CAIE A-Level Physics 15.3 Kinetic Theory of Gases

Practise applying kinetic-theory assumptions, deriving gas-pressure and rms-speed relations and linking average molecular kinetic energy to temperature.

Exam points

state kinetic-theory assumptions and explain molecular collisions as the source of gas pressure
derive and use pV = 1/3 Nmc² and interpret root-mean-square speed
deduce and use average translational kinetic energy Ē = 3/2 kT from the ideal-gas and kinetic-theory relationships

Covered syllabus points

15.3.1—The basic assumptions of the kinetic theory of gases 15.3.2—How molecular movement causes the pressure exerted by a gas and derive and 15.3.3—The root-mean-square speed cr.m.s. is given by c<>2 15.3.4—PV = 1/3 Nm<c2> with pV = NkT to deduce that the average translational

[Maximum number: 4]

(a)

Use one of the basic assumptions of the kinetic theory to explain what can be deduced about the potential energy associated with the random motion of molecules in an ideal gas.

[ 2 ]

(b)

A sample of $0.26 \mathrm{~m}^{3}$ of an ideal gas is at pressure $2.0 \times 10^{5} \mathrm{~Pa}$ and temperature 290 K .

Determine:

[ 2 ]

(i)

the average translational kinetic energy $E_{\mathrm{K}}$ of one molecule of the gas

E_{\mathrm{K}}=\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \mathrm{J}[2]

[ 2 ]

$E_{\mathrm{K}}=(3 / 2) k T$

C1

$\begin{aligned} E_{K} =(3 / 2) \times 1.38 \times 10^{-23} \times 290 =6.0 \times 10^{-21} \mathrm{~J} \end{aligned}$

A1

[Maximum number: 7]

(a)

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

p=\frac{1}{3} \rho\left\langle c^{2}\right\rangle .

[ 4 ]

(i)

State what is meant by the symbol $\left\langle c^{2}\right\rangle$ .

[ 1 ]

(ii)

Use the expression to show that the mean kinetic energy $E_{\mathrm{K}}$ of a gas molecule is given by

E_{\mathrm{K}}=\frac{3}{2} k T

where k is the Boltzmann constant and T is the thermodynamic temperature.

[ 3 ]

p V=N k T or p V=n R T

B1

$\rho=N m / V$

or
$\rho=n N_{\mathrm{A}} m / V \text { and } k=n R / N$

B1

$E_{\mathrm{K}}=1 / 2 m\left\langle c^{2}\right\rangle$ with algebra to (3/2) k T

B1

(b)

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 $\frac{3}{2} R$ , where R is the molar gas constant.

[ 3 ]

no (external) work done or $\Delta U=q$ or w=0

B1

$q=N_{\mathrm{A}} \times(3 / 2) k \times 1.0$

M1

$N_{\mathrm{A}} k=R$ so q=(3 / 2) R

A1

[Maximum number: 5]

(a)

A sealed vessel contains a mass of 0.0424 kg of an ideal gas at $227^{\circ} \mathrm{C}$ . The pressure of the gas is $1.37 \times 10^{5} \mathrm{~Pa}$ and the volume of the gas is $0.640 \mathrm{~m}^{3}$ .

Calculate:

[ 3 ]

(i)

the root-mean-square (r.m.s.) speed v of the molecules of the gas.

\begin{aligned} & v= \\ & \mathrm{ms}^{-1} \end{aligned}

[ 3 ]

$\frac{1}{2} m\left\langle c^{2}\right\rangle=(3 / 2) k T$

C1

$3.34 \times 10^{-27} \times v^{2}=3 \times 1.38 \times 10^{-23} \times 500$

C1

$v=2490 \mathrm{~m} \mathrm{~s}^{-1}$

A1

or

$p V=1 / 3(N m)\left\langle c^{2}\right\rangle$ and N m= mass of gas

(C1)

$0.0424 \times v^{2}=3 \times 1.37 \times 10^{5} \times 0.640$

(C1)

$v=2490 \mathrm{~m} \mathrm{~s}^{-1}$

(A1)

(b)

The gas in (b) is now cooled gradually to absolute zero.

On Fig. 2.1, sketch the variation with thermodynamic temperature T of the r.m.s. speed of the molecules of the gas.

Fig. 2.1

[ 2 ]

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Covered syllabus points

15.3 Kinetic theory of gases question 1

Question (a)

Question (b)

Question (i)

15.3 Kinetic theory of gases question 2

Question (a)

Question (i)

Question (ii)

Question (b)

15.3 Kinetic theory of gases question 3

Question (a)

Question (i)

Question (b)