Gas laws — Boyle, Charles, Gay-Lussac, Avogadro, ideal gas equation connection

Question

State Boyle’s law, Charles’s law, Gay-Lussac’s law, and Avogadro’s law. Show how all four combine to give the ideal gas equation PV = nRT. How do you select the right gas law for a given problem?

(CBSE 11 + JEE Main + NEET — derivation + application)

Solution — Step by Step

Law	Statement	Mathematical Form	Constant
Boyle’s	At constant T, P is inversely proportional to V	$P_1V_1 = P_2V_2$	T, n
Charles’s	At constant P, V is directly proportional to T	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	P, n
Gay-Lussac’s	At constant V, P is directly proportional to T	$\frac{P_1}{T_1} = \frac{P_2}{T_2}$	V, n
Avogadro’s	At constant T and P, V is proportional to n (moles)	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	T, P

Temperature must always be in Kelvin for gas law calculations.

Combining all four:

From Boyle: $V \propto \frac{1}{P}$ (at constant T, n)

From Charles: $V \propto T$ (at constant P, n)

From Avogadro: $V \propto n$ (at constant T, P)

Combining: $V \propto \frac{nT}{P}$

V = \frac{nRT}{P} \quad \Rightarrow \quad \boxed{PV = nRT}

where $R = 8.314$ J mol⁻¹ K⁻¹ (or $0.0821$ L atm mol⁻¹ K⁻¹).

Look at what is constant in the problem:

T and n constant → Boyle’s law (P and V change)
P and n constant → Charles’s law (V and T change)
V and n constant → Gay-Lussac’s law (P and T change)
All four variables changing → use ideal gas equation directly
Comparing two gas samples → use the combined gas law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

A gas occupies 2 L at 300 K and 1 atm. What volume will it occupy at 600 K and 2 atm?

Using combined gas law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

\frac{1 \times 2}{300} = \frac{2 \times V_2}{600}

V_2 = \frac{1 \times 2 \times 600}{300 \times 2} = \mathbf{2 \text{ L}}

graph TD
    A["Gas Problem"] --> B{"What is constant?"}
    B -->|"T, n constant"| C["Boyle: P1V1 = P2V2"]
    B -->|"P, n constant"| D["Charles: V1/T1 = V2/T2"]
    B -->|"V, n constant"| E["Gay-Lussac: P1/T1 = P2/T2"]
    B -->|"Everything changes"| F["PV = nRT or Combined Law"]
    C --> G["P and V are inversely related"]
    D --> H["V and T are directly related"]
    style A fill:#fbbf24,stroke:#000,stroke-width:2px
    style F fill:#86efac,stroke:#000

Why This Works

Each gas law is a special case of the ideal gas equation with certain variables held constant. The ideal gas equation $PV = nRT$ is the master equation that encompasses all four laws. Once you know it, you can derive any individual law by holding the appropriate variables constant.

The gas laws work because gas molecules are far apart and interact weakly — so pressure, volume, and temperature have simple mathematical relationships. These relationships break down at high pressures and low temperatures (where real gas behavior kicks in).

Common Mistake

The number one error: using Celsius instead of Kelvin in gas law calculations. Charles’s law and Gay-Lussac’s law require absolute temperature (Kelvin). Using Celsius gives completely wrong answers. Always convert: $T(K) = T(°C) + 273.15$ . This mistake alone costs students easy marks in both JEE and NEET.

At STP (Standard Temperature and Pressure: 273.15 K, 1 atm), 1 mole of any ideal gas occupies 22.4 L. This is a direct consequence of $PV = nRT$ and is one of the most useful numbers to memorise for quick calculations in gas law problems.

Question

Solution — Step by Step

Why This Works

Common Mistake

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