Physics A Level
Chapter 20: Ideal gases 20.8 Temperature and molecular kinetic energy
Physics A Level
Chapter 20: Ideal gases 20.8 Temperature and molecular kinetic energy
Now we can compare the equation $p = \frac{1}{3}\left( {\frac{{Nm}}{v}} \right)\, \lt \,{c^2}\, \gt $ with the ideal gas equation$pV = nRT$. The lefthand sides are the same, so the two right-hand sides must also be equal:
$\frac{1}{3}\,Nm\, \lt \,{c^2}\, \gt \, = \,nRT$
We can use this equation to tell us how the absolute temperature of a gas (a macroscopic property) is related to the mass and speed of its molecules. If we focus on the quantities of interest, we can see the following relationship:
$m\, \lt \,{c^2}\, \gt \, = \frac{{3nRT}}{N}$
The quantity $\frac{M}{n} = {N_A}$ is the Avogadro constant, the number of particles in 1 mole. So:
$m\, \lt \,c\, \gt \, = \frac{{3RT}}{{{N_A}}}$
It is easier to make sense of this if we divide both sides by 2, to get the familiar expression for kinetic energy:
$\frac{1}{2}m\, \lt \,{c^2}\, \gt \, = \frac{{3RT}}{{2{N_A}}}$
The quantity $\frac{R}{{{N_A}}}$ is defined as the Boltzmann constant, k. Its value is $1.38 \times {10^{ - 23}}\,J\,{K^{ - 1}}$. Substituting k in place of $\frac{R}{{{N_A}}}$ gives
$kinetic\,energy = \frac{1}{2}m\, < \,c\, > \, = \frac{3}{2}kT$
This is the average kinetic energy E of a molecule in the gas, and since k is a constant, the thermodynamic temperature T is proportional to the average kinetic energy of a molecule.
The mean translational kinetic energy of an atom (or molecule) of an ideal gas is proportional to the thermodynamic temperature.
It is easier to recall this as:
$mean\,translational\,kinetic\,energy\,of\,atom\, \propto \,T$
We need to consider two of the terms in this statement. First, we talk about translational kinetic energy.
This is the energy that the molecule has because it is moving from one point in space to another; a molecule made of two or more atoms may also spin or tumble around, and is then said to have rotational kinetic energy – see Figure 20.9.
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Second, we talk about mean (or average) kinetic energy. There are two ways to find the average kinetic energy (k.e.) of a molecule of a gas. Add up all the kinetic energies of the individual molecules of the gas and then calculate the average k.e. per molecule. Alternatively, watch an individual molecule over a period of time as it moves about, colliding with other molecules and the walls of the container and calculate its average k.e. over this time. Both should give the same answer.
The Boltzmann constant is an important constant in physics because it tells us how a property of microscopic particles (the kinetic energy of gas molecules) is related to a macroscopic property of the gas (its absolute temperature). That is why its units are joules per kelvin ($J\,{K^{ - 1}}$). Its value is very small ($1.38 \times {10^{ - 23}}\,J\,{K^{ - 1}}$) because the increase in kinetic energy in J of a molecule is very small for each kelvin increase in temperature.
It is useful to remember the equation linking kinetic energy with temperature as ‘average k.e. is threehalves kT’.
Questions
15) The Boltzmann constant k is equal to $\frac{R}{{{N_A}}}$. From values of R and ${{N_A}}$, show that k has the value $1.38 \times {10^{ - 23}}\,J\,{K^{ - 1}}$.
16) Calculate the mean translational k.e. of atoms in an ideal gas at ${27^ \circ }C$.
17) The atoms in a gas have a mean translational k.e. equal to $5.0 \times {10^{ - 21}}\,J$. Calculate the temperature of the gas in K and in $^ \circ C$.
The root-mean-square speed
You may have wondered how the mean-square speed $ \lt \,{c^2}\, \gt $ compares with the mean speed $ \lt \,c\, \gt $.
The exact relationship depends on the distribution of the speeds of the molecules. If all the molecules have the same speed, then $ \lt \,c\, \gt \, = \sqrt { \lt \,{c^2}\, \gt } $.
But is this always the case?
Imagine three molecules with speeds 10, 20 and $30\,m\,{s^{ - 1}}$; (very low speeds for molecules, but easier for our calculations!).
$Their\,mean\,speed\, \lt \,c\, \gt \, = \frac{{(10 + 20 + 30)}}{3} = 20\,m\,{s^{ - 1}}$
Their square speeds are ${10^2}\,,\,{20^2}$ and ${30^2}$.
