The Ideal Gas Equation

The ideal gas equation combines all three gas laws into a single equation and introduces the important constants R (the molar gas constant) and k (the Boltzmann constant). This lesson also covers the concept of the mole, the Avogadro constant, and the assumptions that define an ideal gas.

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The Equation of State for an Ideal Gas

The ideal gas equation can be written in two forms:

pV = nRT (using moles)

pV = NkT (using number of molecules)

where:

• p = pressure (Pa)
• V = volume (m³)
• n = number of moles (mol)
• R = molar gas constant = 8.31 J mol⁻¹ K⁻¹
• T = absolute temperature (K)
• N = number of molecules
• k = Boltzmann constant = 1.38 × 10⁻²³ J K⁻¹

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Moles, Molecules, and the Avogadro Constant

A mole is the SI unit for amount of substance. One mole of any substance contains exactly 6.022 × 10²³ particles (atoms, molecules, ions, etc.). This number is called the Avogadro constant (Nₐ).

Nₐ = 6.022 × 10²³ mol⁻¹

The relationship between the number of moles (n) and the number of molecules (N) is:

N = nNₐ

The molar mass (M) is the mass of one mole of a substance, measured in kg mol⁻¹ (or g mol⁻¹ in chemistry). For example:

• Hydrogen (H₂): M = 2.0 × 10⁻³ kg mol⁻¹
• Helium (He): M = 4.0 × 10⁻³ kg mol⁻¹
• Nitrogen (N₂): M = 28 × 10⁻³ kg mol⁻¹
• Oxygen (O₂): M = 32 × 10⁻³ kg mol⁻¹

The number of moles is:

n = m/M (mass divided by molar mass)

The mass of a single molecule is:

m_molecule = M/Nₐ

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Relationship Between R and k

The Boltzmann constant is the gas constant per molecule:

k = R/Nₐ = 8.31 / (6.022 × 10²³) = 1.38 × 10⁻²³ J K⁻¹

This can be verified:

pV = nRT = (N/Nₐ)RT = N(R/Nₐ)T = NkT ✓

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Assumptions of an Ideal Gas

An ideal gas is a theoretical model. The assumptions are:

1. The gas consists of a very large number of identical molecules in constant random motion.
2. The volume of the molecules themselves is negligible compared to the volume of the container.
3. There are no intermolecular forces between the molecules (except during collisions).
4. Collisions between molecules and with the container walls are perfectly elastic (kinetic energy is conserved).
5. The duration of a collision is negligible compared to the time between collisions.
6. The molecules obey Newton's laws of motion.

Exam Tip: You must be able to state the assumptions of an ideal gas. Examiners typically award one mark per distinct assumption, up to a maximum of 3–4 marks. The most commonly missed assumptions are the negligible volume of molecules and the negligible collision duration.

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Worked Examples

Worked Example 1 — Using pV = nRT

Question: Calculate the volume occupied by 2.0 mol of an ideal gas at a temperature of 300 K and a pressure of 1.0 × 10⁵ Pa.

Solution:

V = nRT/p = (2.0 × 8.31 × 300) / (1.0 × 10⁵)

V = 4986 / 100 000 = 0.0499 m³ ≈ 0.050 m³ (50 litres)

Worked Example 2 — Using pV = NkT

Question: A container holds 5.0 × 10²⁴ molecules of nitrogen gas at 400 K and a pressure of 2.0 × 10⁵ Pa. Calculate the volume of the container.

Solution:

V = NkT/p = (5.0 × 10²⁴ × 1.38 × 10⁻²³ × 400) / (2.0 × 10⁵)