http://www.ck12.org Chapter 19. Thermodynamics and Heat Engines Version 2
Temperature is anaverage of kinetic energy over degrees of freedom, not a sum. Let’s try to understand why this is in
reference to our monoatomic ideal gas. In the derivation above, volume was constant; so, temperature was essentially
proportional to pressure, which in turn was proportional to the kinetic energydue to translational motionof the
molecules. If the molecules had been able to rotate as well as move around the box, they could have had the same
kinetic energy with slower translational velocities, and, therefore, lower temperature. In other words, in that case,
or assumption that the kinetic energy of the atoms only depends on their velocities, implied between equations
[2] and [3],would not have held.Therefore,the number of degrees of freedom in a substance determines the
proportionality between molecular kinetic energy and temperature: the more degrees of freedom, the more
difficult it will be to raise its temperature with a given energy input.
In solids, degrees of freedom are usually entirely vibrational; in liquids, the situation becomes more complicated.
We will not attempt to derive results about these substances, however.
A note about the above discussion: since the objects at the basis of our understanding of thermodynamics are
atoms and molecules, quantum effects can make certain degrees of freedom inaccessible at specific temperature
ranges. Unlike most cases in your current physics class, where these can be ignored, in this case, quantum effects
can make an appreciable difference. For instance, the vibrational degrees of freedom of diatomic gas molecules
discussed above are, for many gases, inaccessible in very common conditions, although we do not have the means
to explain this within our theory. In fact, this was one of the first major failures of classical physics that ushered in
the revolutionary discoveries of the early 20th century.
Thermal Energy
In light of the above derivation, it should not surprise you that the kinetic energy from motion of molecules
contributes to what is called thethermal energyof a substance. This type of energy is calledsensible energy.
In ideal gases, this is the only kind of thermal energy present.
Solids and liquids also have a different type of thermal energy as well, calledLatent Energy, which is associated
with potential energy of their intermolecular bonds in that specific phase — for example the energy it takes to break
the bonds between water molecules in melting ice (remember, we assumed molecules do not interact in the ideal gas
approximation).
To recap, there are two types ofThermal Energy:
- The kinetic energy from the random motion of the molecules or atoms of the substance, calledSensible
Energy - The intermolecular potential energy associated with changes in the phase of a system (calledLatent Energy).