# Internal Energy

Today is Saturday and you have tickets to go to the football stadium to watch your favourite team's match (you can change football for any sport, do not worry). But you have been reading a lot of StudySmarter articles and you are a full-time physicist now. So, once you sit down with your drink and your snacks, you look at your favourite player and you just imagine him as a particle. And, when you look away, the whole football pitch is no longer a pitch but a space full of physical particles. You want to enjoy the match, of course, but first, a question pops into your mind and you need to answer it. Does a system of actual particles have total kinetic energy, similar as the players on the pitch do?

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Contents

Remember that a system in physics is any part of the universe we would like to study.

The answer to that question is yes. Just like the players on the football pitch, the particles in a system move around. The particles inside this system have certain kinetic energy because of the temperature of the system: particles generally move faster when the temperature of the system is higher. Besides this, particles can also have potential energy, for example, due to the mutual attraction between particles (e.g. if they are electric dipoles).

## Definition of internal energy

The internal energy of a system is the energy found inside. It is the sum of all the microscopic kinetic and potential energies of the particles in the system if the system would be at rest and not in a macroscopic energy potential.

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.

If we heat water, the macroscopic energy of the system does not seem to increase as the water is not moving. However, we know something is happening because the temperature of the water is increasing. As the water temperature increases, the water particles start moving faster and faster, causing their total kinetic energy to increase. Thus, the internal energy of the water increases as the water gets hotter. Meanwhile, the external kinetic energy remains zero.

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

Internal energy is an extensive property: a property of a system that depends on how the system is regarding its size or mass. Its value can be described as the sum of the values of smaller subdivisions of the system.

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.

Thermodynamics is the branch of physics that studies the relationship between heat, work, and other transfers of energy.

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 is the energy transferred to or from a system through a difference in temperature with the environment.

Heat added to or subtracted from a system should not be confused with the temperature of a 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.

The temperature of a system is a measure of the total kinetic energy in the system. When heat flows into a system and no change of state occurs, the internal energy increases and thus the total kinetic energy increases as well. This means that the temperature increases.

## 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.

## 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.

The thermal energy of a system is the sum of all the microscopic kinetic energies of the particles in the system if the system would be at rest.

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.

The internal energy of a system can also be changed by work.

In thermodynamics, we normally talk about expansion and compression. When the volume of a system increases we talk about an expansion and when it decreases we talk about a compression.

Fig. 4: Examples of a) expansion and b) compression of a system.

Doing work on a system will compress a system. How much work is required to compress a system by a certain volume is dictated by the pressure of the system according to the following formula:

$W=-p\Delta V,$

where

• $$W$$ is the work done on the system.
• $$p$$ is the pressure of the system. The standard unit of pressure is the pascal ($$\mathrm{Pa}$$).
• $$\Delta V$$ is the difference in the volume of the system caused by the work being done. This difference is negative if the system is compressed. The standard unit is the cubic metre ($$\mathrm{m}^3$$).

If we do work on the system, we see from the formula that the difference in volume is negative, so we indeed have a compression. Likewise, if the system does work on its environment, the system will expand.

## 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.

Question

Imagine you have a mass of $$m=2\,\,\mathrm{kg}$$ of water. If the temperature of this mass of water is increased from $$20^{\circ}$$C to $$60^{\circ}$$C, how much thermal energy was added to the water? The specific heat capacity of water is $$4182\,\,\mathrm{J/kg\cdot K}$$.

Solution

We just need to apply the equation of the change in the thermal energy:

$\Delta E=mc\Delta \theta$

We note that the temperature difference is $$40\,\,\mathrm{K}$$. If we substitute the given values into the equation, we get the following result:

$\Delta E=2\,\,\mathrm{kg}\cdot 4182\,\,\mathrm{\frac{J}{kg\cdot K}}\cdot 40\,\,\mathrm{K}=3,3\cdot 10^5 \,\,\mathrm{J}$

We conclude that $$3,3\cdot 10^5\,\,\mathrm{J}$$ of thermal energy was added to the water in order for its temperature to be increased as stated in the question. We do not know how this energy was added! It could have been by heat transfer or by work.

