Ideal Gas

Assumptions: a gas contains a large number of identical molecules in random motion; the molecules are point-sized, collisions are perfectly elastic, the intermolecular forces are negligible except during collisions, and the time of collision is negligible compared with the time between collisions.

Derivation: Take $N$ molecules, each of mass $m$, in a cubical container of side $l$ (volume $V=l^3$). Consider one molecule with velocity components $(c_x,c_y,c_z)$, so $c^2=c_x^2+c_y^2+c_z^2$.

When it hits the wall perpendicular to the $x$-axis, it rebounds with $-c_x$, so the change of momentum is

$\Delta p=mc_x-(-mc_x)=2mc_x$

It travels to the opposite wall and back, a distance $2l$, in time $2l/c_x$, so it strikes this wall $\dfrac{c_x}{2l}$ times per second. The rate of change of momentum, i.e. force on the wall from this molecule, is

$F_1=2mc_x\cdot\frac{c_x}{2l}=\frac{mc_{x}^2}{l}$

For all $N$ molecules the total force on that wall:

$F=\frac{m}{l}\left(c_{1x}^2+c_{2x}^2+\dots+c_{Nx}^2\right)=\frac{mN\overline{c_x^2}}{l}$

Pressure $P=F/l^2$:

$P=\frac{mN\overline{c_x^2}}{l^3}=\frac{mN\overline{c_x^2}}{V}$

Since the motion is random, $\overline{c_x^2}=\overline{c_y^2}=\overline{c_z^2}=\tfrac13\overline{c^2}$. Hence

$\boxed{P=\frac13\frac{mN}{V}\overline{c^2}=\frac13\rho\,\overline{c^2}}$

where $\rho=mN/V$ is the density of the gas and $\overline{c^2}$ is the mean square speed. The shape of the container does not matter.

Also, $P V=\tfrac13 mN\overline{c^2}=\tfrac23 N\left(\tfrac12m\overline{c^2}\right)$, which shows pressure is two-thirds of the translational kinetic energy per unit volume.

Three properties of an ideal gas (kinetic theory assumptions):

1. Negligible molecular volume: The volume occupied by the gas molecules themselves is negligible compared to the total volume of the container. The molecules are treated as point masses.

2. No intermolecular forces: There is no force of attraction or repulsion between the gas molecules, except during an elastic collision. Hence the molecules move freely and randomly in all directions.

3. Perfectly elastic collisions: Collisions between the gas molecules, and between the molecules and the walls of the container, are perfectly elastic, so there is no loss of kinetic energy.

Two things the student could do:

1. Use a real gas at low pressure. At low pressure the molecules are far apart, so the intermolecular forces of attraction become negligible and the volume of the molecules themselves is negligible compared with the volume of the container. The gas then obeys $PV = nRT$ almost exactly. (Hydrogen, helium, nitrogen and oxygen are good choices.)

2. Use the gas at high temperature. At a temperature well above its critical temperature (near or above the Boyle temperature), the molecules have enough kinetic energy to overcome the intermolecular attraction, so the gas again behaves nearly ideally.

So the student should take a real gas such as hydrogen or helium and keep it at low pressure and high temperature, where it approximates an ideal gas closely enough for the experiment.

As the car moves, friction between the tyre and the road, and the repeated flexing of the tyre, produce heat. This raises the temperature of the air inside. The volume of the tyre is almost constant, so by the pressure law ($P\propto T$ at constant volume, i.e. $P_1/T_1=P_2/T_2$) the pressure of the air increases. In kinetic theory terms, the molecules move faster and hit the walls harder and more often.

One mole of a gas: the amount of gas that contains Avogadro's number, $6.022\times10^{23}$, of molecules. Its mass in grams equals its molecular mass, and at STP it occupies $22.4$ litres.

Why hydrogen is not found in the atmosphere: Hydrogen molecules are very light, so their average speed is very high: $c_{rms}=\sqrt{3RT/M}\approx1.9\ \text{km s}^{-1}$ at about $300$ K, and higher still in the hot upper atmosphere. The escape velocity from the Earth is $11.2\ \text{km s}^{-1}$. By the Maxwell distribution, a small fraction of the molecules always has speeds much greater than the average, and over geological time these fast hydrogen molecules reach the escape speed and leave the Earth's gravity. Heavier gases such as oxygen and nitrogen move far more slowly and are retained. Free hydrogen is also very reactive, combining with oxygen to form water. Hence almost no free hydrogen remains in the atmosphere.

At STP: $P=1.013\times10^{5}\ \text{Pa}$, $T_0=273$ K, $\rho=1.251\ \text{kg m}^{-3}$.

From the kinetic theory, $P=\tfrac13\rho c^2$, so

$c_0=\sqrt{\frac{3P}{\rho}}=\sqrt{\frac{3\times1.013\times10^{5}}{1.251}}\approx493\ \text{m s}^{-1}$

Since $c\propto\sqrt T$, at $T=127+273=400$ K,

$c=c_0\sqrt{\frac{400}{273}}=493\times1.210\approx5.97\times10^{2}\ \text{m s}^{-1}$

The r.m.s. velocity of nitrogen molecules at $127^\circ$C is about $597\ \text{m s}^{-1}$.

