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Monday, July 22, 2019

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


Question 6
(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 ρc2.

(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]





Reference: Past Exam Paper – June 2017 Paper 42 Q2





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
EK = 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 NA (Avogadro constant) molecules.
KE of 1 mole of molecules = NA × 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

2 comments:

  1. Thank you.

    However, I'd like to ask a question. Is "Heat Capacity" the amount of energy needed to raise the temperature of the whole gas sample by 1K? And not the energy needed to raise the temperature by 1K of 1 mole or 1 kg of the gas sample?

    ReplyDelete
    Replies
    1. yes. heat capacity is for the while gas

      specific heat capacity is for one kg.

      Delete

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