So, their mean-square speed
$ \lt \,{c^2}\, \gt \, = \frac{{({{10}^2} + {{20}^2} + {{30}^2})}}{3} = 467\,{m^2}\,{s^{ - 2}}$
In this case, $\sqrt { \lt \,{c^2}\, \gt } \, = 22\,m\,{s^{ - 1}}$ , which is not the same as the mean speed.
Similarly, the mean of the square of the speeds $ \lt \,{c^2}\, \gt \, = \,467\,{m^2}\,{s^{ - 2}}$ is not the same as the square of the mean of the speeds $( \lt \,{c^2}\, \gt )\, = \,400\,{m^2}\,{s^{ - 2}}$ in the example.
In general, the values for $ \lt \,c\, \gt $ and $\sqrt { \lt \,{c^2}\, \gt } $ are similar but, because they are not the same, we define a special quantity called the root-mean-square speed ${c_{r.m.s.}}$.
This is the square root of the mean-square-speed; that is:
${c_{r.m.s}} = \sqrt { \lt \,{c^2}\, \gt } $
In the example, for the three molecules, ${c_{r.m.s}} = 22\,m\,{s^{ - 1}}$.
Questions
18) Four molecules have speeds $200, 400, 600$ and $800\,m\,{s^{ - 1}}$. Calculate:
a: their mean speed $ \lt \,c\, \gt $
b: the square of their mean speed $ \lt \,c\,{ \gt ^2}$
c: their mean-square speed $ \lt \,{c^2}\, \gt $
d: their root-mean-square speed ${c_{r.m.s.}}$.
19) Calculate the root-mean square speed of the molecules of hydrogen at ${20^ \circ }C$ given that each molecule of hydrogen has mass $3.35 \times {10^{ - 27}}\,kg$.
Mass, kinetic energy and temperature
Since mean $k.e.\, \propto \,T$, it follows that if we double the thermodynamic temperature of an ideal gas (for example, from $300 K$ to $600 K$), we double the mean $k.e.$ of its molecules. It doesn’t follow that we have doubled their speed; because $k.e.\, \propto \,{v^2}$, their mean speed has increased by a factor of $\sqrt 2 $.
Air is a mixture of several gases: nitrogen, oxygen, carbon dioxide, etc. In a sample of air, the mean k.e. of the nitrogen molecules is the same as that of the oxygen molecules and that of the carbon dioxide molecules. This comes about because they are all repeatedly colliding with one another, sharing their energy. Carbon dioxide molecules have greater mass than oxygen molecules; since their mean translational k.e. is the same, it follows that the carbon dioxide molecules move more slowly than the oxygen molecules.
Questions
20) Show that, if the mean speed of the molecules in an ideal gas is doubled, the thermodynamic temperature of the gas increases by a factor of four.
21) A fixed mass of gas expands to twice its original volume at a constant temperature. How do the following change?
a: the pressure of the gas
b: the mean translational kinetic energy of its molecules.
22) Air consists of molecules of oxygen ($molar\,mass = 32\,g\,mo{l^{ - 1}}$) and nitrogen ($molar\,mass = 28\,g\,mo{l^{ - 1}}$). Calculate the mean translational k.e. of these molecules in air at ${20^ \circ }C$. Use your answer to calculate the root-mean-square speed of each type of molecule.
23) Show that the change in the internal energy of one mole of an ideal gas per unit change in temperature is always a constant. What is this constant?
EXAM-STYLE QUESTIONS
1) A gas is enclosed inside a cylinder that is fitted with a freely moving piston.
The gas is initially in equilibrium with a volume ${V_1}$ and a pressure p. The gas is then cooled slowly. The piston moves into the cylinder until the volume of the gas is reduced to ${V_2}$ and the pressure remains at p.
What is the work done on the gas during this cooling? [1]
A: $\frac{1}{2}p({V_2} - {V_1})$
B: $p({V_2} - {V_1})$
C: $\frac{1}{2}p({V_2} + {V_1})$
D: $p({V_2} + {V_1})$
2) An ideal gas is made to expand slowly at a constant temperature.
Which statement is correct? [1]
A: The heat energy transferred to the gas is zero.
B: The internal energy of the gas increases.
C: The work done by the gas is equal to the heat energy added to it.
D: The work done by the gas is zero.
