recall that the Boltzmann constant k is given by k = R / NA

Equation of State

Learning Objective

Recall that the Boltzmann constant k is given by

$$k = \frac{R}{N_{\mathrm A}}$$

Key Concepts

• The ideal gas law relates pressure, volume, temperature and amount of gas: $$pV = nRT$$
• Pressure (p) is the force per unit area exerted by gas molecules on the container walls.
• Volume (V) is the space occupied by the gas.
• Temperature (T) must be in Kelvin for the ideal gas law.
• Amount of substance (n) is measured in moles.
• The universal gas constant R links macroscopic and microscopic descriptions of gases.
• The Avogadro constant N_A is the number of particles per mole.
• The Boltzmann constant k connects temperature to average kinetic energy of a single particle.

Deriving the Boltzmann Constant

Starting from the ideal gas law and substituting n = N/N_A (where N is the total number of particles), we obtain:

$$pV = \frac{N}{N_{\mathrm A}}RT$$

Rearranging gives the microscopic form:

$$pV = NkT$$

Comparing the two expressions shows that

$$k = \frac{R}{N_{\mathrm A}}$$

Important Constants

Constant	Symbol	Value (SI)	Units
Universal gas constant	R	8.314462618	J·mol⁻¹·K⁻¹
Avogadro constant	NA	6.02214076 × 10²³	mol⁻¹
Boltzmann constant	k	1.380649 × 10⁻²³	J·K⁻¹

Applications in A‑Level Physics

1. Calculating the average kinetic energy of a molecule: $$\langle E_{\text{kin}} \rangle = \frac{3}{2}kT$$
2. Deriving the root‑mean‑square speed of gas molecules: $$v_{\text{rms}} = \sqrt{\frac{3kT}{m}}$$ where m is the mass of a single molecule.
3. Understanding the relationship between macroscopic pressure and microscopic collisions.

Quick Revision Checklist

• Can you write the ideal gas law in both macroscopic (pV = nRT) and microscopic (pV = NkT) forms?
• Do you remember the numerical values and units of R, N_A and k?
• Can you derive k = R/N_A from the two forms of the ideal gas law?
• Are you able to use k to calculate average kinetic energy and rms speed?