TOPIC 10 IDEAL GASES

Real gases have complex behaviour, but it is possible to make progress in understanding
gases by developing a simplified model of a gas called an ideal gas.

Balloons, like the one shown above, carry instruments high into the atmosphere to measure
pressure, temperature, wind speed and other variables. The balloon is filled with helium so
that its overall density is less than that of the surrounding air. The result is an upthrust on
the balloon, greater than its weight, so that it rises upwards. As it moves upwards, the
pressure of the surrounding atmosphere decreases so that the balloon expands. The
temperature drops, which tend to make the gas in the balloon, shrink. Under this topic we
will look at the behaviour of gases as their pressure, temperature and volume change.

EQUATION OF STATE

Measuring gases

We are going to picture a container of gas, such as the box shown above. There are four
properties of this gas that we might measure: pressure, temperature, volume and mass. We
will also study how these quantities are related to one another.

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Pressure

This is the normal force exerted per unit area by the gas on the walls of the container.
The pressure is the result of molecular collisions with the walls of the container. Pressure
is measured in pascals, Pa. [1 Pa = 1 N m−2]

Temperature

This is a measure of the average kinetic energy of the gas molecules. This might be
measured in °C, but in practice it is more useful to use the thermodynamic (Kelvin) scale of
temperature. You should recall how these two scales are related:

T (K) = θ (°C) + 273.15

Volume

This is a measure of the space occupied by the gas. Volume is measured in m3.

Mass

This is the amount of substance contained in the gas. It is measured in g or kg. In practice,
it is more useful to consider the amount of gas, measured in moles.

One mole of any substance is the amount of that substance which contains the same
number of particles as there are in 0.012 kg of carbon-12.

One mole of any substance has a mass in grams which is numerically equal to the relative
atomic or molecular mass of the substance. For example, one mole of oxygen (O2) has a
mass of about 32 g. A mole of any substance (solid, liquid or gas) contains a standard
number of particles (molecules or atoms). This number is known as the Avogadro constant,
NA. The experimental value for NA is 6.02 × 1023 mol−1.

We can easily determine the number of atoms in a sample if we know how many moles it
contains. For example:

2.0 mol of helium contains; 2.0 × 6.02 × 1023 = 1.20 × 1024 atoms.

10.0 mol of carbon contains; 10 × 6.02 × 1023 = 6.02 × 1024 atoms.

We will see later that, if we consider equal numbers of moles of two different gases under
the same conditions, their physical properties are the same.

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Boyle’s law

This law relates the pressure p and volume V of a gas. It was discovered in 1662 by Robert
Boyle. If a gas is compressed, its pressure increases and its volume decreases; Pressure
and volume are inversely related.

The pressure exerted by a fixed mass of gas is inversely proportional to its volume,
provided the temperature of the gas remains constant.

�
⟹ �∝ ∴ �� = ��������
�

For solving problems, you may find it more useful to use the equation in this form:

�� �� = �� ��

Here, p1 and V1 represent the pressure and volume of the gas before a change, and p2 and
V2 represent the pressure and volume of the gas after the change. A graph of p against
1/V, shown below, is a straight line passing through the origin, showing direct
proportionality.

Charles law

Boyle’s law requires that the temperature of a gas is fixed. What happens if the
temperature of the gas is allowed to change? The figure below shows the results of an
experiment in which a fixed mass of gas is cooled at constant pressure.

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The gas contracts; its volume decreases. This graph does not show that the volume of a gas
is proportional to its temperature on the Celsius scale. If a gas contracted to zero volume
at 0 °C, the atmosphere would condense on a cold day and we would have a great deal of
difficulty in breathing. However, the graph does show that there is a temperature at which
the volume of a gas does, in principle, shrink to zero. Looking at the lower temperature
scale on the graph, where temperatures are shown in kelvin (K), we can see that this
temperature is 0 K, or absolute zero. (Historically, this is how the idea of absolute zero
first arose). We can represent the relationship between volume V and thermodynamic
temperature T as:

�
⟹ �∝� ∴ = ��������
�

For solving problems, you may find it more useful to use the equation in this form:

�� ��
=
�� ��

Note that this relationship only applies to a fixed mass of gas and to constant pressure.

