# Thermal energy vs. internal energy?

I am tremendously confused about the exact difference between thermal energy and internal energy and this confusion is largely due to the fact that even some of the most credible sources seem to contradict each other as illustrated here:

My copy of "Resnick Halliday" states the following:

"Thermodynamics is the study and application of thermal energy(often called the internal energy) of systems".

"Thermal energy would ideally be the amount of heat required to warm the metal to its temperature, but this quantity is not well-defined, as there are many ways to obtain a given body at a given temperature, and each of them may require a different amount of total heat input. Thermal energy, unlike internal energy, is therefore not a state function."

(I am also confused about the fact that thermal energy is a path function; so, it is not defined for equilibrium states? If it is not possible to answer this question as well, I will post it separately)

Most people seem to think that the inherent confusion leading to this doubt is a confusion between the terms "heat" and "internal energy" (Eg: this question) but no, I fully understand that heat is not something that a system "holds" but is actually energy in transit and that it is NOT defined for an equilibrium state just as work is not defined for the same.

I know that a molecule can possess many types of energy (vibrational, translational, rotational etc.) and that the sum of these energies is defined as the internal energy of the system. Where then, does thermal energy come into the picture? Is it the same as internal energy since the internal energy of a system is technically zero at absolute $0K$? Some websites loosely state that it is the "jiggliness" of the atoms/molecules that constitute the thermal energy of the system. What are they trying to say?!

As you can see, I am in dire need of help. Please share your insights and also, keep in mind that I have only just graduated high school.

Edit: In another page, the same book defines thermal energy as follows;

"It is an internal energy that consists of the kinetic and potential energies associated with the random motions of the atoms, molecules and other microscopic bodies within an object."

So, it is not only the K.E but also the P.E associated with the moving constituents. Other types of energy within the system are those from chemical bonds etc, correct? Please correct me if I am wrong.

Much thanks in advance :) Regards.

• Are there different symbols for thermal and internal energy, in the equations in which they appear? Internal energy $U$ appears in first law. Which equation have you seen thermal energy term appear in?
– Deep
Sep 7 '16 at 5:25
• I haven't. I simply wanted to know the difference.
– user106570
Sep 7 '16 at 5:51
• Well then do what I do: stick to equations, know the names of variables appearing in those equations, and thus avoid getting lost in terminology. This is especially true of Thermodynamics.
– Deep
Sep 7 '16 at 5:57
• I have edited my question. Do check.
– user106570
Sep 7 '16 at 6:01

I am tremendously confused about the exact difference between thermal energy and internal energy

Internal energy $E$ is all contained energy. And there are many ways to contain/store energy - one of them is thermally:

• In chemical bonds, when atoms join and form molecules
• In different phases of matter - ice at $0^\circ$ contains less internal energy than water at at $0^\circ$, because some energy has broken then bonds. Here a phase transition energy also called latent heat is the amount of energy needed to melt or freeze to do this phase change.
• As kinetic energy when particles move within a system
• As potential energies within the system when an object / objects is out of equilibrium (a book on a book-shelf wants to fall, or a compressed spring wants to jump out)
• As thermal energy, which is giving objects their temperature.
• Etc.

So no, internal energy and thermal energy is not at all the same. Internal energy is everything and includes thermal energy.

BUT when talking about ideal gases, their are no chemical interaction, no complex phase structure (because of no interaction), (usually) no significant potential energy. And the kinetic energy of each of the atoms is just the same as thermal energy or temperature. So here the inernal energy equals thermal energy - but only for an ideal gas.

Some websites loosely state that it is the "jiggliness" of the atoms/molecules that constitute the thermal energy of the system. What are they trying to say?!

Let's make this a bit clearer. Thermal energy is what we measure as temperature. So on the macroscale, if is felt as warmth/coldness.

But on the microscale, thermal energy is simply vibrating atoms. In a solid, they vibrate at their spot and more and more violently, as they are heated up. Soon they are heated so much that they vibrate so much that they rip themselves loose from the structure - and the materials is now melting.

This is why the word "jiggliness" is used. Thermal energy and temperature is in fact just microscale kinetic energy of the particles.

• Since "thermal energy is what we measure as temperature", does this mean that it is related to the kinetic energy of the system, since K.E depends only on temperature? And exactly what IS the kinetic energy of molecules for an atom of say, a solid?
– user106570
Sep 6 '16 at 13:30
• @KaumudiHarikumar Uh, that is a tough question, but yes, temperature is in fact just a name for microscale kinetic motion. It is not K.E. that depends on temperture, but rather temperature that is the macroscale way of talking about the phenomenon of microscale kinetic energy giving the feeling of heat. Sep 6 '16 at 13:44
• Kinetic energy of a single atom is found in a more "averaged" fashion. I googled and found this site that says a bit about that: hyperphysics.phy-astr.gsu.edu/hbase/kinetic/molke.html. Kinetic energy of such atom comes from swinging, rotating, vibrating and all other such tiny on-spot motions. All these attribute to the kinetic energy. Sep 6 '16 at 13:47
• @Steeven I didn't understand something you said. On the beginning of your answer, you say an object's potential and kinetic energy does part of Internal Energy. But these forms of enery are mechanical, it does belong to the Total Energy of a Body, but not to its internal energy. And, be ideal gases or not, I think Thermal Energy belong to Internal Energy, which is related to kinetic energy. So, maybe what does exist is kinetic energy, which creates the motion of particles, which can be translated to "thermal energy", and finally, to "temperature". Apr 2 '17 at 12:28
• I don't know if this is a good definition or not, but I call thermal energy the energy that a thermometer measures. In that case, it comprises degrees of freedom that contribute quadratically to the Hamiltonian. Apr 2 '17 at 13:14

