##\alpha_P =\frac{V-b}{TV}## Find equation of State

[GL_Black_Hole]

Homework Statement

The coefficient of thermal expansion and isothermal compressibility of a gas are given by ##\alpha_P =\frac{V-b}{TV}## and ##\kappa_T = \frac{V-b}{PV}## find:
a) The equation of state
b) If the heat capacity at constant volume ##C_V## is constant, what is ##\delta U##?
c) What is the change in enthalpy for a process at constant temperature?

Homework Equations

##\alpha_P = \frac{1}{V} \frac{\partial V}{\partial T}##, ##\kappa_T = - \frac{1}{V} \frac{\partial V}{\partial P}##

The Attempt at a Solution

a) Using the chain rule I can show that ##\frac{\partial T}{\partial P} = \frac{\kappa_T}{\alpha_P} = \frac{T}{P}##, so separating this differential equation gives: ##\int \frac{dT}{T} = \int \frac{dP}{P} ##, so ## T = AP + G(V)##, where ##G(V)## is a function of volume.
But applying the definition of ##\alpha_P## gives ##\frac{\partial T}{\partial V} = \frac{T}{V-b} = G' (V)## so ##G(V) = T ln|V-b|##, giving ##T = AP + T ln|V-b|,## or ## T =\frac{AP}{1+ln|V-b|}##.
Does this make sense? I've never seen an equation of state with a logarithm in it before...

 

[TSny]

Hello.

a) Using the chain rule I can show that ##\frac{\partial T}{\partial P} = \frac{\kappa_T}{\alpha_P} = \frac{T}{P}##

OK

so separating this differential equation gives: ##\int \frac{dT}{T} = \int \frac{dP}{P} ##

OK

so ## T = AP + G(V)##, where ##G(V)## is a function of volume.

This doesn't look correct. The integrations will give logarithms. It doesn't appear that you eliminated the logarithms correctly.