Kinetic Theory • Topic 1 of 3

Kinetic Theory & Gas Laws

A gas has no fixed shape or volume — it fills whatever container it is put in. The kinetic theory of gases explains this behaviour by picturing a gas as a huge number of tiny molecules in constant, random motion. Everything we measure about a gas in the lab — its pressure, temperature and volume — is really a large-scale average of what these countless molecules are doing.

An ideal gas is a simplified model that obeys the gas laws exactly at all pressures and temperatures. The kinetic theory rests on a few key assumptions:

  • A gas consists of a very large number of identical molecules in continuous random motion.
  • The size of a molecule is negligible compared with the average distance between molecules, so the molecules themselves occupy almost no volume.
  • There is no force of attraction or repulsion between molecules except during collisions.
  • Collisions between molecules and with the walls are perfectly elastic — kinetic energy is conserved — and take negligible time.
  • Between collisions a molecule moves in a straight line at constant velocity (no external force).

From these assumptions and from experiment we get the gas laws, each describing how two of the quantities $P$, $V$, $T$ are related when the third is held fixed:

  • Boyle's law (constant $T$): $PV=\text{constant}$, so $P\propto\frac{1}{V}$. Squeeze a gas into half the volume and its pressure doubles.
  • Charles' law (constant $P$): $\frac{V}{T}=\text{constant}$, so $V\propto T$. Heating a gas at fixed pressure makes it expand.
  • Gay-Lussac's law (constant $V$): $\frac{P}{T}=\text{constant}$, so $P\propto T$.
  • Avogadro's law: equal volumes of all gases at the same $T$ and $P$ contain equal numbers of molecules.

Combining these gives the ideal gas equation: $PV=nRT$, where $n$ is the number of moles and $R=8.314\ \text{J}\,\text{mol}^{-1}\,\text{K}^{-1}$ is the universal gas constant. Writing $n=\frac{N}{N_A}$ (with $N$ the number of molecules and $N_A=6.022\times10^{23}$ Avogadro's number) and defining the Boltzmann constant $k_B=\frac{R}{N_A}=1.38\times10^{-23}\ \text{J/K}$, the same law becomes $PV=Nk_BT$. Temperatures here are always in kelvin ($T(\text{K})=t(^\circ\text{C})+273$).

Boyle's law: halving the volume of a gas doubles its pressureV, PV/2, 2PBoyle: PV = constant
Solved Examples
1
Worked Example
A gas occupies 4 L at a pressure of 2 atm. At constant temperature, what volume will it occupy if the pressure is increased to 8 atm?
Solution
  1. Step 1: At constant $T$, Boyle's law gives $P_1V_1=P_2V_2$.
  2. Step 2: Substitute: $2\times4=8\times V_2$, so $V_2=\frac{8}{8}$.
  3. Step 3: Compute: $V_2=1\ \text{L}$.

Answer: $V_2=1\ \text{L}$.

2
Worked Example
A gas at $27^\circ\text{C}$ occupies 300 mL at constant pressure. What volume will it occupy at $127^\circ\text{C}$?
Solution
  1. Step 1: Convert to kelvin: $T_1=27+273=300\ \text{K}$, $T_2=127+273=400\ \text{K}$.
  2. Step 2: Charles' law: $\frac{V_1}{T_1}=\frac{V_2}{T_2}$, so $V_2=V_1\times\frac{T_2}{T_1}=300\times\frac{400}{300}$.
  3. Step 3: Compute: $V_2=400\ \text{mL}$.

Answer: $V_2=400\ \text{mL}$.

3
Worked Example
Calculate the number of moles of an ideal gas occupying 22.4 L at a pressure of $1.013\times10^{5}\ \text{Pa}$ and a temperature of 273 K. ($R=8.314\ \text{J}\,\text{mol}^{-1}\,\text{K}^{-1}$.)
Solution
  1. Step 1: Use $PV=nRT$, so $n=\frac{PV}{RT}$.
  2. Step 2: Substitute (with $V=22.4\times10^{-3}\ \text{m}^3$): $n=\frac{1.013\times10^{5}\times22.4\times10^{-3}}{8.314\times273}$.
  3. Step 3: Numerator $=2269$, denominator $=2270$, so $n\approx1.0\ \text{mol}$.

Answer: $n\approx1\ \text{mol}$ (this is the molar volume at STP).

4
Worked Example
Find the number of molecules in 2 moles of an ideal gas. ($N_A=6.022\times10^{23}$.)
Solution
  1. Step 1: Number of molecules $N=n\times N_A$.
  2. Step 2: Substitute: $N=2\times6.022\times10^{23}$.
  3. Step 3: Compute: $N=1.204\times10^{24}$ molecules.

Answer: $N\approx1.20\times10^{24}$ molecules.

5
Worked Example
A fixed mass of gas at 2 atm and 300 K is heated at constant volume until its pressure becomes 3 atm. Find the new temperature.
Solution
  1. Step 1: At constant $V$, Gay-Lussac's law gives $\frac{P_1}{T_1}=\frac{P_2}{T_2}$.
  2. Step 2: So $T_2=T_1\times\frac{P_2}{P_1}=300\times\frac{3}{2}$.
  3. Step 3: Compute: $T_2=450\ \text{K}=177^\circ\text{C}$.

Answer: $T_2=450\ \text{K}$.

6
Worked Example
Calculate the pressure exerted by 1 mole of an ideal gas confined to a volume of $24.6\times10^{-3}\ \text{m}^3$ at 300 K. ($R=8.314\ \text{J}\,\text{mol}^{-1}\,\text{K}^{-1}$.)
Solution
  1. Step 1: From $PV=nRT$, $P=\frac{nRT}{V}$.
  2. Step 2: Substitute: $P=\frac{1\times8.314\times300}{24.6\times10^{-3}}$.
  3. Step 3: Numerator $=2494$, so $P=\frac{2494}{0.0246}\approx1.01\times10^{5}\ \text{Pa}$.

Answer: $P\approx1.01\times10^{5}\ \text{Pa}$ (about 1 atm).

Key Points

  • Kinetic theory pictures a gas as many tiny molecules in constant random motion; molecular volume and inter-molecular forces are negligible and collisions are perfectly elastic.
  • Boyle's law (constant $T$): $PV=\text{constant}$; Charles' law (constant $P$): $\frac{V}{T}=\text{constant}$; Gay-Lussac's law (constant $V$): $\frac{P}{T}=\text{constant}$.
  • Avogadro's law: equal volumes of gases at the same $T$ and $P$ have equal numbers of molecules.
  • Ideal gas equation: $PV=nRT$ with $R=8.314\ \text{J}\,\text{mol}^{-1}\,\text{K}^{-1}$, or equivalently $PV=Nk_BT$ with $k_B=1.38\times10^{-23}\ \text{J/K}$.
  • Always use absolute temperature in kelvin: $T(\text{K})=t(^\circ\text{C})+273$.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.At constant temperature, the pressure of a fixed mass of gas varies with its volume as:
Explanation: Boyle's law: $PV=\text{constant}$, so $P\propto\frac{1}{V}$ at constant $T$.
Q2.The ideal gas equation can be written as:
Explanation: The ideal gas law is $PV=nRT$, equivalently $PV=Nk_BT$.
Q3.Which of the following is NOT an assumption of the kinetic theory of gases?
Explanation: Kinetic theory assumes no inter-molecular force except during collisions, so persistent strong attraction is not assumed.
Q4.Avogadro's law states that equal volumes of all gases at the same temperature and pressure contain:
Explanation: Avogadro's law: equal volumes at the same $T$ and $P$ contain equal numbers of molecules.
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