David Bauer Physics & Astronomy · UCLA

Physics 5B Chapter 5

Kinetic Theory of Gases

Kinetic theory and molecular speeds

Kinetic theory of gases

Macroscopic properties like temperature and pressure are ultimately due to the motion of and interactions between microscopic particles. But so far, we have made no attempt to connect these two pictures.

A simple approach called kinetic theory allows us to explain the thermodynamic properties of gases from the motion of the gas molecules. In kinetic theory, we assume the molecules move at random, and that they only interact by colliding with one another, so there is no potential energy between molecules.

An analogy: heights in a population

You can think of the distribution of molecular speeds in a gas as analogous to the distribution of heights in a population of people.

A histogram of adult female standing heights. The horizontal axis is height in centimetres, running from 140 to 184 centimetres in 2-centimetre-wide bins; the vertical axis is the number of people in each bin, from 0 to about 120. The blue bars form a single rounded hump that is roughly symmetric about its centre. Counts are near zero at 140 centimetres, climb steadily through the 150s, peak at 115 people in the bin from 160 to 162 centimetres, and fall away again through the 170s to near zero by 184 centimetres. A dashed vertical line near the peak marks the mean height of 162.8 centimetres, and the spread about that mean is about 7 centimetres, so most of the 868 people measured lie between roughly 149 and 177 centimetres. An individual need not have the average height: individual values scatter about a well-defined average, which is exactly how the speeds of the molecules are distributed in a gas.

Data: CDC National Health and Nutrition Examination Survey (NHANES), public domain. Standing heights of 868 female participants aged 20–39.

The Maxwell-Boltzmann distribution

We model the distribution of speeds in an ideal gas at a given temperature $T$ by the Maxwell-Boltzmann distribution, which describes the fraction of molecules with speeds in a given interval.

A histogram of molecular speeds in helium gas at 300 kelvin, predicted by the Maxwell-Boltzmann distribution. The horizontal axis is speed in metres per second, running from 0 to 3000 in bins 100 metres per second wide; the vertical axis is the fraction of the molecules whose speed falls in each bin, from 0 to about 0.08. The blue bars form a single hump, but unlike the symmetric histogram of human heights this one is lopsided: it rises steeply from zero, because no molecule can have a speed below zero, reaches a broad peak of 0.074 of the molecules per bin between 1000 and 1200 metres per second, then falls away gradually into a long tail of fast molecules that thins to a fraction of about 0.001 per bin near 3000 metres per second. Because the bars are fractions of the whole population, they add to one, and the figure describes a gas of any size. Molecular speeds are spread over a wide range, of order a kilometre per second at room temperature, in the same way that heights are spread about their average in a population.

Root-mean-square speed

The average kinetic energy of a single molecule is

\[K_{\mathrm{av}} = \left\langle \frac{1}{2}mv^2 \right\rangle = \frac{1}{2}m\langle v^2\rangle,\]

where $\langle \cdot\rangle$ indicates the average over the speed distribution.

We can write the average kinetic energy in terms of the root-mean-square or rms speed, which is defined as

\[v_{\mathrm{rms}} = \sqrt{\langle v^2\rangle}.\]

The name comes from the fact that we first square the speed, then take the mean, and finally take the square root. In terms of $v_{\mathrm{rms}}$,

\[K_{\mathrm{av}}=\frac12m v_{\mathrm{rms}}^2.\]

For helium at $300\ \mathrm{K}$, the rms speed is about $1370\ \mathrm{m/s}$. This speed lies slightly to the right of the peak because the speed distribution has a long tail of fast molecules.

The same histogram of molecular speeds in helium gas at 300 kelvin, predicted by the Maxwell-Boltzmann distribution, now with the root-mean-square speed marked. The horizontal axis is speed in metres per second, running from 0 to 3000 in bins 100 metres per second wide; the vertical axis is the fraction of the molecules whose speed falls in each bin, from 0 to about 0.08. The blue bars rise steeply from zero, reach a broad peak of 0.074 of the molecules per bin between 1000 and 1200 metres per second, then fall away gradually into a long tail of fast molecules. A dashed vertical line to the right of the peak marks the root-mean-square speed, 1370 metres per second. The long high-speed tail is what pulls the root-mean-square speed above the most probable speed, just as a dashed line marks the mean on the histogram of human heights.

Pressure and temperature from molecular motion

Molecular motion and pressure

A gas exerts a force on its container due to collisions between the gas molecules and the walls of the container, and the gas’s pressure is related to the average speed of the gas molecules. Faster moving particles collide with the walls more frequently and with greater force, resulting in higher pressure.

