Internal Energy

Definition of internal energy

It is important to understand that this internal energy does not have a direct relation with the external energy of the system. This means that, if the system is moving and has kinetic energy, the internal energy of the system does not include this energy that results from the overall movement of the system. Similarly, if we put the whole system at a height ofabove the ground, this macroscopic potential energy does not affect the internal energy of the system. A system can be completely still and have no apparent energy while its internal energy is changing, but on the other hand, a system can be moving while its internal energy is constant.

In general, a change in the internal energy of a system causes either a temperature change or a change of state.

For real systems, we are normally interested in (and thus calculate) the variation of internal energy during a process, such as an increase in temperature.

Internal energy in thermodynamics

In physics, energy is transferred because of changes in temperature, applying forces, etc. The branch that studies this is thermodynamics.

Now, imagine any system you want (and it does not have to be a football stadium this time). Remember that a system in thermodynamics is any part of the universe we want to study, so it can be a human body, a certain amount of a liquid, a plant, or anything else you can think of.

Fig. 1: A thermodynamic system with its boundary, adapted from the image.

The particles with microscopic energies are found inside the system, and the sum of all of these microscopic energies is what we call internal energy.

This leads us to study what happens to the internal energy when some energy is transferred to the system. In our case, we are going to focus on what happens when the temperature is increased. To do so, there needs to be a transfer of energy into the system, so the system either needs to be heated or work must be done on the system.

Heat transfer causes a change in the internal energy of a system. Similarly, applying work to the system increases the system's internal energy.

Fig. 2: Heat transfer between two systems at different temperatures.

A change in the internal energy of a system can either change the potential energy of the particles or the kinetic energy of the particles. If the potential energy is changed, we speak of a change of state. If the kinetic energy is changed, we speak of a temperature change.

Changes in the internal energy

As stated before, a change in the internal energy of a system either causes a temperature change or a change of state. We will look at temperature changes in the next section and focus on changes of state here.

As you may know, we normally distinguish between three states of matter: gas, liquid and solid. If the temperature of a system increases or decreases to a certain point, which depends on the substance we are working with, there could be a change from one state to another. During this change of state, the temperature remains constant, but there is still a change in the internal energy of the system.

First, the internal energy of the system can increase, as the result of the application of some heat or work. These are the three different changes of state regarding increases in internal energy:

• A solid will melt, obtaining a liquid.
• A liquid will evaporate, transforming it into a gas.
• If we have a solid and it turns directly into a gas when increasing the internal energy, we talk about sublimation.

Otherwise, we can decrease the internal energy of a substance when the system starts giving off heat to the outside or does work on its environment:

• A gas will condensate, obtaining a liquid.
• A liquid will freeze, transforming it into a solid.
• If the substance goes from gas to solid without passing through its liquid state, we talk about deposition.

Fig. 3: Increasing and decreasing the temperature we can change the state of matter.

You can learn more about the Changes of State here in StudySmarter.

Equation of the internal energy change

In most cases, a change in internal energy will provoke a change in temperature. In this case, only the total kinetic energy of the particles varies, while the total potential energy stays the same.

In short, the thermal energy can be thought of as the kinetic part of the internal energy. When no change of state occurs during a process, the change in the internal energy is the same as the change in the thermal energy of the system.

The equation relating the change in the thermal energy and the change in temperature of a system is

\[\text{change in thermal energy}=\text{mass}\cdot \text{specific heat capacity}\cdot \text{temperature change}\]

In symbols, this equation becomes

\[\Delta E=mc\Delta \theta\]

where

• \(\Delta E\)is the change in thermal energy of a system. The standard unit is the joule \(\mathrm{J}\).
• \(m\) is the mass of the system. The standard unit is the kilogram (\(\mathrm{kg}\)}.
• \(c\) is called the specific heat capacity. It is defined as the amount of energy needed to increase the temperature of a unit of mass of a certain substance by one unit of temperature. Every substance has its constant specific heat capacity: it is a characteristic, much like density and colour. The standard unit is joule/(kilogram x kelvin), (\(\mathrm{J}/(\mathrm{kg \cdot K})\)). It is always positive.
• \(\Delta \theta\) is the change in temperature of the system. If the final temperature is smaller than the initial temperature, the value will be negative. The standard unit is the kelvin, (\(K\))

As you can see, if the mass of a substance does not change during a process (therefore having a constant value), the temperature of the system would increase if we increased its thermal energy. Given a certain energy input, the temperature change depends on the mass of the system and the specific heat capacity of the material the system is made of. For two systems with two different substances with the same mass and equally modifying the thermal energy of both systems, the variation of the temperature will be different. This is because the two substances will have different values for their specific heat capacity.

Internal energy examples

Now that we understand what thermal energy and internal energy are, let's do some calculations relating thermal energy changes to temperature changes. If there are no changes of state, the thermal energy change is equal to the internal energy change.