Question

6. This problem will further explore the idea of an exact differential. a) Let's consider the molar volume to be a function of pressure and temperature. Write the total differential for molar volume. b) Evaluate this expression for an ideal gas. c) Show that the total differential for the molar volume is an exact differential. d) Show that for a reversible process involving an ideal gas we can write [ RᎢ . 890=1 av „+Cv.md e) Show that this differential is inexact. f) Consider the entropy, defined as dS =- 8 9 rey т ev and show that this function is an exact and s differential.

Answer 1

0.6 (a) Vm = Vm (P. T) dum = (um dP + (DVM ) IT - I alt 1&T p (b) for an ideal gas , PV=RT so that Hence, p2 DTP P COM) - -RT and COMERO Putting egn in egne, dv = (-RI ) d P + (A) IT de dv = ( RJ &T - ( RT ) IP

c) According to dv would be the an Euler exact reciprocal relation differential if paypy - akt/97 ST JP or if - R = R which is true Hence, dv is an exact differential. P2P

(d) To Show : Sarw = RT d Vm + Cv, mdt from 1st law of Thermodynamics, dU = dq - PdV Also, dU = Cud Tu i. Cudt = dg - Pd V da = CudT + P d v Am plong thensughielo deny Pro For ideal gas, PV=RT = P = RI da = RT dv + Crd T o e daren = RI d V m + CrimdT For this to be exact differential, by Euler reciprocal relation this would (e) Г2 (вт) 1 Гэс 1 & (2760) which is not true. Hence, it is inexact differential (f) If we multiply / throughout in egn. (4), we get an exäet differential. ds=8qver (TV) - Cu(t) dT + R dvm

- Now, by Eulen's reciprocity relation (player)/13) - RM JE 0=0 which is true. Hence, ds (T,V) = 8aren is an exact differential.

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