(a) The pressure p and volume V of an ideal gas are related to the density ρ of the gas by the expression

Q#6 (Past Exam Paper – June 2017 Paper 42 Q2)

(a) The pressure p and volume V of an ideal gas are related to the density ρ of the gas by the expression

p = 1/3 ρ〈c²〉.

(i) State what is meant by the symbol 〈c²〉. [1]

(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]

(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 3/2 R, where R is the molar gas constant. [3]

(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. [2]

[Total: 9]

Solution:

(a)

(i) It is the mean square speed.

(ii)

{Ideal gas equation:}

pV = NkT or pV = nRT

{Density = mass / volume = Nm / V

Where N is the number of molecules and m is the mass of 1 molecule}

ρ = Nm / V

or

ρ  = nNAm / V and k = nR / N

{Substituting the formula for density in the expression in part (a) gives:

p = 1/3 Nm <c²> / V

So, pV = 1/3 Nm <c²>

But (also,) pV = NkT

So,

1/3 Nm <c²> = NkT

m <c²> = 3kT

Multiply by ½ ,

½ m <c²> = 3/2 kT

E_K = 3/2 kT}

(b)

(i)

{ΔU = q + w

The volume is constant, so there is no change in volume

External work done, w = pΔV = 0

So, ΔU = q}

no (external) work done or ΔU = q or w = 0

{The thermal energy q required is equal to the change in internal energy.

Internal energy = kinetic energy + potential energy

For an ideal gas, potential energy = 0

So, internal energy = kinetic energy

KE of one molecule = 3/2 kT

1 mole contains N_A (Avogadro constant) molecules.

KE of 1 mole of molecules = N_A × 3/2 kT}

q = NA × (3 / 2)k × 1.0

{since NA × k = R,}

NAk = R so, q = (3 / 2)R

(ii)

{s.h.c = heat capacity / mass

Heat capacity is the heat required to raise the temperature by 1 K. So, this is the energy calculated above.

Mass of 1 mole of nitrogen = 0.028 kg}

specific heat capacity = {(3 / 2) × R} / 0.028

s.h.c. = 450 J kg^-1 K^-1

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