##\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.