In today's lecture we introduced our last mathematical identity for partial derivatives, the cycle rule (also called the triple product rule and the cyclic chain rule). It will be employed in several contexts this quarter, but one useful application is expressing a single partial derivative in terms of the product of two others. Surprising relationships often arise when using this rule (which is valid for any three arbitrary thermodynamic state functions). In some mathematical derivations, it is especially helpful for getting rid of a variable that is difficult to hold constant, such as H or U.
Several thermodynamic partial derivatives go by special names (we have already met the heat capacities CP and CV). Today we discussed the thermal expansion coefficient, the isothermal compressibility and the internal pressure (all, in principle, functions rather than values).
Given an equation of state, we could analytically find functions for these parameters simply by calculating the partial derivative. On the other hand, since every thermodynamic partial derivative implies an experiment, we could also uncover these relationships in the lab when an equation of state is not available (which is most systems except gases). Tables 3.1 and 3.2 show some experimental results at standard conditions for various solids and liquids. (Note that Table 3.1 is mislabeled as isothermal coefficient rather than thermal expansion coefficient).
To further clarify what the internal pressure parameter is actually measuring, we calculated it for both an ideal gas and a van der Waals gas. As expected, since the internal energy U of an ideal gas is a function of T only, its internal pressure is zero. Another way of looking at it: For a gas' internal energy to be altered by changing the volume, that gas would necessarily possess intermolecular forces between its particles. The vdW gas gave a nonzero answer, the correction term for the pressure in the van der Waals equation!
Now that the Box of Mathemagics has been assembled, onward to more relations! Friday will see us relating CP and CV for any arbitrary substance and hopefully mine won't be the only tears of mathematical joy being shed...