Question

A. Show that the ideal gases following PV=nRT satisfy (dp/dv)(dv/dt)(dt/dp)=-1.

B. Show that van see waals fluids following P=(RT/V-b) - a/V^2 satisfy (dp/dv)(dv/dt)(dt/dp)=-1.

C. Use the fundamental relation, dH=TdS + VdP and Maxwell equations to show that (dH^ig/dV) = 0, which means that enthrall of ideal gases us independent of volume.

Answer 1

PV = nRT

P= nRT/V dP/dV= -nRT/V²

V= nRT/P dV/dT= nR/P and T= PV/nR, dT/dP= V/nR

(dP/dV)*(dV/dT)* dT/dP= - nRT*nR*V/{ V²*P*nR)= -nRT*nR*P/ { PV*VnR} , ( since PV= nRT)

=-nRV/nRV= -1

b) P= RT/(V-b)- a/V2

dP/dV= -RT/ (V-b)² + 2a/V³   =   {2a*(V-b)² - V³RT}/ { V3*(V-b)² }    (1A)

(P+a/V²) (V-b)= RT

differentinating V with respet to T

{P+a/V²⁾- (V-b)*2a/V³} dV/dT= R (2A)

but P+a/V²= RT/(V-b), Eq.2A becomes RT/ (V-b)- 2a(V-b)/V³ ={ RTV³-2a(V-b)²}/ (V-b)V³   (3A)

from Eq.2A

dT/dP= (V-b)/R    (4)

(dP/dV) * (dV/dT)* dT/dP={ 2a*(V-b)² - V³RT}*R*(V-b)*V³* (V-b)}/ [ R* {RTV³- 2a(V-b)²}V³*(V-b}²] = -1

C. (dH/dT) _P= Cp

dH= TdS+ VdP

(dH/dT) _P= T (dS/dT)_P =

(dS/dT)_P= Cp/T

from maxwell Equations . (dS/dP)_T= -(dV/dT)P

dH= TdS+ VdP

(dH/dP)T= V+ T*(dS/dP)T= V- T(dV/dT)_P=

H =f (T,P)

dH = (dH/dT)_PdT + (dH/dP)_TdP= CpdT+ [ V- T(dV/dT)_P]dP   (2)

for ideal gas PV= RT and dV/dT= R/P and TdV/dT= RT/P= V

Eq.2 becomes

dH= CpdT+ 0

dH/dV= 0

for ideal gas