Suppose we add thermal energy, by applying a force for a few seconds in the direction shown.
The energy quickly spreads out among all the molecules in the box.
A portion of this energy contributes to the rotations of these molecules.
The remainder of the energy contributes to the motions of these molecules through space.
It is only the motion of these molecules through space, and not the rotation of these molecules, that contributes to the temperature.
The definition of temperature is the average kinetic energy per molecule, not counting the rotation.
Here, the energy associated with the rotation is significant only because we have molecules composed of more than one atom.
This would not be the case if all the molecules consisted of only a single atom.
This is because almost all the mass of a single atom is concentrated in the nucleus.
The nucleus is too small to give the atom a significant moment of inertia.
The energy associated with the atom’s rotation is therefore extremely small and can be neglected.
Suppose we have a system consisting of just a single molecule containing only a single atom.
The thermal energy we add to the system is divided between motion in the X direction, motion in the Y direction, and motion in the Z direction.
We say that this system contains three different "degrees of freedom."
Now let’s consider a system consisting of one molecule containing two atoms.
This system can now also have rotations.
This system has five different "degrees of freedom."
Three of these degrees of freedom are the motions in the X, Y, and Z directions.
The remaining two degrees of freedom are two different types of rotations.
We are not counting the rotation around the axis joining the two atoms.
This is because, as in the case of the rotation of a single atom, the moment of inertia around this axis is very small.
The energy associated with this rotation is therefore also very small and can be neglected.
We have also been neglecting the energy due to the vibrations in the bonds between the atoms.
This energy can actually be significant, and we will talk about these vibrations later in this video.
But for now, for simplicity, let us assume that these vibrations do not exist.
Suppose we add thermal energy as heat.
The thermal energy we add as heat will, on average, be evenly distributed among these five different degrees of freedom.
If we add five units of thermal energy to this system as heat, each of the five different degrees of freedom will gain one unit of energy.
But it is only the three degrees of freedom at the bottom that contribute to increasing the temperature.
We need to add five units of energy as heat in order for three units of energy to be used to increase the temperature.
By contrast, in the case of the molecule with a single atom, we only need to add three units of energy as heat in order for all three units of energy to be used to increase the temperature.
This is somewhat of a simplification, as we are ignoring that some of the energy might go into things unrelated to the temperature, such as exciting an electron in the atom to a higher orbital.
However, these energy transitions can only occur in specific discreet quantities, and the amount of energy needed in order to make even the first transition often exceeds the available thermal energy.
Quantum Mechanics states that the energy associated with rotation is also quantized, with rotations having very small moments of inertia needing very large amounts of energy to make the transition to the next available energy level, and the amount of energy needed in order to make even the first transition often exceeds the available thermal energy.
This is why we can neglect rotations with small moments of inertia as possible "degrees of freedom" to store significant amounts of thermal energy.
If we just consider the degrees of freedom that are actually available for the energy to flow into, the energy will tend to evenly distribute among all the different degrees of freedom.
Let’s consider a system consisting of these two molecules.
This system has 11 different degrees of freedom for the energy to flow into.
This consists of 5 different types of rotations, and 6 different types of motions through space.
If we add energy to this system, it will tend to evenly spread out among all 11 degrees of freedom.
The temperature is defined as the average kinetic energy "per molecule", not counting the rotation.
Therefore, if we add 11 units of energy to this system as heat, each of the 11 degrees of freedom will gain one unit of energy, and the temperature will increase by 3 units of kinetic energy per molecule.
Let us return to the vibrations due to the bonds between the atoms.
We have been ignoring these vibrations up until now, but they can actually play a significant role.
Consider the case of a solid object, where each sphere represents a different molecule consisting of a single atom, and all the molecules have bonds to all the neighboring molecules.
We represent the bonds as springs.
At any given moment, energy is stored both in the kinetic energy of the spheres and in the potential energy of the springs.
The temperature is the average kinetic energy per sphere.
The potential energy does not count as a part of the temperature.
Each different way of storing energy, whether it be kinetic energy or potential energy, counts as a degree of freedom.
The thermal energy that we add as heat will, on average, be equally distributed among all the different degrees of freedom.
At any given moment, only a fraction of the thermal energy we have previously added is now in the kinetic energy of the spheres, while the remainder is in the potential energy of the springs.
Therefore, in a solid material, only a fraction of the thermal energy that we have previously added as heat is being used to increase the temperature of the object.
The term "Heat Capacity" refers to the amount of thermal energy we need to add as heat in order to raise the temperature of a substance by 1 degree.
In this video, we have described why different substances have different heat capacities.
Much more information is available in the video "Thermodynamics and the End of the Universe" and in the other videos on this channel.
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