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4.1: Prelude to Putting the First Law to Work

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    As has been seen in previous chapters, may important thermochemical quantities can be expressed in terms of partial derivatives. Two important examples are the molar heat capacities \(C_p\) and \(C_V\) which can be expressed as

    \[ C_p = \left(\dfrac{\partial H}{\partial T}\right)_p\]


    \[ C_V = \left(\dfrac{\partial U}{\partial T}\right)_V\]

    These are properties that can be measured experimentally and tabulated for many substances. These quantities can be used to calculate changes in quantities since they represent the slope of a surface (\(H\) or \(U\)) in the direction of the specified path (constant \(p\) or \(V\)). This allows us to use the following kinds of relationships:

    \[ dH = \left(\dfrac{\partial H}{\partial T}\right)_p dT\]


    \[ \Delta H = \int \left(\dfrac{\partial H}{\partial T}\right)_p dT\]

    Because thermodynamics is kind enough to deal in a number of state variables, the functions that define how those variable change must behave according to some very well determined mathematics. This is the true power of thermodynamics!

    4.1: Prelude to Putting the First Law to Work is shared under a not declared license and was authored, remixed, and/or curated by Patrick Fleming.

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