"Kinetic Theory of Gases"

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Why does a gas expand to fill its container while a liquid doesn't? Why does heating a gas increase its pressure? The kinetic theory of gases answers these questions by modelling gases as collections of tiny, randomly moving molecules. This chapter connects the microscopic world of molecules to the macroscopic properties of pressure, temperature, and heat capacity.

Key Concepts

10.1 States of Matter and Molecular Behaviour

• Gases fill containers because intermolecular forces are negligible compared to kinetic energy — molecules move freely and spread everywhere.
• Solids have ordered structure because intermolecular forces dominate over kinetic energy — molecules are held in fixed positions.

10.2 Ideal Gas

An ideal gas is a theoretical gas that perfectly follows the kinetic molecular theory:

1. Gas molecules are point masses — their volume is negligible compared to the container
2. No intermolecular forces exist except during collisions
3. All collisions (molecule-molecule and molecule-wall) are perfectly elastic
4. Molecules are in continuous, random motion obeying Newton's laws
5. The gas obeys $PV=nRT$ under all conditions

Real gases approach ideal behaviour at low pressures and high temperatures.

10.3 Ideal Gas Equation

$PV=nRT$

Where:

• $P$ = pressure, $V$ = volume, $n$ = number of moles
• $R=8.314$ J/(mol⋅K) — universal gas constant
• $T$ = absolute temperature in kelvin

In terms of the number of molecules $N$: $PV=N{k}_{B}T$

Where $k_B=\frac{R}{N_A}=1.38\times 10^{-23}$ J/K (Boltzmann constant).

10.4 Pressure Exerted by a Gas

From kinetic theory, the pressure exerted by an ideal gas is:

$P=\frac{1}{3}\rho {\bar{c}}^2$

Where:

• $\rho$ = density of the gas
• ${\bar{c}}^2$ = mean square speed of molecules

Key implications:

• $P\propto \rho$ (at constant temperature, higher density → higher pressure)
• $P\propto {\bar{c}}^2$ (at constant density, faster molecules → higher pressure)
• Compressing a gas increases its pressure (increases density)
• Heating a gas increases its pressure (increases molecular speed)

Also: $P=\frac{1}{3}\frac{mN}{V}{\bar{c}}^2$

10.5 Kinetic Interpretation of Temperature

The average kinetic energy per molecule of an ideal gas:

$\frac{1}{2}m{\bar{c}}^2=\frac{3}{2}{k}_{B}T$

• Temperature is a measure of the average kinetic energy of molecules
• At $T=0$ K, molecular motion theoretically ceases

RMS speed: $c_{\text{rms}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3k_{B}T}{m}}$

10.6 Degrees of Freedom

Degrees of freedom ($f$) are the independent ways a molecule can store energy:

10.7 Law of Equipartition of Energy

Each degree of freedom contributes $\frac{1}{2}{k}_{B}T$ to the average energy per molecule.

Total energy per molecule: $U=\frac{f}{2}{k}_{B}T$

Total energy per mole: $U=\frac{f}{2}RT$

10.8 Specific Heats of Gases

Molar specific heat at constant volume: $C_V=\frac{f}{2}R$

Molar specific heat at constant pressure: $C_P=C_V+R=\left(\frac{f}{2}+1\right)R$

Ratio of specific heats: $\gamma =\frac{C_P}{C_V}=1+\frac{2}{f}$

Specific heat of a substance: The heat required to raise the temperature of unit mass by 1°C (or 1 K). $C=\frac{\Delta Q}{m\times \Delta \theta }\text{[SI: J⋅kg⁻¹⋅K⁻¹]}$

10.9 Thermal Expansion

Coefficient of cubical expansion ($\gamma$): $\gamma =\frac{\Delta V}{V_0\times \Delta \theta }$

• SI unit: K⁻¹
• $V=V_0(1+\gamma \Delta \theta )$ (for small temperature changes)
• $\gamma \approx 3\alpha$ (where $\alpha$ is linear expansion coefficient)

INTEXT QUESTIONS 10.1

Q1. (i) A gas fills a container of any size but a liquid does not. Why?

Ans: In gases, intermolecular forces are extremely weak compared to kinetic energy. Gas molecules move randomly with high speeds and are separated by large distances. Since attractive forces are negligible, gas molecules spread out to fill any available space. In liquids, intermolecular forces are significantly stronger. Liquid molecules stay close together due to these attractive forces — they can flow but maintain a definite volume.

(ii) Solids have more ordered structure than gases. Why?

Ans: In solids, intermolecular forces dominate over kinetic energy. Molecules have very low kinetic energy and are held in fixed positions by strong intermolecular attractions, resulting in a highly ordered, rigid structure. In gases, kinetic energy dominates over intermolecular forces — high kinetic energy allows molecules to move freely in all directions, resulting in a completely disordered structure.

Q2. What is an ideal gas?

Ans: An ideal gas is a theoretical gas that follows the kinetic molecular theory perfectly:

• Gas molecules behave as point masses with negligible volume
• No intermolecular forces exist between molecules except during collisions
• All collisions are perfectly elastic
• Molecules are in continuous random motion
• The gas obeys $PV=nRT$ under all conditions

Real gases behave like ideal gases at low pressures and high temperatures.

