Kinetic Interpretation of Temperature

The Link Between Temperature and Molecular Motion

The kinetic theory of gases gives us a powerful insight: temperature is not a separate, mysterious quantity. It is a direct measure of the average kinetic energy of the molecules in a gas. This section unpacks that connection, starting from the pressure equation we derived earlier and working step by step toward a molecular definition of temperature.

Recall the fundamental pressure equation for an ideal gas:

Here, $N$ is the number of molecules, $m$ is the mass of each molecule, $\bar{v}^2$ is the mean square speed, and $V$ is the volume. The product $N m$ is simply the total mass $M$ of the gas, so we can also write $P = \frac{1}{3} \frac{M \bar{v}^2}{V}$.

Now, multiply both sides by the volume $V$:

The right-hand side contains $m \bar{v}^2$, which is twice the average translational kinetic energy of a single molecule. The average kinetic energy of one molecule, $\bar{K}$, is:

Therefore, $m \bar{v}^2 = 2 \bar{K}$, and we can rewrite the pressure-volume product as:

This is the first bridge between the macroscopic quantity $P V$ and the microscopic average kinetic energy.

Connecting to the Ideal Gas Law

We already know the ideal gas law: $P V = N k_B T$, where $k_B$ is Boltzmann's constant ($1.38 \times 10^{-23} \text{ J/K}$) and $T$ is the absolute temperature. Equating this with the expression above gives:

The number of molecules $N$ cancels out, leaving a clean and profound result:

This is the central equation of this section. It states that the average translational kinetic energy of a molecule in an ideal gas is directly proportional to the absolute temperature. The constant of proportionality is $\frac{3}{2} k_B$.

Root Mean Square Speed

From $\bar{K} = \frac{1}{2} m \bar{v}^2 = \frac{3}{2} k_B T$, we can solve for the root mean square speed, $v_{\text{rms}}$, which is defined as the square root of the mean square speed: $v_{\text{rms}} = \sqrt{\bar{v}^2}$.

Multiplying both sides by 2 and dividing by $m$:

Taking the square root gives:

This is the formula for the root mean square speed of gas molecules. It depends only on the temperature and the mass of a single molecule.

Kinetic Energy per Mole

It is often useful to talk about the kinetic energy of one mole of gas rather than one molecule. One mole contains $N_A$ molecules (Avogadro's number). The total translational kinetic energy of one mole, $U_{\text{trans}}$, is:

But $N_A k_B = R$, the universal gas constant. Therefore:

This is the internal energy due to translational motion for one mole of an ideal monatomic gas. For such a gas (like helium or argon), this is the entire internal energy, since there are no rotational or vibrational modes.

Properties Derived from the Kinetic Interpretation

The kinetic theory leads to several important properties that follow directly from the equations above. Each one is derived step by step.

›Proof

Property (I): The average kinetic energy of a molecule is independent of its mass.

From $\bar{K} = \frac{3}{2} k_B T$, the average kinetic energy depends only on temperature $T$ and Boltzmann's constant $k_B$. The mass $m$ of the molecule does not appear in this equation. Therefore, at a given temperature, a light molecule (like hydrogen) and a heavy molecule (like oxygen) have the same average translational kinetic energy.

This does not mean they have the same speed. Since $\bar{K} = \frac{1}{2} m \bar{v}^2$, if $\bar{K}$ is fixed, a lighter molecule must have a larger $\bar{v}^2$ to compensate. This is why, at the same temperature, hydrogen molecules move faster on average than oxygen molecules.

›Proof

Property (II): The rms speed is proportional to the square root of the absolute temperature and inversely proportional to the square root of the molecular mass.

From $v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}}$, we see:

• If $T$ is doubled, $v_{\text{rms}}$ increases by a factor of $\sqrt{2}$.
• If $m$ is quadrupled (for a fixed $T$), $v_{\text{rms}}$ is halved.

This inverse relationship with mass explains why lighter gases diffuse faster and why sound travels faster in lighter gases at the same temperature.

›Proof

Property (III): At a given temperature, all gases have the same average kinetic energy per molecule.

This is a direct restatement of Property (I). It is a powerful result: if you have a mixture of different gases at the same temperature, every molecule — regardless of its type — has the same $\bar{K} = \frac{3}{2} k_B T$. This is the foundation for understanding phenomena like thermal equilibrium and the equipartition of energy.

›Proof

Property (IV): The pressure of an ideal gas is proportional to the number density and the average kinetic energy.

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