Question 1
1
An ideal gas has volume V and pressure p. For this gas, the product p V is given by the expression where m is the mass of a molecule of the gas.
structured2 marks
Question 1(a)
1(a)
State the meaning of the symbol
structured2 marks
Question 1(a)(i)
1(a)(i)
N,
Mediumstructured1 marks
Answer
number of molecules B1 [1]
Question 1(a)(ii)
1(a)(ii)
\(\left\langle c^{2}\right\rangle\).
Mediumstructured1 marks
Answer
mean square speed B1 [1]
Question 1(b)
1(b)
A gas cylinder of volume \(2.1 \times 10^{4} \mathrm{~cm}^{3}\) contains helium-4 gas at pressure \(6.1 \times 10^{5} \mathrm{~Pa}\) and temperature \(12^{\circ} \mathrm{C}\). Helium-4 may be assumed to be an ideal gas.
structured0 marks
Question 1(b)(ii)
1(b)(ii)
Calculate the root-mean-square (r.m.s.) speed of the helium atoms.
Mediumstructured0 marks
Answer
either \(6.1 \times 10^{5} \times 2.1 \times 10^{-2}=\frac{1}{3} \times 3.25 \times 10^{24} \times 4 \times 1.66 \times 10^{-27} \times\left\langle c^{2}\right\rangle \quad\) C1 \(\left\langle c^{2}\right\rangle=1.78 \times 10^{6} \quad\) C1 \(c_{\text {RMS }}=1.33 \times 10^{3} \mathrm{~m} \mathrm{~s}^{-1} \quad\) A1 or \(\frac{1}{2} \times 4 \times 1.66 \times 10^{-27} \times\left\langle c^{2}\right\rangle=\frac{3}{2} \times 1.38 \times 10^{-23} \times 285\) \(\left\langle c^{2}\right\rangle=1.78 \times 10^{6}\) \(c_{\text {RMS }}=1.33 \times 10^{3} \mathrm{~m} \mathrm{~s}^{-1}\)
Question 1
1
The planet Mars may be considered to be an isolated sphere of diameter \(6.79 \times 10^{6} \mathrm{~m}\) with its mass of \(6.42 \times 10^{23} \mathrm{~kg}\) concentrated at its centre. A rock of mass 1.40 kg rests on the surface of Mars. For this rock,
structured5 marks
Question 1(c)
1(c)
The mean translational kinetic energy \(<E_{\mathrm{K}}>\) of a molecule of an ideal gas is given by the expression where T is the thermodynamic temperature of the gas and k is the Boltzmann constant.
structured5 marks
Question 1(c)(i)
1(c)(i)
Determine the temperature at which the root-mean-square (r.m.s.) speed of hydrogen molecules is equal to the speed calculated in (b). Hydrogen may be assumed to be an ideal gas. A molecule of hydrogen has a mass of 2 u . temperature =
Hardstructured3 marks
Answer
\(1 / 2 \times 2 \times 1.66 \times 10^{-27} \times\left(5.03 \times 10^{3}\right)^{2}=\frac{3}{2} \times 1.38 \times 10^{-23} \times T \quad\) C1 \(T=2030 \mathrm{~K}\) A1
Question 1(c)(ii)
1(c)(ii)
State and explain one reason why hydrogen molecules may escape from Mars at temperatures below that calculated in (i).
Mediumstructured2 marks
Answer
either because there is a range of speeds M1 some molecules have a higher speed A1 or some escape from point above planet surface (M1) so initial potential energy is higher (A1)
Question 2
2
electric potential energy. energy = J
structured3 marks
Question 2(b)
2(b)
The pressure p of an ideal gas is given by the expression where \(\rho\) is the density of the gas.
structured5 marks
Question 2(b)(i)
2(b)(i)
State the meaning of the symbol \(\left\langle c^{2}\right\rangle\).
