AP Physics 2

Gas pressure from atomic collisions

A gas is a very large number of atoms or molecules moving in random directions, colliding with one another and with the walls of their container. The pressure a gas exerts has no mysterious source: it is the net effect of those atoms striking a surface. A single collision with a wall lasts only an instant and exerts a tiny force, but billions of them arrive every microsecond, and the average of their forces is steady.

Each collision can be analysed with conservation of momentum. Take a wall that is fixed in place. An atom of mass $m$ arrives with a velocity component $v_x$ perpendicular to the wall and leaves with $-v_x$ if the collision is elastic, so its momentum component changes by $2mv_x$. The wall must have supplied that impulse, and by Newton's third law the atom gave the wall the same impulse in the opposite direction. The components of velocity parallel to the wall do not change in a smooth wall collision and contribute no force. Collisions between two atoms, or between an atom and a moving piston, are handled the same way, in one or two dimensions, using the vector conservation of momentum and, for elastic collisions, conservation of kinetic energy.

Pressure is the magnitude of the force perpendicular to a surface divided by the area of the surface:

The unit is the pascal, $1\ \mathrm{Pa} = 1\ \mathrm{N/m^2}$. Here $F_\perp$ is the sum of the perpendicular components of the forces that all the atoms exert on the surface. Pressure is larger when atoms hit harder, which means faster atoms with more momentum to give up, and when they hit more often, which means more atoms in each cubic metre of gas. Two changes follow directly.

• Compressing a gas into a smaller volume puts the same atoms into less space, so they strike a wall more frequently and the pressure rises.
• Heating a gas in a rigid container makes the atoms move faster, so each collision exerts a larger impulse and there are more collisions per second, and the pressure rises.

Pressure is not something that exists only at the walls. Any imaginary surface placed inside the gas is crossed and struck by atoms from both sides, and the pressure at that surface is the same kind of quantity. That is why a pressure gauge gives the same reading wherever a small sensor is placed in a gas at rest (at the scale where the weight of the gas is negligible).

Temperature and average kinetic energy

The atoms of a gas do not all move at the same speed, because collisions keep redistributing energy among them. What characterises the temperature of a gas is the average translational kinetic energy per atom. For an ideal gas

where $k_B = 1.38\times10^{-23}\ \mathrm{J/K}$ is Boltzmann's constant and $T$ is the absolute temperature in kelvin. This relationship is what defines temperature on the kinetic-theory picture: doubling the kelvin temperature doubles the average kinetic energy. The Celsius scale cannot be used in this equation, because $0\ ^\circ\mathrm{C}$ does not correspond to zero kinetic energy.

The quantity $v_{\mathrm{rms}}$ is the root-mean-square speed: square each atom's speed, take the average of those squares, then take the square root. It is the speed corresponding to the average kinetic energy, and it is a little larger than the simple average speed, because squaring weights the fast atoms more. Solving for it,

Two consequences follow. At the same temperature, different gases have the same average kinetic energy per atom, so the lighter atoms move faster. And the rms speed grows with the square root of $T$, so doubling the kelvin temperature raises the speed by a factor of $\sqrt{2}$, not 2.

Worked example

Find the rms speed of helium atoms ($m = 4.00\ \mathrm{u} = 6.64\times10^{-27}\ \mathrm{kg}$) at $300\ \mathrm{K}$, and compare it with nitrogen molecules ($28.0\ \mathrm{u} = 4.65\times10^{-26}\ \mathrm{kg}$) at the same temperature.

Both kinds of particle carry the same average kinetic energy, $6.21\times10^{-21}\ \mathrm{J}$, and the helium is faster by the factor $\sqrt{28/4} = 2.6$.

The distribution of speeds

Because the individual atoms have a spread of speeds, a graph of the number of atoms at each speed (or at each kinetic energy) is a useful picture. The Maxwell-Boltzmann distribution is the curve a gas in thermal equilibrium follows. It starts at zero, rises to a peak, then falls away in a long tail towards high speeds. A few atoms have speeds far above the average, and almost none have speeds near zero.

The course does not ask for the equation of the curve, only for how its features relate to temperature.

• When the gas is heated, the peak moves to a higher speed, the curve flattens and broadens, and the tail reaches further. The average kinetic energy and the rms speed both increase.
• The area under the curve is the total number of atoms (or, if the curve is normalised, 1), so heating the gas at fixed particle number does not change the area. The curve is lower at its peak because the same area is spread over a wider range of speeds.
• For two gases at the same temperature, the lighter atoms have the curve with the peak at a higher speed and a broader spread. Both curves correspond to the same average kinetic energy.
• The fraction of atoms with speeds above some fixed high value rises sharply with temperature, which is why hotter gases react and evaporate faster.

Linking pressure and temperature

The same picture gives the pressure of $N$ atoms in a volume $V$. Averaging the wall impulses over all the atoms, and using the fact that the motion is random so that one third of the kinetic energy is associated with each of the three directions, gives

Substituting $K_{\mathrm{avg}} = \tfrac{3}{2}k_BT$ gives $PV = Nk_BT$, which is the ideal gas law in its particle form. The gas law is therefore not an extra assumption: it follows from treating the atoms as point-like particles that collide elastically with the walls.

Several qualitative comparisons follow from $P \propto (N/V)\,K_{\mathrm{avg}}$.

The last row shows that pressure depends on the average kinetic energy and not on the speed alone: heavier atoms move more slowly but hit harder, and the two effects cancel.