The Meaning of Temperature — Theory & Guide
In kinetic–molecular theory, the absolute temperature \(T\) of an ideal gas
measures the average translational kinetic energy of its molecules.
Temperature does not track a single molecule; it summarizes the ensemble.
\[
\big\langle \varepsilon_{\text{trans}} \big\rangle
= \tfrac{3}{2}\,k_\mathrm{B}T
\qquad\text{and}\qquad
\big\langle E_{\text{trans}} \big\rangle
= \tfrac{3}{2}\,RT
\]
Constants (SI)
\(k_\mathrm{B}=1.380\,649\times10^{-23}\ \mathrm{J\,K^{-1}}\),
\(R=8.314\,462\,618\ \mathrm{J\,mol^{-1}\,K^{-1}}\),
\(1\ \mathrm{eV}=1.602\,176\,634\times10^{-19}\ \mathrm{J}\).
Why the factor \(\tfrac{3}{2}\)? — Equipartition
Each quadratic degree of freedom contributes
\( \tfrac{1}{2}k_\mathrm{B}T \) per molecule (or \( \tfrac{1}{2}RT \) per mole)
to the thermal energy. A translating molecule in 3-D has three quadratic terms
\( \tfrac{1}{2}mu_x^2,\ \tfrac{1}{2}mu_y^2,\ \tfrac{1}{2}mu_z^2 \Rightarrow f=3 \).
\[
\langle \varepsilon \rangle = \tfrac{f}{2}\,k_\mathrm{B}T
\]
\[
\langle E \rangle = \tfrac{f}{2}\,RT
\]
Degrees of freedom in common gases
At ordinary temperatures, vibrational modes are typically “frozen out.”
| Type |
Usual \(f\) |
Per molecule \(\langle \varepsilon \rangle\) |
Per mole \(\langle E \rangle\) |
Internal energy \(U\) for \(n\) mol |
| Monatomic (e.g., He, Ne) |
3 |
\(\tfrac{3}{2}k_\mathrm{B}T\) |
\(\tfrac{3}{2}RT\) |
\(U=\tfrac{3}{2}nRT\) |
| Diatomic, room \(T\) |
\(\approx 5\) (3 trans + 2 rot) |
\(\tfrac{5}{2}k_\mathrm{B}T\) |
\(\tfrac{5}{2}RT\) |
\(U=\tfrac{5}{2}nRT\) |
| Polyatomic nonlinear, room \(T\) |
\(\approx 6\) (3 trans + 3 rot) |
\(3\,k_\mathrm{B}T\) |
\(3\,RT\) |
\(U=3\,nRT\) |
Links to molecular speeds
From equipartition, the mean of the squared speed satisfies
\[
\tfrac{1}{2}m \langle u^2 \rangle = \tfrac{3}{2}k_\mathrm{B}T
\]
\[
u_{\mathrm{rms}} = \sqrt{\langle u^2 \rangle} = \sqrt{\dfrac{3k_\mathrm{B}T}{m}}
\quad\text{or}\quad
u_{\mathrm{rms}} = \sqrt{\dfrac{3RT}{M}}
\]
Here \(m\) is the mass of one molecule and \(M\) the molar mass in \(\mathrm{kg\,mol^{-1}}\).
Consequently, \(u_{\mathrm{rms}} \propto \sqrt{T}\) and decreases with increasing \(M\).
Scaling insights
- Doubling \(T\) doubles the average translational energy.
- Speeds scale as \(u \propto \sqrt{T}\); e.g., raising \(T\) by a factor of 4 doubles \(u_{\mathrm{rms}}\).
- At fixed \(T\), lighter gases have broader and faster Maxwell–Boltzmann distributions.
Unit checks
\[
\left[\tfrac{3}{2}RT\right] = \mathrm{J\,mol^{-1}}
\]
\[
\left[\tfrac{3}{2}k_\mathrm{B}T\right] = \mathrm{J}
\]
When the simple picture fails
- Low \(T\): rotational and vibrational modes may be “frozen,” reducing effective \(f\).
- High \(T\): vibrational modes activate, increasing \(f\); dissociation/ionization may occur.
- Non-ideal gases: strong interactions (high \(P\), low \(T\)) deviate from ideal-gas assumptions.