The relationship above is an expression of Charles’s law, named after the French physicist
Jacques Charles.

The volume of a fixed mass of a gas, at constant pressure, is directly proportional to

its absolute temperature.

If we combine Boyle’s law and Charles’s law, we can arrive at a single equation for a fixed
mass of gas:

��
= ��������
�

Real and ideal gases

The relationships between p, V and T that we have considered above are based on
experimental observations of gases such as air, helium, nitrogen, etc., at temperatures and
pressures around room temperature and pressure. In practice, if we change to more
extreme conditions, such as low temperatures or high pressures, gases start to deviate
from these laws as the gas atoms exert significant electrical forces on each other. For
example, the figure below shows what happens when nitrogen is cooled down towards
absolute zero.

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At first, the graph of volume against temperature follows a good straight line. However, as
it approaches the temperature at which it condenses, it deviates from ideal behaviour, and
at 77 K it condenses to become liquid nitrogen. Thus, we have to attach a condition to the
relationships discussed above. We say that they apply to an ideal gas. When we are dealing
with real gases, we have to be aware that their behaviour may be significantly different
from the ideal equation.

Ideal gas equation

So far, we have seen how p, V and T are related. It is possible to write a single equation
relating these quantities which takes into account the amount of gas being considered. If
we consider n moles of an ideal gas, we can write the equation in the following form:

pV = nRT

This equation is called the ideal gas equation or the equation of state for an ideal gas.
It relates all four of the variable quantities discussed at the beginning of this topic.

The constant of proportionality R is called the universal molar gas constant. Its
experimental value is: R = 8.31 J mol−1 K−1. Note that it doesn’t matter what gas we are
considering – it could be a very ‘light’ gas like hydrogen, or a much ‘heavier’ one like carbon
dioxide. So long as it is behaving as an ideal gas, we can use the same equation of state with
the same constant R.

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KINETIC THEORY OF GASES

Particles of a gas

We picture the particles of a gas as being fast-moving. They bounce off the walls of their
container (and off each other) as they travel around at high speed, seen above. How do we
know that these particles are moving like this? It is much harder to visualise the particles
of a gas than those of a solid, because they move about in such a disordered way, and most
of a gas is empty space. The movement of gas particles was investigated in the 1820s by a
Scottish botanist, Robert Brown. He was using a microscope to look at pollen grains
suspended in water, and saw very small particles moving around inside the water. He then
saw the same motion in particles of dust in the air. It is easier in the laboratory to look at
the movement of tiny particles of smoke in air.

Fast molecules

For air, at standard temperature and pressure (STP, 0 °C and 100 kPa), the average speed
of the molecules is about 400 m s−1. At any moment, some are moving faster than this and
others more slowly. If we could follow the movement of a single air molecule, we would find
that, some of the time, its speed was greater than this average; at other times it would be
less. The velocity (magnitude and direction) of an individual molecule changes every time it
collides with anything else.

This value for molecular speed is reasonable. It is comparable to (but greater than) the
speed of sound in air (approximately 330 m s−1 at STP). Very fast-moving particles can
easily escape from the Earth’s gravitational field. The required escape velocity is about 11
km s−1. Since we still have an atmosphere, on average the air molecules must be moving
much more slowly than this value.

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Path of a particle

The erratic motion of particles in water that Brown observed comes about because the
particles are constantly bombarded by the much smaller, faster water molecules. This
motion came to be known as Brownian motion, and it can be observed in both liquids and
gases.

Modelling gases – the kinetic model

A gas is made of particles (atoms or molecules). Its pressure arises from collisions of the
particles with the walls of the container; more frequent or harder collisions give rise to
greater pressure. Its temperature indicates the average kinetic energy of its particles;
the faster they move, the greater their average kinetic energy and the higher the
temperature.

The kinetic theory of gases is a theory which links these microscopic properties of
particles (atoms or molecules) to the macroscopic properties of a gas.