Internal energy of a body is a concept devised for bodies in thermodynamic equilibrium. Its most important, defining properties are:

• It is a function of equilibrium state variables, usually denoted as $U$. For ideal gas, we can use variables $p,V$ and the function is $U(p,V) = fpV$ where $f$ is a number dependent on the chemical composition of the gas (roughly 3/2 for rarified helium). It does not matter how the system got into that state $p,V$, the internal energy of this state is always the same.

• Any increase of internal energy $U$ can be expressed as sum of heat and work accepted (this follows from the 1st law of thermodynamics).

It is important that the parts of the body are macroscopically at rest; such frame of reference always exists if the body is in thermodynamic equilibrium. From the point of view of mechanics, the internal energy is then the value of total energy of the system (kinetic+potential) in this frame. This means that energy of rotation or translation of the body as a whole does not count, because it is "transformed out" by using the proper frame of reference.

For example, internal energy of a cube of ideal gas of molar number $n$ is $fnRT$ where $T$ is thermodynamic temperature of the gas (again, measured in the reference frame where the gas is at rest). When we observe the same amount of gas from a frame that moves with respect to the gas, total energy of the gas is increased by the kinetic energy of the motion of the gas as a whole, but internal energy is still $fnRT$.

Thermal energy is less precise notion, it is probably being used with several different meanings. My guess is it can be used to name any part of internal energy or energy transferred that is somehow related to temperature. So it can be the kinetic energy of the gas molecules, since this can be related to temperature of the gas; or it can be the energy exchanged during natural heat exchange, since this involves difference of temperatures.

I was thinking hard about the same thing today. This is the only logical defination of heat/thermal energy I could derive.

Internal energy is mathematically well defined (as the sum of kinetic and potential energies of the particales constituting the system). Work done is mathematically well defined. But heat, the third term in the 1st law equation, is not clearly defined.

So, we can use the 1st law to define 'heat' mathematically.

1st law of thermadynamics states that for any thermodynamic system, following is true-

delta E(int) = Q - W

[ E(int) = internal energy of the system , Q = Heat , W = Work done by gas on its surroundings ]

Let W(on) be the work done on the thermodynamic system by its surroundings.

So, clearly, W(on) = - W

The first law can now be written as-

delta E(int) = Q + W(on)

Internal energy is the sum of kinetic and potential energies of all the particles constituting the thermodynamic system.

So, delta E(int) = delta K(sys) + delta U(sys)

Also, W(on) = delta K(sys)

['sys' refers to the thermodynamic system under consideration]

The first law now implies-

delta K(sys) + delta U(sys) = Q + delta K(sys)

implies- Q = delta U(sys)

So, heat is nothing but the change in potential energy of the thermodynamic system.

Heat is change in thermal energy.

So, thermal energy is the net potential energy of the system.

while, internal energy is the net potential energy + the net kinetic energy of the system.

So, that's the difference.

p.s - I had proper formulas written with mathematical symbols by using www.codecogs.com. I dont understand why it triggered the spam filter. I had to remove them.

• The proper way to write mathematics for Physics Stack Exchange is using MathJax markup, not by trying to import content from off site. Apr 6 '17 at 3:02

The confusion comes out from different definitions of Thermal Energy, e.g. internal energy, atom/molecular total kinetic energy. You can use following definition, which I think is accurate.

Thermal energy The kinetic energy associated with the random motions of the molecules of a material or object; often used interchangeably with the terms heat and heat energy. Measured in joules, calories, or Btu.

As you can see, it only includes kinetic energy but not other energy such as vibrational or potential. When writing a book or a paper, some people skip a few steps with some loss in vigorousness. It is understandable.

knzhou answered a similar question. He defined thermal energy in the following way:

Thermal energy refers to the energy that a body has because of thermal excitations (e.g. because the individual atoms/molecules are moving/rotating/vibrating). Typically the amount is $\frac{1}{2}kT$ per degree of freedom by the equipartition theorem.

By contrast, here's his description of internal energy:

Internal energy is a catch-all term that means "energy that a body has that isn't due to something external". That is, it doesn't include gravitational potential energy (the gravitational field is external), or overall kinetic energy. It includes thermal energy, but can also include other things (like if the system is elastic and is being stretched).

And finally, heat is:

Heat refers to the transfer of thermal energy between two systems. Sometimes when people are being sloppy they talk about the "amount of heat" a system has, which means its thermal energy.

So to answer the first question, the difference between thermal energy and internal energy is that internal energy includes all energy except that which is due to external interactions, whereas thermal energy only includes a subset of internal energy that is due to thermal excitations. To answer the last question, chemical energy is part of the internal energy because chemical energy is not the result of external interactions. However, chemical energy is not included in thermal energy because chemical energy is not a result of thermal excitations.

Here's the link to Kevin Zhou's post:

Confusion between thermal energy and heat