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Temperature Is Motion: Speed and the Distribution

Compare two formulas for PV and temperature reveals its deepest identity: it is molecular kinetic energy in disguise. Then meet the Maxwell-Boltzmann distribution — a gas has not one speed, but a whole bell of them.

What temperature really measures

We now hold two expressions for the same quantity PV: the experimental PV = Nk_BT and the kinetic PV = \tfrac{1}{3}Nm\overline{v^2}. Set them equal, cancel the N, and rearrange. Out drops one of the most illuminating equations in physics.

\tfrac{1}{2} m \overline{v^2} = \tfrac{3}{2} k_B T

The average translational kinetic energy of a gas molecule is exactly (3/2)k_BT. Temperature IS molecular motion — nothing more, nothing less.

How fast are the molecules?

Solve the boxed equation for the speed and you get the root-mean-square speed, the speed of a molecule with the average kinetic energy. Written with the molar mass M it is easy to evaluate.

v_{\mathrm{rms}} = \sqrt{\overline{v^2}} = \sqrt{\dfrac{3 k_B T}{m}} = \sqrt{\dfrac{3 R T}{M}}

The rms speed rises with temperature and falls with molecular mass. Heavier molecules move more sluggishly at the same temperature.

Put in numbers for nitrogen (M = 0.028 kg/mol) at room temperature (T = 300 K): v_{\mathrm{rms}} = \sqrt{3(8.314)(300)/0.028} \approx 517 m/s. The air molecules around you are hurtling past at roughly the speed of a rifle bullet — which is also, not coincidentally, close to the speed of sound, since sound is carried by exactly these molecular motions. Lighter helium, thirteen times less massive per mole, moves about \sqrt{7}\approx 2.6 times faster.

Equipartition: energy shared evenly

Why the factor of \tfrac{3}{2}? Because a molecule can move in three independent directions — three degrees of freedom — and random collisions share energy evenly among them. This is the equipartition theorem: at equilibrium, every independent way a molecule can hold energy gets, on average, the same slice.

\langle E \rangle = \tfrac{1}{2} k_B T \quad \text{per degree of freedom}

Each degree of freedom carries ½k_BT of energy on average. Three translational directions give (3/2)k_BT — and diatomic molecules, which can also spin, get more.

Not one speed, but a whole spread

The rms speed is a single representative number, but the molecules do not all move at it. Ceaseless collisions constantly reshuffle energy, some molecules crawling, a few streaking, most somewhere in between. The exact shape of this spread is the Maxwell-Boltzmann distribution — one of the crown jewels of nineteenth-century physics.

The distribution of molecular speeds, skewed with a long high-speed tail. Raise the temperature and the whole curve flattens and slides right — more fast molecules — while the area beneath (the total number of molecules) stays fixed.

v_p = \sqrt{\dfrac{2 k_B T}{m}} \;<\; \bar v = \sqrt{\dfrac{8 k_B T}{\pi m}} \;<\; v_{\mathrm{rms}} = \sqrt{\dfrac{3 k_B T}{m}}

Three speeds live on the curve: the most-probable v_p (its peak), the mean v̄, and the rms speed. They differ modestly and always fall in this order — the long high-speed tail pulls the rms highest.

How far between collisions

A molecule at 500 m/s does not fly in a straight line for long — it slams into a neighbour almost immediately. The average distance it travels between collisions is its mean free path \lambda, set by how crowded the gas is (the number density N/V) and how big the molecules are (diameter d).

\lambda = \dfrac{1}{\sqrt{2}\,\pi d^2 (N/V)}

The mean free path shrinks as the gas gets denser or its molecules larger. For ordinary air it is about 70 nanometres — a few hundred molecular diameters.

Seventy nanometres sounds tiny, and it is — yet it is hundreds of times a molecule's own size, confirming how empty a gas really is. Dividing the speed by the mean free path, an air molecule suffers several billion collisions every second. That furious collision rate is precisely why a gas reaches equilibrium so fast and why the neat averages of this guide are so reliable.