. Phillips, J. M. J. Chem. Educ. 1987, 64, 674. 4. Rodriguez, J.; Brainard, A. A. J. Chem. Educ. 1989, 66, 495. 5. Pogliani, L.; La Mesa, C. J. Chem. Educ. 1992, 69, 808. 6. Pogliani, L. J. Chem. Inf. Comput. Sci. 1998, 38, 130. 7. Pogliani, L. J. Chem. Educ. 2001, 78, 680. 8. Pogliani, L. MATCH - Commun. Math. Comput. Chem. 2003, 47, 153. 9. Bridgman, P. W. Phys. Rev. 1914, 3, 273. 10. Shaw, A. N. Phil. Trans. Roy. Soc. London 1935, A334, 299. 11. Pitzer, K. S. Thermodynamics, 3rd ed.; McGraw-Hill: New York, 1995. 12. Kyame, J. J. Am. J. Phys. 1956, 25, 67. 13. Carrol, P. J.; Kyame, J. J. Am. J. Phys. 1962, 30, 282. 14. Wiederkehr, R. R. V. J. Chem. Phys. 1962, 37, 1192.
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