When you think of the word “energy,” you probably think about something like the kinetic energy of a moving object, or maybe the potential energy something might possess due to gravity.

However, on the microscopic scale, the *internal energy* an object possesses is more important than these macroscopic forms of energy. This energy ultimately results from the motion of molecules, and it’s generally easier to understand and calculate if you consider a closed system that’s simplified, such as an ideal gas.

## What Is the Internal Energy of a System?

Internal energy is the total energy of a closed system of molecules, or the sum of the molecular kinetic energy and potential energy in a substance. The macroscopic kinetic and potential energies don’t matter for internal energy – if you move the whole closed system or change its gravitational potential energy, the internal energy remains the same.

As you would expect for a microscopic system, calculating the kinetic energy of the multitude of molecules and their potential energies would be a challenging – if not practically impossible – task. So in practice, the calculations for internal energy involve averages rather than the painstaking process of directly calculating it.

One particularly useful simplification is treating a gas as an “ideal gas,” which is assumed to have no intermolecular forces and hence essentially no potential energy. This makes the process of calculating the internal energy of the system much simpler, and it isn’t far from accurate for many gases.

Internal energy is sometimes called thermal energy, because temperature is essentially a measure of the internal energy of a system – it’s defined as the average kinetic energy of the molecules in the system.

## Internal Energy Equation

The internal energy equation is a state function, which means its value at a given time depends on the state of the system, not how it got there. For internal energy, the equation depends on the number of moles (or molecules) in the closed system and its temperature in Kelvins.

The internal energy of an ideal gas has one of the simplest equations:

Where *n* is the number of moles, *R* is the universal gas constant and *T* is the temperature of the system. The gas constant has the value *R* = 8.3145 J mol^{−}^{1} K^{−}^{1}, or around 8.3 joules per mole per Kelvin. This gives a value for *U* in joules, as you would expect for a value of energy, and it makes sense in that higher temperatures and more moles of the substance lead to a higher internal energy.

## The First Law of Thermodynamics

The first law of thermodynamics is one of the most useful equations when dealing with internal energy, and it states that the change in internal energy of a system equals the heat added to the system minus the work done by the system (or, *plus* the work done *on* the system). In symbols, this is:

This equation is really simple to work with provided you know (or can calculate) the heat transfer and work done. However, many situations simplify things even further. In an isothermal process, the temperature is constant, and since internal energy is a state function, you know the change in internal energy is zero. In an adiabatic process, there is no heat transfer between the system and its surroundings, so the value of *Q* is 0, and the equation becomes:

An isobaric process is one that occurs at a constant pressure, and this means that the work done is equal to the pressure multiplied by the change in volume: *W* = *P*∆*V*. Isochoric processes occur with a constant volume, and in these cases *W* = 0. This leaves the change in internal energy as equal to the heat added to the system:

Even if you can’t simplify the problem in one of these ways, for many processes, there is no work done or it can be easily calculated, so finding the amount of heat gained or lost is the main thing you’ll need to do.

#### References

- Georgia State University: HyperPhysics: Ideal Gas Law
- Georgia State University: HyperPhysics: Internal Energy
- Chemistry LibreTexts: Internal Energy
- Physics LibreTexts: Work, Heat, and Internal Energy
- Georgia State University: HyperPhysics: First Law of Thermodynamics
- Lumen: The First Law of Thermodynamics