Measurable Quantities

The following are the measurable quantities: \(P,V,T,C_P, C_V, \beta, \kappa\).

  • Specific heats \[\begin{align*} C_V &= \left(\frac{\partial Q}{\partial T}\right)_V \\ C_P &= \left(\frac{\partial H}{\partial T}\right)_P \end{align*}\]

  • Definitions of Volume Expansivity (\(\beta\)) and Isothermal Compressibility (\(\kappa\)):
    Let us consider \(V\) as a function of \(T\) and \(P\). i.e., \[V = V(T,P)\] Differentiating, \[dV = \left(\frac{\partial V}{\partial T}\right)_P dT + \left(\frac{\partial V}{\partial P}\right)_TdP\] i.e., \[\frac{dV}{V} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P dT + \frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T dP\] Using \(\beta\) and \(\kappa\), we have \[\frac{dV}{V} = \beta dT - \kappa dP\] where

    \[\beta =\text{Volume expansivity} = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P\] and \[\kappa = \text{Isothermal compressibility} = -\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T\] As with increase in \(P\), \(V\) decreases, \((\partial V/\partial P)_T\) is negative. Hence to have positive value for \(\kappa\), there is a negative sign added in the definition.

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