Change of Variable

Sometimes it is more useful to be able to measure some quantity, such as

$$\input{equations/cp-def}$$ (22-4)

or

$$\input{equations/cv-def}$$ (22-5)

under different conditions than those indicated by their natural variables.

It would be easier to measure $C_V$ at constant $P$, $T$, so a change of variable would be useful.

To change variables, a useful scheme using Jacobians can be employed:¹

$$\input{equations/jacob-def}$$ (22-6)

$$\input{equations/jacob-rules}$$ (22-7)

To see where the last rule comes from:

For example,

$$\input{equations/cv-examp}$$ (22-8)

Using the Maxwell relation: $\ensuremath{ \left( \frac{\partial{S}}{\partial{P}} \right)... ...} = - \ensuremath{ \left( \frac{\partial{V}}{\partial{T}} \right)_{P} }$:

$$\input{equations/cv-examp-res}$$ (22-9)