The table above shows the assumptions on which the theory is based. On the basis of these
assumptions, it is possible to use Newtonian mechanics to show that pressure is inversely
proportional to volume (Boyle’s law), volume is directly proportional to thermodynamic

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(kelvin) temperature (Charles’s law), and so on. The theory also shows that the particles of
a gas have a range of speeds – some move faster than others.

Explaining pressure

The figure below shows the movement of a single molecule of air in a box. It bounces
around inside, colliding with the various surfaces of the box.

At each collision, it exerts a small force on the box. The pressure on the inside of the box
is a result of the forces exerted by the vast number of molecules in the box. Two factors
affect the force, and hence the pressure that the gas exerts on the box:

1. the number of molecules that hit each side of the box in one second;
2. the force with which a molecule collides with the wall.

If a molecule of mass m hits the wall head-on with a speed v it will rebound with a speed v
in the opposite direction. The change in momentum of the molecule is 2mv. Since force is
equal to rate of change of momentum, the higher the speed of the molecule the greater
the force that it exerts as it collides with the wall. Hence the pressure on the wall will
increase if the molecules move faster.

If the piston in a bicycle pump is pushed inwards but the temperature of the gas inside is
kept constant, then more molecules will hit the piston in each second but each collision will
produce the same force, because the temperature and therefore the average speed of the
molecules is the same. The increased rate of collisions alone means that the force on the
piston increases and thus the pressure rises. If the temperature of the gas in a container
rises then the molecules move faster and hit the sides faster and more often; both of
these factors cause the pressure to rise.

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Kinetic theory and gas pressure

 Kinetic theory states that the molecules of a gas moves continuously at random and
often collides with the wall of the container.
 When a molecule collides with the wall of the container it undergoes change in
momentum.
 The rate of change in momentum means that a force acts on the molecules. By
Newton’s third law of motion an equal but opposite force acts on the wall.
 Pressure is the average force acting per unit area as a result of impact of molecules
of the gas on the wall of the container.

Pressure due to an ideal gas

Let N = number of gas molecules in the cube

m = mass of each molecule

c = 3 – D velocity of a molecule

(Note: a molecule can move in 3-dimensions. i.e. x, y, and z).

Thus, the velocity of a molecule at any time is made up of 3-components;

a) a velocity in the x – component;

b) a velocity in the y – component;
c) a velocity in the z - component

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 �
Hence, let � = � = speed of any molecule in the x, y, and z directions as shown in the
�
diagram.

Also, let the entire N – molecules be contained in the cubical vessel of length, l.

Consider the force exerted on the face X of the cube due to component u. Just before
impact, the momentum of the molecule due to u is mu. After an elastic impact, the
momentum is – mu, since the momentum reverses. So, momentum change on impact = mu – (–
mu) = 2mu.

The distance travelled by the molecule to the face X and back to the opposite face = 2l.
Hence, time (t) taken for the molecule (m) to cover 2l distance is

�������� 2�
�= =
�������� �

The time rate of change of momentum at X = force exerted at X [recall F = m (v – u)/t]

Thus, force, F exerted on face is;

2�� ���
�= =
2� �
�

Therefore, Pressure, P exerted on X is;

��2
����� � � � ���
�= = 2 = �
���� �� � � �

This is the pressure exerted on face X by a single molecule moving in the x – direction. The
total pressure exerted by all the N – molecules moving in the x – direction is;

��21 ��22 ��23 ��2�

�� = 3 + 3 + 3 + −−−+ 3
� � � �
� �
�� = � ��
+ ��� + ��� + −−−+ ��� −−−− (�)
�

Where u1, u2, u3,.…., uN = x – component velocities for all the N – molecules.

Let �� = the average or mean square velocity or speed of all the molecules in the x –
direction.

