Derivation of (22.12). Consider an arbitrary function of...

Derivation of (22.12). Consider an arbitrary function of state $f$ expressed as a function of $p$ and $T$ (for a closed system): $\mathrm{d} f=\left.\frac{\partial f}{\partial T}\right|_p \mathrm{~d} T+$ $\left.\frac{\partial f}{\partial p}\right|_T \mathrm{~d} p$. Divide this expression by $\mathrm{d} T$ under the constraint that both $T$ and $p$ change but $\mathrm{d} p / \mathrm{d} T$ is fixed. Hence derive equation (22.12).

Key Concepts

Chain Rule in Constrained Systems

The chain rule in multivariable calculus allows one to compute the derivative of a function along a specific path or constraint by accounting for how all independent variables change with respect to a chosen variable. In this context, dividing the total differential by the change in temperature under a fixed rate of change of pressure (dp/dT) provides a means to derive a relation that incorporates the constraints into the differential expression, leading to the desired equation.

Partial Derivatives

Partial derivatives measure the rate at which a function changes with respect to one variable while other variables are held constant. In the context of state functions, using partial derivatives allows one to isolate the influence of one variable, such as temperature, on the function, which is crucial for applying the chain rule when additional constraints on the variables are present.

Total Differential

The total differential of a function of several variables represents the change in the function in terms of the changes in its independent variables. For a function expressed in terms of pressure and temperature, the total differential is the sum of the partial derivatives with respect to each variable multiplied by the corresponding differential. This concept is vital for expressing infinitesimal changes in state functions in thermodynamic systems.

State Functions

A state function is a property of a system that depends only on the current state of the system and not on the path taken to reach that state. In thermodynamics, recognizing a variable as a state function allows one to apply differential calculus techniques effectively since its differential can be expressed solely in terms of independent variables like pressure and temperature.

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