Question

Imagine we have $$m=0,5\,\,\mathrm{kg}$$ of a substance, and we want to figure out what material it is. We decide that we can measure its specific heat capacity and then look up what material has that specific heat capacity. We heat the material, increasing the internal energy by $$2500\,\,\mathrm{J}$$. Likewise, we see no change of state happen, and we measure a temperature change of $$10^{\circ}$$C. What is the specific heat capacity of this material? Which material do we have?

Solution

There was no change of state, so the change in internal energy is the change in thermal energy. Again, we need to use the equation of the change in the thermal energy, but this time we need to isolate the specific heat capacity, as follows:

$\Delta E=mc \Delta \theta \ \rightarrow \ c=\dfrac{\Delta E}{m \Delta \theta}$

Now, we can substitute the values into the equation:

$c=\dfrac{2250\,\,\mathrm{J}}{0,5\,\,\mathrm{kg}\cdot 10\,\,\mathrm{K}}=450\,\,\mathrm{\frac{J}{kg \cdot K}}$

The specific heat capacity is $$c=450\,\, \mathrm{J/kg \cdot K}$$. If we look at a table of specific heat capacity, we will find that iron has this specific heat capacity, so we most likely have iron.

## Internal Energy - Key takeaways

• Thermodynamics is the branch of physics that studies the relationship between heat, work, and other transfers of energy.
• A thermodynamic system is a bounded part of the universe.
• The internal energy of a system is the sum of all the microscopic kinetic and potential energies of the particles in the system.
• Increasing the internal energy of a system will either cause an increase in temperature or a change of state.
• The thermal energy of a system is the sum of all the microscopic kinetic energies of the particles in the system.
• Increasing the thermal energy of a system will make the particles of the system move faster (as their kinetic energy increases), which causes an increase in temperature.
• The change in temperature of a system as a result of a thermal energy change can be computed with the following formula: $$\Delta E=mc \Delta \theta$$.
• The specific heat capacity of a substance is defined as the amount of energy that is needed to increase the temperature of one unit mass of the substance by one unit of temperature.

## References

1. Fig. 2- Heat Flow (https://commons.wikimedia.org/wiki/File:Heat_flow_hot_to_cold.png) by BlyumJ is licensed by CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)
2. Fig. 3- States of matter (https://commons.wikimedia.org/wiki/File:States-of-matter-template.svg) by Enoshd is licensed by CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)
3. Fig. 4- Thermal expansion of a volume (https://commons.wikimedia.org/wiki/File:Thermal-expansion-volume.svg) by MikeRun is licensed by CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)

#### Flashcards in Internal Energy 28

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What type of energy is internal energy?

It is the sum of potential energy and kinetic energy within a system.

What is internal energy?

The internal energy of a system is the sum of all the microscopic kinetic and potential energies of the particles in the system. We can say it is the energy inside the system.

What is the change in internal energy?

A change of internal energy happens when the internal energy of a system increases or decreases when there is a heat transfer between the system and its surroundings and/or work is done between on or by the system.

How to calculate the internal energy of a gas?

When there is a change in the temperature of a gas, we can compute the change in internal energy as follows:

∆E = m c T.

The left-hand side ∆E is the change in internal energy. The first term of the right-hand side m is the mass of the gas, c is the specific heat capacity of the gas, and ∆T is the change in temperature.

What is the internal energy formula?

The formula to compute the change in internal energy when there is a temperature change is:

E = m c T.

The left-hand side ∆E is the change in internal energy. The first term of the right-hand side m is the mass of the substance, c is the specific heat capacity of the substance, and ∆T is the change in temperature.

## Test your knowledge with multiple choice flashcards

The temperature and the internal energy of the system do not change during a change of state. True or false?

The internal energy of a system is the sum of all the microscopic energies of the particles inside it.

If we have two different systems of two different substances with the same mass, is there an equal temperature change if we change the thermal energy by the same amount in both systems?

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