IMAT Physics · Heat and Thermodynamics

Ideal Gases and Kinetic Theory

Pressure, volume and kelvin temperature of a fixed amount of gas are tied by pV = nRT, and temperature is a measure of the molecules' average kinetic energy.

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Why this matters

Five past questions sit here, two of them in the ministry papers: Boyle's law as a pressure change and as a graph, a temperature found after a gas cools at fixed volume, and what does or does not change for gas molecules at a given temperature.

Concept 1 of 2: The ideal gas equation and the three gas laws

Gas pressure comes from molecules hitting the walls. Squeeze the same molecules into half the space and they hit each wall twice as often, so the pressure doubles. Heat them and they hit harder and more often, so the pressure rises unless the gas is allowed to expand. All of this only works with kelvin, because the molecules' energy is proportional to the kelvin temperature.

Definition

An ideal gas obeys $pV = nRT$ exactly; real gases come close at low pressure and high temperature.
For a fixed amount of gas, $pV/T$ is constant, so $\dfrac{p_1V_1}{T_1} = \dfrac{p_2V_2}{T_2}$ , with $T$ always in kelvin.
Boyle's law (constant $T$ ): $pV$ is constant. Halve the volume and the pressure doubles. A graph of $p$ against $V$ is a falling curve (a hyperbola), steep at small volume and flattening out; $p$ against $1/V$ is a straight line through the origin.
Charles's law (constant $p$ ): $V \propto T$ .
Gay-Lussac's law (constant $V$ ): $p \propto T$ .

Ideal gas equation

pV = nRT \qquad \frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}

$p$ pressure, in Pa
$V$ volume, in m³
$n$ amount of gas, in mol
$R$ gas constant, 8.31 J mol⁻¹ K⁻¹
$T$ absolute temperature, in K

Worked example

A gas occupies 2.0 L at

2.0 \times 10^5\ \text{Pa}

and 27 °C. It is compressed to 1.0 L and its temperature rises to 177 °C. What is its new pressure?

Practice this conceptself-check · 4 quick reps

Gas laws need kelvin, never Celsius

Proportions such as

p \propto T

hold only for absolute temperature. Heating from 10 °C to 20 °C does not double the pressure; it raises it by about 3.5%, from 283 K to 293 K.

Boyle's law is not a straight line on a p against V graph

At constant temperature,

p = \text{constant}/V

, a curve that falls ever less steeply. A straight line through the origin appears only when

p

is plotted against

1/V

Concept 2 of 2: Kinetic theory: temperature and molecular speed

At one temperature every kind of gas molecule carries the same average kinetic energy. A heavy molecule with that energy must move more slowly than a light one, so light molecules are fast and heavy ones slow. In an ideal gas the molecules do not attract one another, so their kinetic energy is all the internal energy there is: fix the temperature and you fix the internal energy.

Definition

The mean kinetic energy of a gas molecule depends only on the kelvin temperature, not on the type of gas.
At the same temperature, lighter molecules have a larger mean square speed $\langle v^2 \rangle$ : the rms speed $v_{\text{rms}} = \sqrt{\langle v^2 \rangle}$ goes as $1/\sqrt{M}$ .
The rms speed goes as $\sqrt{T}$ : four times the kelvin temperature doubles it.
The internal energy of a fixed amount of ideal gas depends only on its temperature. An ideal gas compressed or expanded at constant temperature keeps the same internal energy.

Mean kinetic energy of a molecule

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

$m$ mass of one molecule, in kg
$\langle v^2 \rangle$ mean of the squared speeds
$k_B$ Boltzmann constant, 1.38 × 10⁻²³ J K⁻¹
$T$ absolute temperature, in K

Worked example

Helium (molar mass 4 g/mol) and argon (40 g/mol) are mixed at 300 K. Find the mean kinetic energy of a molecule, and the ratio of the rms speed of helium to that of argon.

Practice this conceptself-check · 3 quick reps

Heavier molecules do not have more kinetic energy

At the same temperature all gas molecules have the same mean kinetic energy, whatever their mass. The heavier ones simply move more slowly, with a smaller mean square speed.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

The ideal gas equation and the three gas laws
Ideal gas equation
$pV = nRT \qquad \frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}$
Kinetic theory: temperature and molecular speed
Mean kinetic energy of a molecule
$\overline{E_k} = \tfrac{1}{2} m \langle v^2 \rangle = \tfrac{3}{2} k_B T$

Watch out for (3)

Gas laws need kelvin, never Celsius
→ The ideal gas equation and the three gas laws
Boyle's law is not a straight line on a p against V graph
→ The ideal gas equation and the three gas laws
Heavier molecules do not have more kinetic energy
→ Kinetic theory: temperature and molecular speed

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