What is a state function?

What is a state function? I've been briefly introduced to the idea in an introductory module, and I'm confused by the whole idea, I understand the 'path dependence' from line integrals, but is it about the arguments that the function itself takes, I.E. it cannot be simply expressed in terms of the state variables, or is it about the form of the function, because they should generally be linked by their time dependence?

State function (or a function of state) is independent on the path, by which one arrives to this state. In other words, as its name says, it depends only on the state itself (more precisely on the state variables). Thus, e.g., if the state of a thermodynamic system is fully described by pressure and volume, any function that depends only on pressure and volume (such as the internal energy of an ideal gas) is a state function. On the other hand, heat or work are not state functions, since we can arrive to the same state by different paths, i.e., via transferring different amounts of heat and work.

• Would heat or work depend on different variables other than the state variables? Commented May 2, 2022 at 11:43
• @user1007028 more precisely, heat and work would depend on how exactly the state variables change between, say $p_1, V_1$ and $p_2,V_2$, i.e., they will depend on the path. Whereas the change of the state function would be simply $U(p_2,V_2)-U(p_1,V_1)$. Path can be considered a function of some external parameters. I guess my statement is imprecise, I will rewrite it. Commented May 2, 2022 at 11:48
• So if a quantity N is a state function we can know the value of N for a particular state of the system, and it doesn't matter how you get there, a certain 'state' has a certain corresponding values of the state variables? So for non-state functions of time how will this change the function we can represent it with? Commented May 2, 2022 at 12:38
• A state is defined by the values of the corresponding state variables. There is no such a thing as non-state functions: there are state functions and everything that is not state functions. Of course, all of them can be functions of something in purely mathematical sense. Commented May 2, 2022 at 12:49
• Commented May 2, 2022 at 12:51

In my view, your confusion is understandable.

On the one hand, Wikipedia uses the term "state function" to describe a mathematical function (e.g., ideal gas equation) that describes the relationship between "state variables" (pressure, volume, temperature, mass, etc.) of a system in equilibrium. This is my preference.

Then the article goes on to call the state variables themselves "state quantities" or "state functions", tending to blur the difference between the function that relates the properties of a system at equilibrium with the variables (properties) themselves.

But when it does the latter it explicitly cites the examples of internal energy, enthalpy, and entropy, seemingly to draw a distinction between these properties of a system and other properties such as pressure, volume, temperature, etc.. And yet they are all system properties.

Rather than getting hung up on the inconsistencies, the important thing to remember is the change in properties (internal energy, enthalpy, entropy, temperature, pressure, volume, etc.) of a system between equilibrium states, and the mathematical function that relates the properties, is independent of the path between the states.

Hope this helps.