Q3. How is pressure related to density of molecules?

Ans: From kinetic theory: $P=\frac{1}{3}\rho {\bar{c}}^2$

Where $P$ = pressure, $\rho$ = density, ${\bar{c}}^2$ = mean square speed of molecules.

This shows:

• Pressure is directly proportional to density of the gas
• Pressure is directly proportional to mean square speed of molecules
• At constant temperature, higher density → higher pressure (compression)
• At constant density, higher molecular speed → higher pressure (heating)

Q4. What is meant by specific heat of a substance?

Ans: Specific heat capacity is the amount of heat required to raise the temperature of unit mass of a substance by 1°C (or 1 K).

$C=\frac{\Delta Q}{m\times \Delta \theta }$

• SI unit: J⋅kg⁻¹⋅K⁻¹
• It is an intensive property (independent of amount)
• Different substances have different specific heats
• Water has one of the highest: 4184 J⋅kg⁻¹⋅K⁻¹
• For gases, $C_P$ and $C_V$ differ because the amount of heat required depends on the process.

Q5. Define coefficient of cubical expansion.

Ans: The coefficient of cubical expansion ($\gamma$) is the fractional change in volume per degree change in temperature.

$\gamma =\frac{\Delta V}{V_0\times \Delta \theta }$

• SI unit: K⁻¹ or °C⁻¹
• For small temperature changes: $V=V_0(1+\gamma \Delta \theta )$
• Related to linear expansion: $\gamma =3\alpha$

Terminal Exercise

1. State the postulates of the kinetic theory of gases.

2. Derive the expression for pressure exerted by an ideal gas: $P=\frac{1}{3}\rho {\bar{c}}^2$.

3. Using kinetic theory, derive the ideal gas equation $PV=nRT$.

4. Show that the average kinetic energy per molecule of an ideal gas is $\frac{3}{2}{k}_{B}T$.

5. Explain the concept of degrees of freedom. Find the degrees of freedom for monatomic, diatomic, and polyatomic gases.

6. State and explain the law of equipartition of energy.

7. Using equipartition, derive $C_V=\frac{3}{2}R$, $\frac{5}{2}R$, and $3R$ for monatomic, diatomic, and polyatomic gases respectively.

8. Define $C_P$ and $C_V$. Show that $C_P-C_V=R$ (Mayer's relation).

9. Calculate the RMS speed of oxygen molecules at 27°C. ($M_{{\text{O}}_2}=32$ g/mol, $R=8.31$ J/mol⋅K)

10. A gas occupies 2 L at 27°C and 1 atm pressure. If it is heated to 127°C at constant pressure, find its new volume.

11. Explain why: (a) $C_P>C_V$ for gases, (b) $\gamma$ is maximum for monatomic gases.

12. The density of a gas at NTP is 1.25 kg/m³. If the RMS speed of molecules is 500 m/s, calculate the pressure of the gas.

Worked Examples

Example 1: RMS Speed

Problem: Find the RMS speed of nitrogen molecules at 0°C. ($M=28$ g/mol, $R=8.31$ J/mol⋅K)

Solution: $c_{\text{rms}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3\times 8.31\times 273}{0.028}}$ $=\sqrt{\frac{6805.89}{0.028}}=\sqrt{243,067}\approx 493\text{ m/s}$

Example 2: Kinetic Energy

Problem: Find the average kinetic energy of a helium atom at 300 K. ($k_B=1.38\times 10^{-23}$ J/K)

Solution: For monatomic gas ($f=3$): $K{E}_{\text{avg}}=\frac{3}{2}{k}_{B}T=\frac{3}{2}\times 1.38\times 10^{-23}\times 300$ $=6.21\times 10^{-21}\text{ J}$

Example 3: Specific Heats

Problem: For a diatomic gas, find $C_V$, $C_P$, and $\gamma$. ($R=8.31$ J/mol⋅K)

Solution: For diatomic gas, $f=5$: $C_V=\frac{5}{2}R=\frac{5}{2}\times 8.31=20.78\text{ J/mol⋅K}$ $C_P=C_V+R=20.78+8.31=29.09\text{ J/mol⋅K}$ $\gamma =\frac{C_P}{C_V}=\frac{7}{5}=1.40$

Common Mistakes

1. Confusing ideal and real gases: Ideal gas has zero molecular volume and no intermolecular forces — real gases approach this only at low P and high T.
2. Forgetting that $T$ in gas equations must be in kelvin: Always convert °C to K by adding 273.
3. Using RMS speed as average speed: RMS speed > average speed; $c_{\text{rms}}=\sqrt{\frac{3RT}{M}}$, while average speed $=\sqrt{\frac{8RT}{\pi M}}$.
4. Counting wrong degrees of freedom: Diatomic gases have 5 (not 6) at room temperature because vibration is not excited.
5. Mixing up $C_P$ and $C_V$: $C_P$ is always greater than $C_V$ because work is done against the atmosphere when heating at constant pressure.

Quick Revision