�21 + �22 + �23 + −−−+ �2�

�2 =
�

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 ⟹ ��2 = �21 + �22 + �23 + −−−+ �2�

Thus, from (1),

�
�� = ��� −−−−− (�)
��

Also, let �� = mean square speed in the y – direction

�� = mean square speed in the z – direction

�
⟹ �� = �� + �� + �� [Recall � = � ]
�

= mean square speed for all the molecules

Because the motion of molecules in all direction is equally probable, coupled with the fact
that the molecules do not pile up in any corner of the container their mean velocities in all
directions is the same. That is,

�2 = �2 = �2

⟹ �� = ���

� �
⟹ �� = �
�

Hence, from (2)

�� 1 2
�= �
�3 3
�
∴ �� = ����
�

The equation also suggests that pressure p is proportional to the average value of the
speed squared. This is because, if a molecule is moving faster, not only does it strike the
container harder, but it also strikes the container more often.

KINETIC ENERGY OF A MOLECULE

From,
�
�� = ��� and �� = ����
�

The left-hand sides are the same, so the two right-hand sides must also be equal:

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 �
��� = ����
�

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:

����
��� =
�

The quantity N/n= NA is the Avogadro constant, the number of particles in 1 mole. So:

���
��� =
��

It is easier to make sense of this if we divide both sides by 2, to get the familiar
expression for kinetic energy:

� ���
��� =
� ���

The quantity R/NA is defined as the Boltzmann constant, k. Its value is 1.38 × 10−23 J K−1.
Substituting k in place of R/NA gives;

� ���
��� =
� �
�
The quantity � ��� is the average kinetic energy E of a molecule in the gas, and k is a
constant. Hence, 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.

We need to consider two of the terms in this statement. Firstly, we talk about
translational kinetic energy. This is the energy that the molecule has because it is moving
along; a molecule made of two or more atoms may also spin or tumble around, and is then
said to have rotational kinetic energy, as shown below.

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Secondly, we talk about mean (or average) translational kinetic energy. There are two ways
to find the average translational 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).

Temperature and kinetic energy

 Molecules in gases moved about randomly at high speed

 They collide with one another and with the walls of their container.
 Collisions with the walls give rise to the pressure of the gas on the container.
 When a thermometer is place in the container, the molecules collide with it and
imparting their kinetic energy to the thermometer.
 At higher temperature, the molecules move faster or with greater kinetic energy.
They give more kinetic energy to the bulb and the mercury rises higher.
 Hence the reading on the thermometer is an indication of the kinetic energy of the
gas molecules

EXAMPLES

1) Nitrogen consists of molecules N2. The molar mass of nitrogen is 28 g mol−1. For 100

g of nitrogen, calculate:

a) the number of moles

b) the volume occupied at room temperature and pressure. (r.t.p. = 20 °C, 1.01 ×

105 Pa)

2) Calculate the volume of 5.0 mol of an ideal gas at a pressure of 1.0 × 105 Pa and a

temperature of 200 °C.

3) A sample of gas contains 3.0 × 1024 atoms. Calculate the volume of the gas at a

temperature of 300 K and a pressure of 120 kPa.

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4) The quantity Nm is the total mass of the molecules of the gas, i.e. the mass of the

gas. At room temperature, the density of air is about 1.29 kg m−3 at a pressure of

105 Pa.

a) Use these figures to deduce the value of �� for air molecules at room

temperature.

b) Find a typical value for the speed of a molecule in the air by calculating �� .

How does this compare with the speed of sound in air, approximately 330 m s−1?

5) 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.

6) Air consists of molecules of oxygen (molar mass = 32 g mol−1) and nitrogen (molar

mass = 28 g mol−1). Calculate the mean translational k.e. of these molecules in air at

20 °C. Use your answer to estimate a typical speed for each type of molecule.

7) 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?

8) Calculate the average speed of helium molecules at room temperature and pressure.

(Density of helium at room temperature and pressure = 0.179 kg m−3)

Comment on how this speed compares with the average speed of air molecules at the

same temperature and pressure.

9) A truck is to cross the Sahara Desert. The journey begins just before dawn when

the temperature is 3 °C. The volume of air held in each tyre is 1.50 m3 and the

pressure in the tyres is 3.42 × 105 Pa.

a) Explain how the air molecules in the tyre exert a pressure on the tyre walls.

b) Calculate the number of moles of air in the tyre.

c) By midday the temperature has risen to 42 °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.

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ii. Calculate the increase in the average translational kinetic energy of an

air molecule due to this temperature rise.

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