# 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

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

and

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

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:

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

and

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

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.