The pressure p and volume V of an ideal gas are related to the density ρ of the gas by the expression p = ¹/₃ρ(c²).

1. (a)(i)State what is meant by the symbol 〈c²〉.[1]
2. (a)(ii)Use the expression to show that the mean kinetic energy Eₖ of a gas molecule is given by Eₖ = ³/₂ kT where k is the Boltzmann constant and T is the thermodynamic temperature.[3]
3. (b)(i)An ideal gas containing 1.0 mol of molecules is heated at constant volume. Use the expression in (a)(ii) to show that the thermal energy required to raise the temperature of the gas by 1.0 K has a value of ³/₂R, where R is the molar gas constant.[3]
4. (b)(ii)Nitrogen may be assumed to be an ideal gas. The molar mass of nitrogen gas is 28 g mol⁻¹. Use the answer in (b)(i) to calculate a value for the specific heat capacity, in J kg⁻¹ K⁻¹, at constant volume for nitrogen. specific heat capacity = ................................................... J kg⁻¹ K⁻¹ [2][2]