Relations for Thermodynamic Properties in terms of Measurable Quantities

  • Relation for \(dU\) is obtained by considering \(U\) as a function of \(T\) and \(V\). \[dU = C_V dT + \left[ T\left(\frac{\partial P}{\partial T}\right)_V - P \right] dV\]

  • Relation for \(dH\) is obtained by considering \(H\) as a function of \(T\) and \(P\). \[dH = \left[ V- T\left(\frac{\partial V}{\partial T}\right)_P \right]dP + C_PdT\]

  • Relations for \(dS\) \[\begin{align*} \left(\frac{\partial S}{\partial T}\right)_V &= \frac{C_V}{T} \\ \left(\frac{\partial S}{\partial T}\right)_P &= \frac{C_P}{T} \end{align*}\] By considering \(S\) as a function of \(T\) and \(V\), it can be shown that, \[dS = \frac{C_V}{T}dT + \left(\frac{\partial P}{\partial T}\right)_V dV\] Similarly, by considering \(S = S(T,P)\) \[dS = \frac{C_P}{T}dT - \left(\frac{\partial V}{\partial T}\right)_P dP\]