Mediumstructured1 marks
Answer
\(\left(<c^{2}>\right.\) is the) mean / average square speed B1 [1]
Question 2(b)(ii)
2(b)(ii)
Use the expression to show that the mean kinetic energy \(<E_{\mathrm{K}}>\) of the atoms of an ideal gas is given by the expression Explain any symbols that you use.
Hardstructured4 marks
Answer
\(\rho=N m / V\) with N explained B1 so, \(p V=1 / 3 N m<c^{2}>\) B1 and p V=N k T with k explained B1 so mean kinetic energy \(/<E_{\mathrm{K}}>=1 / 2 m<c^{2}>=3 / 2 k T\) B1 [4]
Question 2(c)
2(c)
Helium-4 may be assumed to behave as an ideal gas. A cylinder has a constant volume of \(7.8 \times 10^{3} \mathrm{~cm}^{3}\) and contains helium-4 gas at a pressure of \(2.1 \times 10^{7} \mathrm{~Pa}\) and at a temperature of 290 K . Calculate, for the helium gas,
structured2 marks
Question 2(c)(ii)
2(c)(ii)
the mean kinetic energy of the atoms, mean kinetic energy = J
Mediumstructured2 marks
Answer
mean kinetic energy =3 / 2 kT
Question 2
2
In a sample of gas at room temperature, five atoms have the following speeds: For these five atoms, calculate, to three significant figures,
structured4 marks
Question 2(a)
2(a)
the mean speed, mean speed = \(\mathrm{ms}^{-1}[1]\)
Easystructured1 marks
Answer
mean speed \(=1.44 \times 10^{3} \mathrm{~m} \mathrm{~s}^{-1}\)
Question 2(b)
2(b)
the mean-square speed, mean-square speed = \(\mathrm{m}^{2} \mathrm{~s}^{-2}[2]\)
Easystructured2 marks
Answer
evidence of summing of individual squared speeds mean square speed \(=2.09 \times 10^{6} \mathrm{~m}^{2} \mathrm{~s}^{-2}\)
Question 2(c)
2(c)
the root-mean-square speed. root-mean-square speed = \(\mathrm{ms}^{-1}[1]\)
Easystructured1 marks
Answer
root-mean-square speed \(=1.45 \times 10^{3} \mathrm{~m} \mathrm{~s}^{-1}\) (allow ECF from (b) but only if arithmetic error)
Question 2
2
9 marks
Question 2(a)
2(a)
The kinetic theory of gases is based on some simplifying assumptions. The molecules of the gas are assumed to behave as hard elastic identical spheres. State the assumption about ideal gas molecules based on
structured3 marks
Question 2(a)(i)
2(a)(i)
the nature of their movement,
Easystructured1 marks
Answer
either random motion or constant velocity until hits wall/other molecule B1
Question 2(a)(ii)
2(a)(ii)
their volume.
Easystructured2 marks
Answer
(total) volume of molecules is negligible M1 compared to volume of containing vessel A1 or radius/diameter of a molecule is negligible (M1) compared to the average intermolecular distance (A1)
Question 2(b)
2(b)
A cube of volume V contains N molecules of an ideal gas. Each molecule has a component \(c_{\mathrm{X}}\) of velocity normal to one side S of the cube, as shown in Fig. 2.1. The pressure p of the gas due to the component \(c_{X}\) of velocity is given by the expression where m is the mass of a molecule. Explain how the expression leads to the relation where \(<c^{2}>\) is the mean square speed of the molecules.
Hardstructured3 marks
Answer
either molecule has component of velocity in three directions or \(\quad c^{2}=c_{X}{ }^{2}+c_{Y}{ }^{2}+c_{Z}{ }^{2} \quad\) M1 random motion and averaging, so \(\left\langle c_{\mathrm{X}}{ }^{2}\right\rangle=\left\langle c_{\mathrm{Y}}{ }^{2}\right\rangle=\left\langle c_{\mathrm{Z}}{ }^{2}\right\rangle \quad\) M1 \(\left\langle c^{2}\right\rangle=3\left\langle c_{x}^{2}\right\rangle\) A1 so, \(p V=1 / 3 N m<c^{2}>\) A0
Question 2(c)
2(c)
The molecules of an ideal gas have a root-mean-square (r.m.s.) speed of \(520 \mathrm{~m} \mathrm{~s}^{-1}\) at a temperature of \(27^{\circ} \mathrm{C}\). Calculate the r.m.s. speed of the molecules at a temperature of \(100^{\circ} \mathrm{C}\).
