Thermodynamic Relations

Perfect Liquids/Solids:

$d u = C_{v} d T + \frac{\partial u}{\partial v} |_{T} d v ⟶ d u \approx d h \approx C d T$

d u = T d s - P d v ⟶ d s = \frac{d u}{T} + \frac{P}{T} d v ⟶ d s = C \frac{d T}{T} ⟶ Δ s = s_{2} - s_{1} = C \ln (\frac{T_{2}}{T_{1}})

d u = C_{v} d T + \frac{\partial u}{\partial v} |_{T} d v = C_{v} d T

d h = C_{p} d T + \frac{\partial h}{\partial P} |_{T} d P = C_{p} d T

d s = \frac{d u}{T} + \frac{P}{T} d v ⟶ d s = C_{v} \frac{d T}{T} + \frac{R}{W} \frac{d v}{v} ⟶ Δ s = C_{v} \ln (\frac{T_{2}}{T_{1}}) + \frac{R}{W} \ln (\frac{v_{2}}{v_{1}})

d s = \frac{d h}{T} - \frac{v}{T} d P ⟶ d s = C_{p} \frac{d T}{T} - \frac{R}{W} \frac{d P}{P} ⟶ Δ s = C_{p} \ln (\frac{T_{2}}{T_{1}}) - \frac{R}{W} \ln (\frac{P_{2}}{P_{1}})

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