3) a: State how many atoms there are in:
i- a mole of helium gas (a molecule of helium has one atom) [1]
ii- a mole of chlorine gas (a molecule of chlorine has two atoms) [1]
iii- a kilomole of neon gas (a molecule of neon has one atom). [1]
b: A container holds four moles of carbon dioxide of molecular formula $C{O_2}$.
Calculate:
i- the number of carbon dioxide molecules there are in the container [1]
ii- the number of carbon atoms there are in the container [1]
iii- the number of oxygen atoms there are in the container. [1]
[Total: 6]
4) A bar of gold-197 has a mass of $1.0 kg$. Calculate:
a: the mass of one gold atom in kg. [1]
b: the number of gold atoms in the bar [1]
c: the number of moles of gold in the bar. [2]
(An atom of gold contains 197 nucleons and has a mass of $197 u$.)
[Total: 4]
5) A cylinder holds $140\,{m^3}$ of nitrogen at room temperature and pressure. Moving slowly, so that there is no change in temperature, a piston is pushed to reduce the volume of the nitrogen to $42\,{m^3}$.
a: Calculate the pressure of the nitrogen after compression. [2]
b: Explain the effect on the temperature and pressure of the nitrogen if the piston is pushed in very quickly. [1]
[Total: 3]
6) The atmospheric pressure is $100 kPa$, equivalent to the pressure exerted by a column of water 10 m high. A bubble of oxygen of volume $0.42\,c{m^3}$ is released by a water plant at a depth of $25 m$. Calculate the volume of the bubble when it reaches the surface. State any assumptions you make. [4]
7) A cylinder contains $4.0 \times {10^{ - 2}}\,{m^3}$ of carbon dioxide at a pressure of $4.8 \times {10^5}\,Pa$ at room temperature.
Calculate:
a: the number of moles of carbon dioxide [2]
b: the mass of carbon dioxide. [2]
(Molar mass of carbon dioxide $= 44 g$ or one molecule of carbon dioxide has mass $44 u$.)
[Total: 4]
8) Calculate the volume of 1 mole of ideal gas at a pressure of $1.01 \times {10^5}\,Pa$ and at a temperature of ${0^ \circ }C$. [2]
9) A vessel of volume $0.20\,{m^3}$ contains $3.0 \times {10^{26}}$ molecules of gas at a temperature of ${127^ \circ }C$. Calculate the pressure exerted by the gas on the vessel walls. [3]
10) a: Calculate the root-mean-square speed of helium molecules at room temperature and pressure. (Density of helium at room temperature and pressure $ = 1.179\,kg\,{m^{ - 3}}$.) [3]
b: Compare this speed with the average speed of air molecules at the same temperature and pressure. [2]
[Total: 5]
11) A sample of neon is contained in a cylinder at ${27^ \circ }C$. Its temperature is raised to ${243^ \circ }C$.
a: Calculate the kinetic energy of the neon atoms at:
i- ${27^ \circ }C$ [3]
ii- ${243^ \circ }C$. [1]
b: Calculate the ratio of the speeds of the molecules at the two temperatures. [2]
[Total: 6]
12) A truck is to cross the Sahara desert. The journey begins just before dawn when the temperature is ${3^ \circ }C$. The volume of air held in each tyre is $1.50\,{m^3}$ and the pressure in the tyres is $3.42 \times {10^5}\,Pa$.
a: Explain how the air molecules in the tyre exert a pressure on the tyre walls. [3]
b: Calculate the number of moles of air in the tyre. [3]
c: By midday the temperature has risen to ${42^ \circ }C$.
i- Calculate the pressure in the tyre at this new temperature. You may assume that no air escapes and the volume of the tyre is unchanged. [2]
ii- Calculate the increase in the average translational kinetic energy of an air molecule due to this temperature rise. [2]
[Total: 10]
13) The ideal gas equation is $pV = \frac{1}{3}Nm\, \lt {c^2}\, \gt $.
a: State the meaning of the symbols $N, m$ and $ \lt {c^2}\, \gt $. [3]
b: A cylinder of helium-4 contains gas with volume $4.1 \times {10^4}\,c{m^3}$ at a pressure of $6.0 \times {10^5}\,Pa$ and a temperature of ${22^ \circ }C$. You may assume helium acts as an ideal gas and that a molecule of helium-4 contains 4 nucleons, each of mass $1.66 \times {10^{ - 27}}\,kg$.