Mediumstructured3 marks
Answer
\(\left\langle c^{2}\right\rangle \propto T\) or \(c_{\text {rms }} \propto \sqrt{T} \quad\) C1 temperatures are 300 K and 373 K C1 \(c_{\text {rms }}=580 \mathrm{~m} \mathrm{~s}^{-1} \quad \mathrm{~A} 1\) (Do not allow any marks for use of temperature in units of \({ }^{\circ} \mathrm{C}\) instead of K )
Question 2
2
the period of Deimos in its orbit about Mars. period = hours
structured3 marks
Question 2(a)
2(a)
One assumption of the kinetic theory of gases is that gas molecules behave as if they are hard, elastic identical spheres. State two other assumptions of the kinetic theory of gases. 1. 2.
Easystructured2 marks
Answer
e.g. moving in random (rapid) motion of molecules/atoms/particles no intermolecular forces of attraction/repulsion volume of molecules/atoms/particles negligible compared to volume of container time of collision negligible to time between collisions (1 each, max 2)
Question 2(b)
2(b)
Using the kinetic theory of gases, it can be shown that the product of the pressure p and the volume V of an ideal gas is given by the expression where m is the mass of a gas molecule.
structured3 marks
Question 2(b)(i)
2(b)(i)
State the meaning of the symbol 1. N, 2. \(\left\langle c^{2}\right\rangle\).
Easystructured1 marks
Answer
1. number of (gas) molecules 2. mean square speed/velocity (of gas molecules)
Question 2(b)(ii)
2(b)(ii)
Use the expression to deduce that the mean kinetic energy \(<E_{\mathrm{K}}>\) of a gas molecule at temperature T is given by the equation where k is a constant.
Mediumstructured2 marks
Answer
either p V=N k T or p V=n R T and links n and k and \(\left\langle E_{\mathrm{K}}\right\rangle=1 / 2 m\left\langle c^{2}\right\rangle\) clear algebra leading to \(\left\langle E_{K}\right\rangle=\frac{3}{2} k T\)
Question 2
2
A student suggests that, when an ideal gas is heated from \(100^{\circ} \mathrm{C}\) to \(200^{\circ} \mathrm{C}\), the internal energy of the gas is doubled.
structured3 marks
Question 2(a)
2(a)
3 marks
Question 2(a)(ii)
2(a)(ii)
By reference to one of the assumptions of the kinetic theory of gases and your answer in (i), deduce what is meant by the internal energy of an ideal gas.
Mediumstructured3 marks
Answer
no intermolecular forces no potential energy internal energy is kinetic energy (of random motion) of molecules (reference to random motion here then allow back credit to (i) if M1 scored)
Question 2
2
The product of the pressure p and the volume V of an ideal gas is given by the expression where m is the mass of one molecule of the gas.
structured3 marks
Question 2(a)
2(a)
State the meaning of the symbol
structured1 marks
Question 2(a)(ii)
2(a)(ii)
\(\left\langle c^{2}\right\rangle\).
Mediumstructured1 marks
Answer
\(\left\langle c^{2}\right\rangle\) : mean square speed/velocity
Question 2(b)
2(b)
The product p V is also given by the expression Deduce an expression, in terms of the Boltzmann constant k and the thermodynamic temperature T, for the mean kinetic energy of a molecule of the ideal gas.
Mediumstructured2 marks
Answer
\(p V=1 / 3 N m\left\langle c^{2}\right\rangle=N k T\) (mean) kinetic energy \(=1 / 2 m<c^{2}>\) algebra clear leading to \(\frac{1}{2} m\left\langle c^{2}\right\rangle=(3 / 2) k T\)