Class 12 > Unit # 15:Molecular Theory of Gases > Gas Laws

Q.3 State Boyle’s Law, Charles’s Law, and Avogadro’s Law. Derive the General Gas Equation using these gas laws.

1. Boyle’s Law

Statement: The volume of a given mass of an ideal gas is inversely proportional to its applied pressure, provided the temperature remains constant.

Mathematically, if $P$ represents the pressure exerted on the gas and $V$ represents its volume:

$$V \propto \frac{1}{P} \quad \text{(at constant } T\text{)}$$ $$V = \frac{\text{constant}}{P} \implies PV = \text{constant}$$

If a gas at a constant temperature undergoes a change from an initial state $(P_1, V_1)$ to a final state $(P_2, V_2)$:

$$P_1V_1 = P_2V_2$$

2. Charles’s Law

Statement: The volume of a given mass of an ideal gas is directly proportional to its absolute temperature, provided the pressure is kept constant.

Mathematically, if $V$ is the volume and $T$ is the absolute temperature (in Kelvin):

$$V \propto T \quad \text{(at constant } P\text{)}$$ $$V = \text{constant} \cdot T \implies \frac{V}{T} = \text{constant}$$

If a gas at constant pressure changes from an initial state $(V_1, T_1)$ to a final state $(V_2, T_2)$:

$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$

3. Avogadro’s Law

Statement: The volume of a given mass of an ideal gas is directly proportional to the number of moles of the gas, provided the temperature and pressure are kept constant.

Mathematically, if $V$ is the volume and $n$ represents the number of moles of gas:

$$V \propto n \quad \text{(at constant } P \text{ and } T\text{)}$$ $$V = \text{constant} \cdot n \implies \frac{V}{n} = \text{constant}$$

For two states at equal temperature and pressure conditions:

$$\frac{V_1}{n_1} = \frac{V_2}{n_2}$$

Derivation of the General Gas Law

To establish a single relation governing any ideal gas, we combine the three fundamental expressions derived above:

• According to Boyle's Law: $V \propto \frac{1}{P}$  — (i)
• According to Charles's Law: $V \propto T$   — (ii)
• According to Avogadro's Law: $V \propto n$  — (iii)

Combining equations (i), (ii), and (iii) into a unified joint proportionality statement:

$$V \propto \frac{n \cdot T}{P}$$

To eliminate the proportionality sign, we introduce a constant of proportionality, $R$, known as the Universal or General Gas Constant:

$$V = R \left(\frac{nT}{P}\right)$$ $$PV = nRT$$

Alternatively, the equation can be rearranged for checking variable changes across multiple states:

$$\frac{PV}{nT} = R \implies \frac{P_1V_1}{n_1T_1} = \frac{P_2V_2}{n_2T_2}$$

The Universal Gas Constant (R): In Standard International (S.I.) units, the value of $R$ for an ideal gas is: $$R = 8.314 \text{ J / (mol}\cdot\text{K)}$$ This equation is fundamental to the macroscopic study of gases and is called the General Gas Equation or the Ideal Gas Law.