16 From Temperature to Thermal Energy

Brokk Toggerson

Learning Objectives

By the end of this section, you should be able to…

  • Define thermal energy.
  • Compare and contrast thermal energy and temperature.
  • Given two of average, total, or number of entries, find the third.


“All of the energy present before a change occurs always exists in some form after the change is completed.”

Statement of Conservation of Energy

As discussed both in the last chapter and in-class during the first unit, is the average kinetic energy per . This is fine and good, but, as we saw in the last chapter, it is only the total energy is what is constant for a given process. The total energy associated with temperature is called the . We, therefore, need to be able to connect the average energy associated with temperature to the total thermal energy present if we are to apply the conservation of energy correctly.

Going from an Average to a Total

For our unit on entropy, we had to explore Means/Averages and Other Measures of Center (Appendix B). In that case, we were interested in how to determine the average of a set of data in order to categorize those data. In the case of relating temperature and thermal energy, however, we have a slightly different problem: we have the average (temperature) and need to get the total amount of thermal energy. Fortunately, if we know the number of atoms/molecules, we can determine the total.

We know the definition of average when all elements have equal probability:

    \[\langle x \rangle = \frac{1}{N} \sum_{i=0}^N x_i\]

where \langle x_i \rangle is our average value, N is the number of entries and \sum_{i=0}^N x_i is the total of all the values. This last point is what is important: \sum_{i=0}^N x_i is the total!

Thus, if we know the average \langle x \rangle and the number of entries N, we can determine the total:

    \[\langle x \rangle = \frac{1}{N} \sum_{i=0}^N x_i\]

    \[N \langle x \rangle = \sum_{i=0}^N x_i\]

Recognizing \sum_{i=0}^N x_i as the total T, we see that

    \[T = N \langle x \rangle\]

exactly what we were looking for! In fact, this equation can also be manipulated in other ways. For example, if I know the average \langle x \rangle and the total \sum_{i=0}^{N}, then I can find the number of entries.

Example: Average and number to total


If the temperature of 5 \, \mathrm{mols} of a monatomic ideal gas is 293 \, \mathrm{K} (about room temperature), then, as we will see in class, the average energy per molecule comes out to be 0.038 \, \mathrm{eV}.

How many Joules of energy are in the gas as a whole, i.e. what is the total thermal energy?


Following the discussion on units in the What is Energy?, the first step is to convert from {\rm eV} to {\rm J}:

    \[0.038 \, \mathrm{eV} \times \left( \frac{1.602 \times 10^{-19} \, \mathrm{J}}{\rm eV} \right) = 6.07 \times 10^{-21} \, \mathrm{J}\]

Now we can work out the total thermal energy. In addition to being related to temperature, we know that the average energy \langle E \rangle could be determined by simply adding up the energies of all the atoms and then dividing by the number:

    \[\langle E \rangle = \frac{1}{N_{\rm atoms}} \sum_{\rm atoms} E\]

This \sum_{\rm atoms} E is the total as described above. We can solve for this total by simply multiplying the number of atoms across

    \[N_{\rm atoms} \cdot \langle E \rangle = \sum_{\rm atoms} E\]

    \[{\rm Total \, Energy} = N_{\rm atoms} \cdot \langle E \rangle\]

The second-to-last step is we need to convert from mols to number of atoms (that is what we are dividing by after all!)

    \[N = \left( 5 \, \mathrm{mol} \right) \times \left( 6.022 \times 10^{23} \, \frac{\rm atoms}{\rm mol} \right)\]

    \[N = 3.01 \times 10^{24} \, \mathrm{atoms}\]

Finally, we can substitute our values and get the total thermal energy:

    \[{\rm Total \, Energy} = \left( 3.01 \times 10^{24} \right) \times \left( 6.07 \times 10^{-21} \, \mathrm{J} \right)\]

    \[{\rm Total \, Energy} = 1.83 \times 10^{3} \, \mathrm{J} = 1.83 \, \mathrm{kJ}\]

Or, looking at our energy touchstone examples, the total amount of energy present in one-quarter of a food Calorie.


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