We can use Newton’s second law to derive this relationship quantitatively. This derivation shows that the average force depends on the square of the velocity component perpendicular to the wall. Intuitively, increasing the velocity has two effects: stronger collisions (larger $\Delta v_x$) and more frequent collisions (smaller $\Delta t$).

Since the molecules move at random, there is no preferred direction for the velocity components:

\[\langle v_x^2\rangle=\langle v_y^2\rangle=\langle v_z^2\rangle=\frac{1}{3}\langle v^2\rangle.\]

Averaging over the molecules, the average force per molecule on one wall of a container of length $L$ is

\[F_{\mathrm{av}}=\frac{m\langle v_x^2\rangle}{L}=\frac{m v_{\mathrm{rms}}^2}{3L}.\]

The total force is $N$ times this average force. Dividing the force by the wall area $A$, we get the pressure in terms of the rms speed:

\[pV=\frac{1}{3}Nm v_{\mathrm{rms}}^2.\]

Temperature, speed, and kinetic energy

Combining the molecular form of the ideal gas law, $pV = N k_B T$, with our pressure-speed relationship, we get

\[\frac{1}{3} N m v_{\mathrm{rms}}^2 = N k_B T.\]

Rearranging, we get the rms speed in terms of the temperature and the molecular mass $m$:

\[\boxed{v_{\mathrm{rms}}=\sqrt{\frac{3k_BT}{m}}.}\]

This also tells us that the average kinetic energy of a single molecule is

\[K_{\mathrm{av}}=\frac{3}{2}k_BT.\]

This is an important result! It tells us that for a gas, the temperature is directly proportional to the average kinetic energy.

rms speed and temperature

Check your understanding: Compare the rms speed of a helium atom at room temperature to the rms speed of a nitrogen molecule at room temperature. The molar mass of helium is $M_{\mathrm{He}} = 4.00\ \mathrm{g/mol}$ and the molar mass of nitrogen is $M_{\mathrm{N_2}} = 28.0\ \mathrm{g/mol}$. What is the ratio of the rms speeds of the two molecules?

Answer

The rms speed of a molecule of mass $m$ in a gas at temperature $T$ is

\[v_{\mathrm{rms}} = \sqrt{\frac{3k_BT}{m}}.\]

Here $m$ is the mass of one molecule, not the molar mass. One mole contains $N_A$ molecules, so $M = N_A m$, or $m = M/N_A$. Both gases are at room temperature, so $T$ is the same for both. In the ratio, the factors $3$, $k_B$, $T$, and $N_A$ all cancel:

\[\frac{v_{\mathrm{He}}}{v_{\mathrm{N_2}}} = \sqrt{\frac{M_{\mathrm{N_2}}}{M_{\mathrm{He}}}} = \sqrt{\frac{28.0\ \mathrm{g/mol}}{4.00\ \mathrm{g/mol}}} = \sqrt{7} = 2.65.\]

A helium atom moves about $2.65$ times as fast as a nitrogen molecule at the same temperature. The two have the same average kinetic energy, so the lighter atom must move faster.

The speed distribution changes with temperature

The speed distribution depends on the temperature. At higher temperature, the peak in the distribution shifts to higher speed, and the distribution spreads out more.

Three Maxwell-Boltzmann speed distributions for helium, drawn as continuous curves on one set of axes at 300, 600, and 1200 kelvin. The horizontal axis is speed in metres per second from 0 to 6000; the vertical axis, labeled fraction of molecules, runs from 0 to 0.08, each value being the fraction of the molecules per 100 metres per second of speed. Each curve is labeled with its temperature, T equals 300, 600, or 1200 kelvin, in the curve's own colour. Each curve rises from zero, passes through a single peak, and falls away with a long high-speed tail. Raising the temperature slides the whole distribution to the right and flattens it. The 300 kelvin curve, a solid blue line, is the tallest and narrowest, peaking at a fraction of 0.074 near 1100 metres per second and reaching zero by about 3500. The 600 kelvin curve, a dashed purple line, peaks lower and further right, at 0.053 near 1580. The 1200 kelvin curve, a dash-dotted red line, is the lowest and widest of the three, peaking at only 0.037 near 2230 and not falling to zero until about 6000. The three curves enclose the same total area, because every molecule has some speed no matter how hot the gas is: as the peak drops, the distribution spreads out to compensate. The root-mean-square speed rises with temperature, from 1370 to 1930 to 2730 metres per second, in proportion to the square root of the temperature.