Ideal gases: (g) the Boltzmann constant; k = R / NA

The idea

The Boltzmann constant k relates the macroscopic molar gas constant R to the number of particles in one mole, the Avogadro constant NA. Because R = 8.31 J mol⁻¹ K⁻¹ applies to a mole of particles, dividing by NA = 6.02 × 10²³ mol⁻¹ gives the gas constant per single particle: k = R / NA ≈ 1.38 × 10⁻²³ J K⁻¹. This lets you write gas equations per molecule rather than per mole, for example pV = NkT, where N counts molecules. Check units: J mol⁻¹ K⁻¹ divided by mol⁻¹ leaves J K⁻¹, the unit of k. In calculations, use k when the question gives a number of molecules or atoms, and R when it gives an amount in moles.

What earns the marks

• States k = R / NA and identifies k as the Boltzmann constant, the gas constant per particle.
• Uses R = 8.31 J mol⁻¹ K⁻¹ and NA = 6.02 × 10²³ mol⁻¹ to obtain k ≈ 1.38 × 10⁻²³ J K⁻¹.
• Explains that dividing the molar constant by Avogadro's number converts a per-mole quantity into a per-particle quantity.
• Applies k in particle-based equations such as pV = NkT, distinguishing N (number of molecules) from n (number of moles).
• Checks the unit: J mol⁻¹ K⁻¹ ÷ mol⁻¹ = J K⁻¹, consistent with energy per kelvin per particle.

Common mistakes and how to put them right

• Multiplying R by NA instead of dividing.

  Put it right: k = R / NA, which gives a much smaller value.

• Confusing N (number of molecules) with n (number of moles).

  Put it right: Use N with k and n with R.

• Treating k as having units J mol⁻¹ K⁻¹.

  Put it right: k has units J K⁻¹ because it is a per-particle constant.

• Using NA = 6.02 × 10²³ without its unit mol⁻¹.

  Put it right: Include mol⁻¹ so the division cancels correctly.

Exam tips

• Write the defining equation k = R / NA before substituting numbers, so the method is visible.
• Decide from the wording whether the question supplies moles or molecules, then choose R or k accordingly.
• Carry the unit mol⁻¹ with NA and cancel units explicitly to confirm J K⁻¹.
• Keep the value of k to the precision given in the data sheet rather than rounding early.