📂Thermal Physics

What is a State Function in Thermophysics?

Definition¹

A state function or state variable is a property that has a fixed value independent of the path taken and can be measured macroscopically.

Explanation

Let’s explain this more mathematically. Consider a function $f(\mathbf{x})$ that has a value in three dimensions. When $\mathbf{x}$ changes from $\mathbf{x}_{1}=a$ to $\mathbf{x}=b$, if the difference in the value of $f$ is independent of the path, then $f$ is called a state function.

$$ \Delta f = \int _{a} ^{b} df = f(b) - f(a) = \text{constant} $$

Here, $df$ is the total differential of $f$. In other words, physical quantities expressed by total differentials are state functions. Examples of state functions include volume, pressure, temperature, internal energy, etc. Examples of non-state functions include the total work done on the system and the total heat flow into the system.

On the other hand, if it is not a total differential, it is called an inexact differential. Inexact differentials are denoted by $\delta f$ or $d\! \! \bar{}f$. For instance, consider $f = xy$ defined in two dimensions.

$$ df = \dfrac{\partial f}{\partial x} dx + \dfrac{\partial f}{\partial y} dy = y dx + x dy $$

Let’s think about just the first term.

$$ d \! \! \bar{} g = y dx $$

Then, since there is no term for $dy$, $dg$ is an inexact differential and is denoted as above by $d\!\!\bar{} g$. Hence, a physical quantity represented by $d\!\!\bar{} g$ has different values depending on the path, so it is not a state function.

$$ \Delta g = \int _{a} ^{b} d\!\!\bar{} g = \text{not constnat} $$

Also, equations expressed in terms of state functions are called equations of state. An example is the ideal gas equation.

$$ Pv = nRT $$