Study Material

# Internal energy

We explain what internal energy is, its function, how to calculate it and we give several examples.

## What is internal energy?

The internal energy of an object comes from the random movement of the atoms and molecules that compose it. Even when the object is perfectly at rest, inside it, at the microscopic level, the particles that compose it move continuously without rest.

Since these particles are in motion, they have kinetic energy, which is part of their internal energy. The other contribution to internal energy is the potential energy that comes from intermolecular forces, responsible for maintaining the cohesion of the substance.

These intermolecular forces are generally electromagnetic in origin, but gravitational and nuclear forces are also involved.

Now, depending on the type of movement of the particles, the kinetic energy can be:

• Translational
• Vibrational
• Rotational

Temperature measures only the translational kinetic energy of the particles. In principle, the higher the temperature, the higher the internal energy, but this also depends on the mass. An example clarifies the issue: a glass of warm water has less internal energy than a lake at room temperature, even though the temperature of the glass is higher, and that is because there is more water in the lake than in the glass.

In any case, the internal energy of a substance is not perceptible to the eye, it does not depend on its movement, nor does it depend on its relative position with respect to other objects. For example, the internal energy of a glass of water at room temperature is the same whether the glass is on a table or if it is on the floor.

The internal energy, which is usually denoted as U, is a function that depends on the thermodynamic variables of the system, such as the temperature T and the volume V. Therefore, it can be expressed mathematically as:

U = U (T, V)

Actually, the absolute value of the internal energy of a system is not relevant, what matters is its variation, which is denoted by the Greek letter Δ (“delta”):

ΔU = initial U – final U

However, internal energy can be varied by adding or removing heat to the system. By adding heat it is possible for the system to do a certain job, such as a piston in an engine cylinder.

### The first law of thermodynamics From the first law of thermodynamics it is established that the variation of the internal energy of a system is equivalent to:

ΔU = Q – W

Where Q is the amount of heat that is transferred to the system and W is the work the system does, if any. Everything is measured in joules in the International System of Units.

The following can be deduced from the previous expression:

• ΔU> 0 means that the system increases its internal energy
• ΔU <0 the internal energy of the system decreases
• Q> 0 the system absorbs heat
• W> 0 the system does work.

## How to calculate internal energy?

### The Monatomic Ideal Gas

In the monatomic (single atom) ideal gas model, the particles do not interact with each other, so the translational kinetic energy is the only contribution to the internal energy U of the gas. There is a theorem, called the equipartition theorem , which states that each degree of freedom has an energy equal to:

Where T is the temperature and R is the ideal gas constant, whose value in International System units is:

8.314472 J / mol ∙ K

A degree of freedom represents the possibility of moving along a certain direction in space. An ideal gas particle can move in all three directions of space, therefore the energy of n moles of gas is:

### Heat quantity

Assuming that heat is added to a system and it does no work, this heat is used only to increase the internal energy of the system. The quantity of heat Q must be proportional to the mass m of the system and to the change in temperature:

Q ∝ m⋅ΔT

The constant of proportionality depends on the substance and is called specific heat . Calling this constant c, the heat is:

Q = mc⋅ΔT

How heat is invested in increasing internal energy:

Q = mc⋅ΔT = ΔU

ΔU corresponds to the variation in internal energy.

## Examples of internal energy

### 1. Temperature variations

Changes in the temperature of substances in turn generate changes in their internal energy. This is what happens, for example, when cooking. By heating food, the agitation of its molecules increases, the kinetic energy increases and therefore its internal energy.

### 2. Internal energy of two substances at the same temperature

Two substances at the same temperature do not necessarily have the same internal energy, since it depends on two contributions: kinetic energy and potential energy.

If the substances are at the same temperature, it can be stated that the translational kinetic energy of their particles is the same, but the potential energy is different, since it depends on the configuration of the constituent atoms.

### 3. Thermal expansion Substances expand when heated, since the increase in internal energy causes their particles to vibrate with greater amplitude, so the dimensions increase

A visible consequence of heating substances is the increase in their dimensions, since the average distance between their atoms increases.

The intermolecular forces can be simulated through springs that join the atoms, an increase in temperature increases the amplitude of the vibration, consequently, the separation between atoms is greater, resulting in the expansion of the object.

### 4. Compressed gases

Compressed gases can store internal energy, since their density increases with decreasing volume and thus their interactions with the walls of the container container. In this way, a compressed gas is capable of doing work such as moving a piston.

### 5. Batteries

Chemical reactions occur in a battery capable of generating an electric current as soon as the circuit is closed. This chemical potential energy is considered as part of the internal energy of the system.

### 6. System that absorbs heat and does work

When a system absorbs heat, its internal energy increases. But if you do work at the same time, you can use the first law of thermodynamics to find out by how much the internal energy changes. For example, assuming that a system absorbs 175 J of heat and does work equivalent to 62 J, the change in its internal energy is:

ΔU = Q – W = 175 J – 62 J = 113 J.