14) a: State what is meant by an ideal gas. [2]
b: A cylinder contains $500 g$ of helium-4 at a pressure of $5.0 \times {10^{ - 527}}\,Pa$ and at a temperature of ${27^ \circ }C$. You may assume that the molar mass of helium-4 is $4.0 g$.
Calculate:
i- the number of moles of helium the cylinder holds [1]
ii- the number of molecules of helium the cylinder holds [1]
iii- the volume of the cylinder. [3]
[Total: 7]
One assumption of the kinetic theory of gases is that molecules undergo 15) a: elastic collisions with the walls of their container.
i- Explain what is meant by a perfectly elastic collision. [1]
ii- State three other assumptions of the kinetic theory. [3]
b: A single molecule is contained within a cubical box of side length $0.30 m$.
Determine:
i- the amount of gas in mol [3]
ii- the number of molecules present in the gas [2]
iii- the root-mean-square speed of the molecules. [3]
[Total: 11]
14) a: State what is meant by an ideal gas. [2]
b: A cylinder contains $500 g$ of helium-4 at a pressure of $5.0 \times {10^5}\,Pa$ and at a temperature of ${27^ \circ }C$. You may assume that the molar mass of helium-4 is $4.0 g$.
Calculate:
i- the number of moles of helium the cylinder holds [1]
ii- the number of molecules of helium the cylinder holds [1]
iii- the volume of the cylinder. [3]
[Total: 7]
One assumption of the kinetic theory of gases is that molecules undergo
15) a: perfectly elastic collisions with the walls of their container.
i- Explain what is meant by a perfectly elastic collision. [1]
ii- State three other assumptions of the kinetic theory. [3]
b: A single molecule is contained within a cubical box of side length $0.30 m$.
The molecule, of mass $2.4 \times {10^{ - 26}}\,kg$, moves backwards and forwards parallel to one side of the box with a speed of $400\,m\,{s^{ - 1}}$. It collides elastically with one of the faces of the box, face P.
i- Calculate the change in momentum each time the molecule hits face P. [2]
ii- Calculate the number of collisions made by the molecule in $1.0 s$ with face P. [2]
iii- Calculate the mean force exerted by the molecule on face P. [2]
[Total: 10]
16) a: A cylinder contains $1.0 mol$ of an ideal gas. The gas is heated while the volume of the cylinder remains constant. Calculate the energy required to raise the temperature of the gas by ${1.0^ \circ }C$. [2]
b: Calculate the root-mean-square speed of a molecule of hydrogen-1 at a temperature of ${100^ \circ }C$.
($Mass\,of\,a\,Hydrogen\,molecule = 3.34 \times {10^{ - 27}}\,kg$.) [3]
c: Calculate, for oxygen and hydrogen at the same temperature, the ratio
$\frac{{root\,mean\,square\,speed\,of\,a\,hydrogen\,molecule}}{{root\,mean\,square\,speed\,of\,an\,oxygen\,molecule}}$
($Mass\,of\,an\,oxygen\,molecule = 5.31 \times {10^{ - 26}}\,kg$.) [2]
[Total:
SELF-EVALUATION CHECKLIST
After studying the chapter, complete a table like this:
| I can | See topic… | Needs more work | Almost there | Ready to move on |
| use molar quantities and understand that one mole is an amount of substance containing ${N_A}$ particles, where ${N_A}$ is the Avogadro constant | 20.3 | |||
| understand that an ideal gas obeys $PV \propto T$ T where T is the thermodynamic temperature | 20.4 | |||
| recall and use the equation of state for an ideal gas expressed as $pV = nRT$, where n = amount of substance (number of moles), and as $pV = NkT$, where N = number of molecules | 20.6 | |||
| state the basic assumptions of the kinetic theory of gas | 20.7 | |||
| explain how molecular movement causes the pressure exerted by a gas and derive the relationship: $pV = \frac{1}{3}Nm\,\, \lt \,{c^2}\,\, \gt $, where $ < \,{c^2}\,\, > $ is the mean-square speed |
20.7 | |||
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understand that the root-mean-square speed ${c_{r.m.s.}}$ is given by: $\sqrt { < \,{c^2}\,\, > } $ |
20.8 | |||
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recall that the Boltzmann constant k is given by: $k = \frac{R}{{{N_A}}}$ |
20.8 | |||
| compare with $pV = \frac{1}{3}Nm\,\, \lt \,{c^2}\, \gt $ with $pV = NkT$ to deduce that the average translational kinetic energy of a molecule is $\frac{3}{2}kT$. |
20.8 |