Thermodynamics filled in class notes_Part_49

# Thermodynamics filled in class notes_Part_49 - 105 4.1....

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Unformatted text preview: 105 4.1. EXACT DIFFERENTIALS AND STATE FUNCTIONS Now, if the algebraic deﬁnition of Eq. (4.13) holds, what amounts to the deﬁnition of the partial derivative gives the parallel result that dy = ∂y ∂x1 dx1 + xj ,j =1 ∂y ∂x2 xj ,j =2 dx2 + · · · + ∂y ∂xN dxN . (4.15) xj ,j =N Now, combining Eqs. (4.14) and (4.15) to eliminate dy , one gets ψ1 dx1 + ψ2 dx2 + · · · + ψN dxN = ∂y ∂x1 dx1 + xj ,j =1 ∂y ∂x2 xj ,j =2 dx2 + · · · + ∂y ∂xN dxN . xj ,j =N (4.16) Rearranging, one gets 0= ∂y ∂x1 xj ,j =1 − ψ1 dx1 + ∂y ∂x2 xj ,j =2 − ψ2 dx2 + · · · + ∂y ∂xN xj ,j =N − ψN dxN . (4.17) Since the variables xi , i = 1, . . . , N , are independent, dxi , i = 1, . . . , N , are all independent in Eq. (4.17), and in general non-zero. For equality, one must require that each of the coeﬃcients be zero, so ψ1 = ∂y ∂x1 , ψ2 = xj ,j =1 ∂y ∂x2 ,..., ψN = xj ,j =2 ∂y ∂xN . (4.18) xj ,j =N So when dy is exact, one says that each of the ψi and xi are conjugate to each other. From here on out, for notational ease, the j = 1, j = 2, . . . , j = N will be ignored in the notation for the partial derivatives. It becomes especially confusing for higher order derivatives, and is fairly obvious for all derivatives. If y and all its derivatives are continuous and diﬀerentiable, then one has for all i = 1, . . . , N, and k = 1, . . . , N that ∂2y ∂2y = . (4.19) ∂xk ∂xi ∂xi ∂xk Now from Eq. (4.18), one has ψk = ∂y ∂xk , xj ψl = ∂y ∂xl . (4.20) xj Taking the partial of the ﬁrst of Eq. (4.20) with respect to xl and the second with respect to xk , one gets ∂2y ∂2y ∂ψl ∂ψk = = , . (4.21) ∂xl xj ∂xl ∂xk ∂xk xj ∂xk ∂xl CC BY-NC-ND. 18 November 2011, J. M. Powers. 106 CHAPTER 4. MATHEMATICAL FOUNDATIONS OF THERMODYNAMICS Since by Eq. (4.19) the order of the mixed second partials does not matter, one deduces from Eq. (4.21) that ∂ψk ∂ψl = . (4.22) ∂xl xj ∂xk xj This is a necessary and suﬃcient condition for the exact-ness of Eq. (4.12). It is a generalization of what can be found in most introductory calculus texts for functions of two variables. For the Gibbs equation, (4.1), du = −P dv + T ds, one has y → u, x1 → v, x2 → s, ψ1 → −P ψ2 → T. (4.23) and one expects the natural, or canonical form of u = u(v, s). (4.24) Here, −P is conjugate to v , and T is conjugate to s. Application of the general form of Eq. (4.22) to the Gibbs equation (4.1) gives then ∂T ∂v s =− ∂P ∂s . (4.25) v Equation (4.25) is known as a Maxwell relation. Moreover, specialization of Eq. (4.20) to the Gibbs equation (4.1) gives −P = If the general diﬀerential dy = • The path integral yB − yA = ∂u , ∂v s N i=1 B A T= ∂u ∂s . (4.26) v ψi dxi is exact, one also can show N i=1 ψi dxi is independent of the path of the integral. • The integral around a closed contour is zero: N dy = ψi dxi = 0. (4.27) i=1 • The function y can only be determined to within an additive constant. That is, there is no absolute value of y ; physical signiﬁcance is only ascribed to diﬀerences in y . In fact now, other means, extraneous to this analysis, can be used to provide absolute values of key thermodynamic variables. This will be important especially for ﬂows with reaction. CC BY-NC-ND. 18 November 2011, J. M. Powers. ...
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