- #1

steroidjunkie

- 18

- 1

## Homework Statement

Find: (a) Equation of state $$f (p, V, T)$$ and (b) Adiabatic equation in dependence on volume and pressure. Internal energy $$U(V, S)=\frac{1}{aV} ln(\frac{S}{\gamma})$$ where a and ##\gamma## are positive constants.

## Homework Equations

(a) ##dU=TdS-pdV \rightarrow##

##T=(\frac{dU}{dS})_V##

##p=-(\frac{dU}{dV})_{S}##

##T=(\frac{d}{dS})_V \cdot \frac{1}{aV} ln(\frac{S}{\gamma})=\frac{1}{aV} \cdot \frac{\gamma}{S} \cdot \frac{1}{\gamma}=\frac{1}{a \gamma V} ##

##p=-(\frac{d}{dV})_S \frac{1}{aV} ln(\frac{S}{\gamma})=- \frac{1}{a} \cdot (- \frac{1}{V^2}) \cdot ln(\frac{S}{\gamma})=\frac{1}{aV^2} ln(\frac{S}{\gamma}) ##

(b) ##pV^{\gamma}=NkT##

##\gamma=?##

##\gamma=\frac{C_p}{C_V}##

##C_p=(\frac{dU}{dT})_p##

##C_V=(\frac{dU}{dT})_V##

## The Attempt at a Solution

(b) ##C_p=(\frac{d}{dT})_p \frac{1}{aV} ln(\frac{S}{\gamma})##

##T=\frac{1}{a \gamma V} \rightarrow T \gamma=\frac{1}{a V}##

##C_p=(\frac{d}{dT})_p T \gamma ln(\frac{S}{\gamma})##

##p=\frac{1}{aV^2} ln(\frac{S}{\gamma}) \rightarrow paV^2=ln(\frac{S}{\gamma})##

##C_p=(\frac{d}{dT})_p T \gamma paV^2=\gamma paV^2##

##C_V=(\frac{d}{dT})_V \frac{1}{aV} ln(\frac{S}{\gamma})##

##C_V=(\frac{d}{dT})_V T \gamma ln(\frac{S}{\gamma})##

##C_V=(\frac{d}{dT})_V T \gamma paV^2=\gamma paV^2##

##\gamma=\frac{C_p}{C_V}= \frac{\gamma paV^2}{\gamma paV^2}=1##

I need help with (b) part of the problem. I know this is not correct and I assume I did something wrong while substituting, but I have no idea what. If you know something please post.

Thanks.