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Question 3.11: using Equation (3.43),(∂H/∂P)T=[(∂U/∂V)T+ P](∂V/∂P)T+V show ...

Using Equation (3.43),

\left( \frac{\partial H}{\partial P} \right)_{T}=\left[ \left( \frac{\partial U}{\partial V} \right)_{T}+ P \right]\left( \frac{\partial V}{\partial P} \right)_{T} + V

 

Show that \mu _{J-T}=0 for an ideal gas. 

 

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\mu_{J-T}=-\frac{1}{C_{P}}\left( \frac{\partial H}{\partial P} \right)_{T}=-\frac{1}{C_{P}}\left[ \left( \frac{\partial U}{\partial V} \right)_{T}\left( \frac{\partial V}{\partial P} \right)_{T} + P \left( \frac{\partial V}{\partial P} \right)_{T}+V\right]

 

=-\frac{1}{C_{P}}\left[ 0+ P\left( \frac{\partial V}{\partial P} \right)_{T}+V\right]

 

=-\frac{1}{C_{P}}\left[P \left( \frac{\partial \left[nRT/P \right] }{\partial P} \right)_{T} + V\right]=-\frac{1}{C_{P}}\left[ -\frac{nRT}{P}+V \right]=0

In this calculation, we have used the result that \left( \partial U/\partial V\right)_{T}=0 for an ideal gas.

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