State Functions — Definition & Examples

A function of state, or state function, relates several state variables together. In essence, a state function is a system where only the start and end points matter. If you are a more math oriented person, you've probably already learned a bit about state functions. State functions are exact differential equations, and crop up a lot with conservative fields and simply connected spaces.

If you're not a math oriented person, here is a physical example to help give you an idea. Imagine we have a block that currently has a gravitational potential energy of 0 joules. If we wanted to put it in a position where it has an energy of 10 joules, it wouldn't make a difference whether we carried it up the stairs or took the elevator. The energy is only concerned with the start position, and the end position. Take note, however, that the work put in to moving the block is different depending on the path. This is because work is an inexact differential, and can be viewed as the total amount of energy transferred. In general, inexact differentials are path dependent

In our study of thermodynamics and thermochemistry, we are primarily interested in 4 state functions: internal energy, $U$, entropy, $S$, gibbs free energy, $G$, and enthalpy, $H$. The mathematical definitions are listed below. $$dS=\frac{1}{T}dU+\frac{P}{T}dV-\frac{\mu}{T}dN$$ $$dU=TdS-PdV+\mu dN$$ $$dH=TdS+VdP+\mu dN$$ $$dG=-SdT+VdP+\mu dN$$ V is the volume, T is the temperature, P is the pressure, $\mu$ is the chemical potential, and N is the number of molecules. In the majority of cases we will only be considering closed systems. In a closed system there is no exchange of particles, so the $\mu dN$ terms are always $0$.

State Functions: What are state functions?

Be Aware: In thermochemistry, variables such as internal energy, entropy, Gibbs free energy, and enthalpy are path-independent.

(video) State Functions

By David Vanden Bout

In this video, Dr. McCord explains the definition of state functions. In addition, Dr. McCord uses the state function variable, temperature, to demonstrate how the state functions path-independence characteristics. Path-dependent variables such as heat and works are also compared with state function variables.

In our last lesson, we defined the system and the surroundings. In chemistry, our system is typically our chemical reaction. So, it includes every atom, molecule, reactant, product, transition state, intermediate, and solvent. We've seen that our system and surroundings can transfer energy. For example, let's say one of our products was a gas. As the reaction generates the gas, it lifts a piston. We would say the system did work on the surroundings. If the piston compressed the gas, the surroundings did work on the system. The other type of energy transfer was heat. It was the exchange of kinetic energy caused by molecular collisions.

In each case, one of the system's physical properties changes. In the piston examples, the volume changed. When energy gets exchanged via heat, we see a temperatures change. The physical properties inform us about the molecular level. In this lesson, we'll seek to understand why.

State Functions

As we've studied chemistry, we've seen a lot of equations. Often, we learn them in class, in our textbook, or from an online explanation. Then, we immediately use them to solve practice problems. But we don't think about what those equations are tellings us. Almost all the equations describe a change in our system's state. A great example is the gas laws. I've written them out so you can see them.

Boyle's Law: $P_{1}V_{1}=P_{2}V_{2}$

Charles' Law: $\frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}$

Avagadro's Law: $\frac{V_{1}}{n_{1}}=\frac{V_{2}}{n_{2}}$

Gay-Lussac's Law: $\frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}$

Combined Gas Law: $\frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}}$

Ideal gas Law: $PV=nRT$

For the gas laws, our system was a gas in a container. When we changed one of the variables, another variable got altered. The result was that we maintained equilibrium. Both sides of the equation were equal. But, the initial and final states were different. So, we call these types of equations state functions. A state function is an equation that describes a thermodynamic system. But, it only depends on the system's current conditions (the current state). State functions are composed of state variables. A state variable is a measurable component of the system. The state variables also only depend on the current conditions.

Let's explore state variables in more detail. We know that state variables only depend on the current conditions. But what's the implication? The change in a state variable is always the difference between the final and initial states. So, we write it mathematically as $\Delta X= X_{final}-X_{initial}$. For example, let's say we heated neon gas from $5^{o}C$ to $10^{o}C$. Our change in temperature is always $5^{o}C$! It does not matter if we heat it slowly or rapidly. We could heat it to $100^{o}C$ and then cool it to $10^{o}C$. The difference between our initial and final temperatures is still $5^{o}C$.

So, what's the point? Again, these variables only depend on the initial and final states. The fancy term is that they are path-independent. We don't care how we ended up there. Below, I've listed many of the state variables. It might be helpful to think through why each of these is path-independent. Further down, we'll explore path-dependent functions.

State Variables You'll Encounter in Thermodynamics

• Mass

• Moles

• Pressure

• Volume

• Temperature

• Density

• Energy

• Internal Energy

• Gibbs Free Energy

• Enthalpy

• Entropy

Path Functions

Let's consider a scenario where the path matters. I live on the corner of my block. Let's say I wanted to go to the end of my street. I have two options. I could walk straight down it, or I could walk around the block. In my case, that would mean I would walk the other three sides of a square. Mathematically, my first option would be $A \rightarrow D$. My other option is $A\rightarrow B + B\rightarrow C + C\rightarrow D$.

At first, it may seem like my scenario is path independent. And based on the final location, it is. I end up at the end of my block either way. But, let's consider my energy expenditure. I would burn far more calories walking around the block. Why? Because I'm walking a greater distance. So, energy expenditure is path-dependent.

Where have we seen energy movement in thermodynamics? We have two concepts: work and heat. Both of these are path functions. Path functions depend on how we get to our final state. But isn't pressure-volume work composed of two state variables? Yes, pressure and volume are both state variables. But remember our definition of work. Work is a force applied over a length. So, by definition, work depends on the path (length). Heat is a little less intuitive. But we know that heat is a transfer of energy. So, we can't specify the amount of heat at a specific moment or set of conditions. We can only define the heat for a process. Consider a chemical reaction. We can't stop it halfway and determine the heat left. We can only find the heat released or absorbed at the end. As a side note, that's how we know if a reaction is endothermic or exothermic.

Why does it Matter?

At this point, are you wondering why any of this matters? It turns out we can manipulate path and state functions. How? By keeping some parameters constant. For example, later, we'll study calorimetry. A bomb calorimeter keeps the system at a constant volume. So, there is no pressure-volume work. In that case, the change in internal energy only equals heat. What does that mean? We created a scenario where heat is a state function. We can also turn heat into a state function at constant pressure. We'll study that when we investigate enthalpy. Finally, we'll also study Hess's law. It manipulates path independence to calculate enthalpy from known quantities.