Ideal gas equation and the universal gas constant
Combining the gas laws gives the ideal gas equation PV = μRT, where μ is the number of moles and R = 8.31 J mol⁻¹ K⁻¹.
Ideal and real gases
An ideal gas is a gas in which the atoms or molecules do not attract or repel one another. It obeys the gas laws exactly. Real gases, such as air, oxygen and hydrogen, have small forces between their molecules. They behave almost like an ideal gas at low pressure and high temperature. Then the molecules are far apart, so these forces hardly matter.
Deriving the ideal gas equation
- 1Boyle's law: V ∝ 1/P, at constant T and n.
- 2Charles's law: V ∝ T, at constant P and n.
- 3Avogadro's law: V ∝ n, at constant P and T, where n is the number of atoms or molecules.
- 4Combining: V ∝ nT/P, so PV ∝ nT. Writing the constant as k: PV = nkT. Here k is the Boltzmann constant, 1.381 × 10⁻²³ J K⁻¹.
- 5The number of molecules n = μN. Here μ is the number of moles and N is the Avogadro number, 6.023 × 10²³ per mole. So PV = μNkT.
- 6★Put R = N × k. Then PV = μRT. This is the ideal gas equation, also called the equation of state of a gas.
★ key step
The universal gas constant
R = N × k ≈ 6.023 × 10²³ × 1.38 × 10⁻²³ ≈ 8.31 J mol⁻¹ K⁻¹ (with k rounded to 1.38 × 10⁻²³ J K⁻¹). R is called universal because it has the same value for all gases. For one mole of any gas (μ = 1), the equation becomes PV = RT.
Volume of one mole of a gas
Find the volume of 1 mole of an ideal gas at 0 °C and a pressure of 1.013 × 10⁵ Pa. (R = 8.31 J mol⁻¹ K⁻¹)
- Given: μ = 1 mol, T = 0 + 273 = 273 K, P = 1.013 × 10⁵ Pa.
- Ideal gas equation: PV = μRT, so V = μRT/P.
- V = 1 × 8.31 × 273 ÷ (1.013 × 10⁵) m³.
- 8.31 × 273 = 2268.6, and 2268.6 ÷ 101300 = 0.0224 m³ (rounded).
V ≈ 0.0224 m³ = 22.4 litres. This is the molar volume of a gas at STP (0 °C and normal pressure).
Number of moles in a cylinder
A cylinder of volume 0.0831 m³ holds a gas at a pressure of 3 × 10⁵ Pa and a temperature of 300 K. How many moles of gas does it contain? (R = 8.31 J mol⁻¹ K⁻¹)
- Given: V = 0.0831 m³, P = 3 × 10⁵ Pa, T = 300 K.
- From PV = μRT: μ = PV/(RT).
- PV = 3 × 10⁵ × 0.0831 = 24930 J. RT = 8.31 × 300 = 2493 J mol⁻¹.
- μ = 24930 ÷ 2493 = 10 mol.
The cylinder holds 10 moles of gas.
A cycle pump pushes more air into the tube (more moles in the same volume), so the pressure rises. Hospital oxygen cylinders are kept away from heat, because a rise in temperature raises the pressure in a fixed volume.
2 marks: “What is an ideal gas?” A gas whose atoms or molecules do not attract or repel one another. Real gases behave like it at low pressure and high temperature.
2 marks: “Why is R called the universal gas constant? Give its value.” It has the same value for all gases; R = 8.31 J mol⁻¹ K⁻¹.
7 marks: “Derive the ideal gas equation from the gas laws. Find the volume of 1 mole of a gas at 273 K and 1.013 × 10⁵ Pa.” V ∝ 1/P, V ∝ T, V ∝ n; PV = nkT; n = μN; R = Nk; PV = μRT; V = 8.31 × 273 ÷ (1.013 × 10⁵) ≈ 0.0224 m³ = 22.4 L.
Boyle + Charles + Avogadro = PV = μRT. R = N × k links the mole to the single molecule.