CAIE A-Level Physics A2: Kinetic Theory of Gases

[Maximum number: 6]

(a)

The kinetic theory of gases is based on a number of assumptions about the molecules of a gas.

State the assumption that is related to the volume of the molecules of the gas.

[ 2 ]

(b)

An ideal gas occupies a volume of $2.40 \times 10^{-2} \mathrm{~m}^{3}$ at a pressure of $4.60 \times 10^{5} \mathrm{~Pa}$ and a temperature of $23^{\circ} \mathrm{C}$ .

[ 3 ]

(i)

Each molecule has a diameter of approximately $3 \times 10^{-10} \mathrm{~m}$ .

Estimate the total volume of the gas molecules.

volume =m³

[ 3 ]

volume of one atom $=d^{3}\left(=2.7 \times 10^{-29} \mathrm{~m}^{3}\right)$
volume of all atoms $=2.7 \times 10^{-29} \times 2.7 \times 10^{24}$ $=7 \times 10^{-5} \mathrm{~m}^{3}$
or

volume of one atom $=(4 / 3) \pi r^{3}\left(=1.41 \times 10^{-29} \mathrm{~m}^{3}\right)$
volume of all atoms $=2.7 \times 10^{24} \times 1.41 \times 10^{-29}$ $=4 \times 10^{-5} \mathrm{~m}^{3}$

(c)

By reference to your answer in (b)(ii), suggest why the assumption in (a) is justified.

[ 1 ]

numerical comparison between answer to (b)(ii) and $2.4 \times 10^{-2}\left(\mathrm{~m}^{3}\right)$ showing (b)(ii) is much less than $2.4 \times 10^{-2}\left(\mathrm{~m}^{3}\right)$

[Maximum number: 10]

(a)

One of the assumptions of the kinetic theory of gases is that all collisions involving molecules of the gas are elastic.

[ 2 ]

(i)

State two other assumptions of the kinetic theory of gases.
1.
2.

[ 2 ]

(b)

A molecule of an ideal gas has mass m and is contained in a cubic box of side length L. The molecule is moving with velocity u towards the face of the box that is shaded in Fig. 3.1.

Fig. 3.1

The molecule collides elastically with the shaded face and the face opposite to it alternately.
Deduce expressions, in terms of m, u and L, for:

[ 4 ]

(i)

the magnitude of the change in momentum of the molecule on colliding with a face

change in momentum =

[ 1 ]

(ii)

the time between consecutive collisions of the molecule with the shaded face
time =

[ 1 ]

(iii)

the average force exerted by the molecule on the shaded face

force =

[ 1 ]

force = change in momentum / time = 2 m u /(2 L / u)
$=m u^{2} / L$

(iv)

the pressure on the shaded face if the force in (iii) is exerted over the whole area of the face.

pressure =

[ 1 ]

$\begin{aligned} \text { pressure } =\text { force } / \text { area }=\left(m u^{2} / L\right) / L^{2} =m u^{2} / L^{3} \end{aligned}$

(c)

When the model described in (b) is extended to three dimensions, and to a gas containing N molecules, each of mass m, travelling with mean-square speed $\left\langle c^{2}\right\rangle$ , it can be shown that

p V=\frac{1}{3} N m\left\langle c^{2}\right\rangle

where p is the pressure exerted by the gas and V is the volume of the gas.
Use this expression, together with the equation of state of an ideal gas, to show that the average translational kinetic energy $E_{\mathrm{K}}$ of a molecule of an ideal gas is given by

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

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

[ 2 ]

p V=N k T
$N k T=\frac{1}{3} N m\left\langle c^{2}\right\rangle$ leading to $1 / 2 m\left\langle c^{2}\right\rangle=(3 / 2) k T$ and $1 / 2 m\left\langle c^{2}\right\rangle=E_{\mathrm{K}}$

(d)

The mass of a hydrogen molecule is $3.34 \times 10^{-27} \mathrm{~kg}$ .

Use the expression for $E_{\mathrm{K}}$ in (c) to determine the root-mean-square (r.m.s.) speed of a molecule of hydrogen gas at $25^{\circ} \mathrm{C}$ .

r.m.s. speed =

\mathrm{m} \mathrm{~s}^{-1}

[ 2 ]

$1 / 2 \times 3.34 \times 10^{-27} \times\left\langle c^{2}\right\rangle=(3 / 2) \times 1.38 \times 10^{-23} \times(25+273)$
r.m.s. speed $=1.9 \times 10^{3} \mathrm{~m} \mathrm{~s}^{-1}$

[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 ]

$\frac{1}{2} m\left\langle c^{2}\right\rangle=(3 / 2) k T$ $3.34 \times 10^{-27} \times v^{2}=3 \times 1.38 \times 10^{-23} \times 500$ $v=2490 \mathrm{~m} \mathrm{~s}^{-1}$
or

$p V=1 / 3(N m)\left\langle c^{2}\right\rangle$ and N m= mass of gas
$0.0424 \times v^{2}=3 \times 1.37 \times 10^{5} \times 0.640$ $v=2490 \mathrm{~m} \mathrm{~s}^{-1}$

(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 ]

[Maximum number: 3]

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:

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

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

$\frac{1}{2} m\left\langle c^{2}\right\rangle=(3 / 2) k T$ $3.34 \times 10^{-27} \times v^{2}=3 \times 1.38 \times 10^{-23} \times 500$ $v=2490 \mathrm{~m} \mathrm{~s}^{-1}$
or

$p V=1 / 3(N m)\left\langle c^{2}\right\rangle$ and N m= mass of gas
$0.0424 \times v^{2}=3 \times 1.37 \times 10^{5} \times 0.640$ $v=2490 \mathrm{~m} \mathrm{~s}^{-1}$

[Maximum number: 2]

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:

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]

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

CAIE A-Level Physics A2 15.3 Kinetic Theory of Gases Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

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Question (b)

Question (i)

Question (c)

Question 2

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Question (b)

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Question (ii)

Question (iii)

Question (iv)

Question (c)

Question (d)

Question 3

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Question 5