If x, y are functions of state, give an argument to prove...

If $x, y$ are functions of state, give an argument to prove that $$ \frac{\mathrm{?} Q_{(\mathrm{p})}}{\mathrm{d} x}=\left.\frac{\mathrm{?} Q_{(\mathrm{p})}}{\mathrm{d} y} \frac{\partial y}{\partial x}\right|_{(\mathrm{p})}, $$ where the subscript (p) indicates the path along which the changes are evaluated.

If $x, y$ are functions of state, give an argument to prove that
$$
\frac{\mathrm{?} Q_{(\mathrm{p})}}{\mathrm{d} x}=\left.\frac{\mathrm{?} Q_{(\mathrm{p})}}{\mathrm{d} y} \frac{\partial y}{\partial x}\right|_{(\mathrm{p})},
$$
where the subscript (p) indicates the path along which the changes are evaluated.

Key Concepts

Chain Rule

The chain rule is a fundamental principle in calculus which states that if a quantity Q depends on an intermediate variable y, which in turn depends on another variable x, then the derivative of Q with respect to x can be found by multiplying the derivative of Q with respect to y by the derivative of y with respect to x. This argument applies regardless of whether Q is a state function or a path function, as long as the dependencies are differentiable.

State Functions versus Path Functions

In thermodynamics, state functions are properties that depend solely on the current state of the system, while path functions depend on the specific route or process taken between states. Even though Q (heat) is generally a path function, when state variables like x and y are used to parameterize the process, local derivatives of Q along that path can be related via the chain rule, linking the state variable changes to the path-dependent changes in Q.

Parameterization of Thermodynamic Paths

The process of describing a path by expressing one state variable as a function of another (for example, y as a function of x) allows the changes in path-dependent quantities like Q to be analyzed using calculus. This parameterization makes it possible to express the derivative of a path function with respect to one state variable as the product of its derivative with respect to another state variable and the derivative of the latter with respect to the first.

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