Question

7. a) Starting with the expression for heat added to a system: dq = Cydt + (P + (**).) av and letting S = S(VT), show that the second law of thermodynamics requires that all equations of state must have the form: (P+ **)n) = ctf(V) where c is a constant and f(V) is a function only of V. b) Show that this is true for an ideal gas and a van der Waal's gas. c) For a Virial gas (to second order) this is not so obvious since: (9), = ) where B = B(T) is the second Virial coefficient. Show that in order to be consistent with the second law of thermodynamics B(T) = e-where e and g are constants.

Answer 1

a) dq = wdT+ (P+ CD) av S = SCV,T) as = ( 2 ), dr + (os), at Equating as = dq one obtains (23) - + [P+CC), ] -0 From Maxwell's relatior, ), - o *), sefcu). . By vitre of Equo & , one can write Pt Cre)- et fc) — (b) for ideal gas, PV = R1 (24) = Rp = e flv) ( = RI-P=0. E - ECT) For vander waal's gus, (P + 2 / 2) (V-6) = RT - P = R1 - az (f), = Pb Ee for) (O) = RIT-P = Qiz E = E(1,v)

(e) vinal gas : PE BI (07. BCT) Ezt...) Of), = (14 BCF) ERO...): + RT [t de cat -...] from Eqr , one gets Pue) - The C1* Bers)Ís* ---) + R5 [draci ..] = P + B charge (), preces This imfries