Thermodynamics

Partial Differential RelationsNow let us rewrite Eq. 12–3 aswhere(12–4)Taking the partial derivative of M with respect to y and of N with respect tox yieldsThe order of differentiation is immaterial for properties since they are continuouspoint functions and have exact differentials. Therefore, the two relationsabove are identical:a 0M(12–5)0y b a 0Nx 0x b yThis is an important relation for partial derivatives, and it is used in calculusto test whether a differential dz is exact or inexact. In thermodynamics, thisrelation forms the basis for the development of the Maxwell relations discussedin the next section.Finally, we develop two important relations for partial derivatives—thereciprocity and the cyclic relations. The function z z(x, y) can also beexpressed as x x(y, z) if y and z are taken to be the independent variables.Then the total differential of x becomes, from Eq. 12–3,Eliminating dx by combining Eqs. 12–3 and 12–6, we haveRearranging,a 0M0y b xdz M dx N dyM a 0z0x b andN a 0zy0y b x02 z0Nanda0x 0y 0x b ydx a 0x0y b dy a 0xz 0z b dzy02 z0y 0xdz ca 0z0x b a 0xy 0y b a 0zz 0y b d dy a 0xx 0z b a 0zy 0x b dzyca 0z0x b a 0xy 0y b a 0zz 0y b d dy c 1 a 0xx0z b a 0zy 0x b d dzy(12–6)(12–7)Chapter 12 | 655The variables y and z are independent of each other and thus can be variedindependently. For example, y can be held constant (dy 0), and z can bevaried over a range of values (dz 0). Therefore, for this equation to bevalid at all times, the terms in the brackets must equal zero, regardless ofthe values of y and z. Setting the terms in each bracket equal to zero givesa 0x0z b a 0zy 0x b 1 S a 0xy 0z b ya 0z0x b a 0xy 0y b a 0xz 0y b S a 0xx 0y b a 0yz 0z b a 0zx 0x b 1y110z>0x2 y(12–8)(12–9)