Real gases obey $PV=nRT$ more closely at low pressure and high temperature.

At low pressure the gas is dilute, so the molecular volume is negligible compared with the container's volume. At high temperature the molecules move fast, so the attractive forces between them have little effect. These conditions match the assumptions of an ideal gas.

The average translational kinetic energy of the molecules, $\tfrac12m\overline{c^2}=\tfrac32kT$. Temperature is proportional to this mean kinetic energy.

(a) At constant volume, when the temperature of a fixed mass of gas is raised, the pressure and the r.m.s. speed of the molecules (also the mean kinetic energy per molecule and the internal energy) increase.

(b)(i) From $pV=nRT$ with $p=3.0\times10^{5}$ Pa, $V=1.0\times10^{-2}\ \text{m}^3$, $T=290$ K, $R=8.31\ \text{J mol}^{-1}\text{K}^{-1}$: $n=\frac{pV}{RT}=\frac{3.0\times10^{5}\times1.0\times10^{-2}}{8.31\times290}=\frac{3000}{2410}\approx1.2\ \text{mol}$

(ii) $m=nM=1.245\times2.9\times10^{-2}\approx3.6\times10^{-2}\ \text{kg}\ (\approx36\ \text{g})$

(c)(i) Same. The mean translational kinetic energy per molecule is $\tfrac32kT$, which depends only on the temperature and not on the type of molecule.

(ii) Not the same. $\tfrac12m\overline{c^2}=\tfrac32kT$ gives $c_{rms}=\sqrt{3kT/m}$, so lighter nitrogen molecules ($M=28$) have a higher r.m.s. speed than the heavier oxygen molecules ($M=32$).

(a) Boyle's law: for a fixed mass of gas at constant temperature, the pressure is inversely proportional to the volume, $P\propto\dfrac1V$ (or $PV=$ constant).

(b) Take a fixed mass of gas (amount $n$) changing from state $(P_1,V_1,T_1)$ to $(P_2,V_2,T_2)$ in two steps.

Step 1. Change pressure from $P_1$ to $P_2$ at constant temperature $T_1$. By Boyle's law the volume becomes $V'$:

$P_1V_1=P_2V'\ \Rightarrow\ V'=\frac{P_1V_1}{P_2}$

Step 2. Heat from $T_1$ to $T_2$ at constant pressure $P_2$. By Charles' law the volume changes from $V'$ to $V_2$:

$\frac{V'}{T_1}=\frac{V_2}{T_2}\ \Rightarrow\ V'=\frac{V_2T_1}{T_2}$

Equating the two expressions for $V'$:

$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$

So $\dfrac{PV}{T}$ is constant for a given mass of gas. For $n$ moles the constant is $nR$, where $R$ is the universal gas constant:

$PV=nRT$

(a) Properties of an ideal gas assumed by the kinetic theory

1. Negligible molecular volume: The molecules of an ideal gas are considered as point masses — the total volume occupied by the molecules themselves is negligibly small compared with the volume of the container.
2. No intermolecular force: There is no force of attraction or repulsion between the molecules except during a collision; hence a molecule moves with a uniform velocity between two successive collisions.
3. Perfectly elastic collisions: The molecules are in continuous random motion and their collisions with one another and with the walls of the container are perfectly elastic, so there is no loss of kinetic energy.

(Other accepted assumptions: the molecules obey Newton's laws of motion; the time of collision is negligible compared with the time between collisions; the number of molecules is very large; the average kinetic energy of a molecule is directly proportional to the absolute temperature of the gas.)

(b) Two things the student could do

1. Use a real gas under conditions where it behaves almost ideally: Choose a real gas (such as hydrogen, helium or nitrogen) and keep it at low pressure and high temperature. Under these conditions the molecules are far apart, so the intermolecular forces and the volume of the molecules themselves become negligible — the gas then obeys the ideal gas equation $PV = nRT$ closely enough for the experiment.

2. Apply corrections to the ideal gas equation: Use the real gas but correct the ideal gas equation for the finite size of the molecules and the intermolecular attraction by using the van der Waals equation $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$ so that the calculated values agree with the observed behaviour of the real gas.

The tube has an almost fixed volume. When the temperature rises in summer, the pressure of the air inside rises, since $P\propto T$ at constant volume. The hot road and sunlight heat the tube, and the rubber also weakens when hot. If the pressure exceeds what the tube can withstand, it bursts.

From kinetic theory, $p=\dfrac13\rho c^2$, so $c=\sqrt{\dfrac{3p}{\rho}}$.

At STP ($T_0=273\ \text{K}$, $p=1.013\times10^5\ \text{Pa}$, $\rho=0.1785\ \text{kg m}^{-3}$):

$c_0=\sqrt{\frac{3\times1.013\times10^5}{0.1785}}\approx1.305\times10^{3}\ \text{m s}^{-1}$

At $30^\circ$C ($T=303\ \text{K}$): since $c\propto\sqrt{T}$,

$c=c_0\sqrt{\frac{T}{T_0}}=1305\times\sqrt{\frac{303}{273}}=1305\times1.0535\approx1.37\times10^{3}\ \text{m s}^{-1}$

The rms speed of helium molecules at $30^\circ$C is about $1.37\times10^{3}\ \text{m s}^{-1}$.