Question 3.8: Evaluate، (∂H/∂P)T for an ideal gas.

Thermodynamics, Statistical Thermodynamics, & Kinetics

Evaluate, $\left({\partial}H/{\partial}P \right)_{T}$ for an ideal gas.

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$\left(\partial P/\partial T\right)_{V}= \left( \partial\left[ nRT/V \right] ⁄ \partial T\right)_{V} = nR/V$ and $\left(\partial V/\partial P\right)_{T}=\left( d\left[ nRT /P\right]/dp\right)_{T}=-nRT/P^{2}$ for an ideal gas. therefore,

\left( \frac{\partial H }{\partial P} \right)_{T}=T\left( \frac{\partial P }{\partial T} \right)_{V}\left( \frac{\partial V }{\partial P} \right)_{T}+ V=T\frac{nR}{V}\left( – \frac{nRT}{P^{2}}\right)+V=-\frac{nRT}{P}\frac{nRT}{nRT}+V=0

This result could have been derived directly from the definition H = U+PV. For an ideal gas, $U=U\left( T \right)$ only and $PV=$ . Therefore, $H=H\left( T \right)$ for an ideal gas and $\left(\partial H/\partial P\right)_{T}=0$

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