B3-3. The Ideal Gas Equations

1. The Ideal Gas Law

By combining Boyle's, Charles's, and Gay-Lussac's laws, we get a single equation of state that relates all macroscopic variables of an ideal gas simultaneously.

Macroscopic Form (Using Moles):

$$pV = nRT$$
  • $p$: Pressure (Pa)
  • $V$: Volume (m³)
  • $n$: Number of moles (mol)
  • $R$: Ideal Gas Constant ($8.31$ J K⁻¹ mol⁻¹)
  • $T$: Absolute Temperature (K)

Microscopic Form (Using Number of Molecules):

By substituting $n = \dfrac{N}{N_A}$ and recognising that $\dfrac{R}{N_A}$ is the Boltzmann constant ($k_B$), the equation can be rewritten in terms of the exact number of molecules.

$$pV = N k_B T$$
  • $N$: Total number of molecules
  • $k_B$: Boltzmann Constant ($1.38 \times 10^{-23}$ J K⁻¹)

Example 1: Ideal Gas Equation

Problem: Calculate the volume occupied by $0.50$ moles of an ideal gas at a pressure of $1.01 \times 10^5$ Pa and a temperature of $0^\circ\text{C}$.

Solution:

  • Step 1: Convert variables to SI units.
    $T = 0 + 273.15 = 273.15$ K
  • Step 2: Rearrange the macroscopic equation for $V$.
    $$pV = nRT \implies V = \dfrac{nRT}{p}$$
  • Step 3: Calculate.
    $$V = \dfrac{0.50 \times 8.31 \times 273.15}{1.01 \times 10^5} \approx \mathbf{0.0112 \text{ m³}}$$