11. Ideal gases

Measuring gases• Temperature

should be measured in thermodynamic scale - kelvins

• VolumeThis is a measure of the space occupied by the gas. Volume

is measured in

• MassThis is measured in kg. In practice, it is more useful to

consider the amount of gas measured in moles.

• PressureThis is the force exerted normally per unit area by the gas

on the walls of the container. Pressure is measured in pascals.

The mole definition

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.

Boyle’s law

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

Boyle’s law

Charles’s law

The volume of a given mass of an ideal gas is directly proportional to its temperature on the absolute temperature scale (in Kelvin) if pressure and the amount of gas remain constant

Charles’s law

Charles’s law and Boyle’s law

𝑃𝑉𝑇

=𝑐𝑜𝑛𝑠𝑡

𝑃1𝑉 1

𝑇1

=𝑃2𝑉 2

𝑇2

𝑉 ∝𝑇𝑃

Avogadro’s Law

Avogadro’s law states that equal volumes of different gases at the same pressure and

temperature will contain equal numbers of particles.

For example, if there are 2 moles of O2 in 50 cm3 of oxygen gas, then there will be 2 moles of N2 in 50 cm3 of nitrogen gas and 2 moles of CO2 in 50 cm3 of carbon dioxide gas at the same temperature and pressure.

Using this principle, the volume that a gas occupies will depend on the number of moles of the gas.

Molar volumes of gases

If the temperature and pressure are fixed at convenient standard values, the molar volume of a gas can be determined.

At standard temperature and pressure (273 K and 100 kPa), 1 mole of any gas

occupies a volume of 2.27 × 10-2 m3.

This is the molar volume.

12

The Ideal Gas EquationSummarizing the Gas Laws

Boyle: V 1/P (constant n, T)Charles: V T (constant n, P)Avogadro: V n (constant P, T).

Combined:

Ideal gas equation

R = ideal gas constant

𝑉 ∝𝑛𝑇𝑃

𝑉=𝑅𝑛𝑇𝑃→

𝑃𝑉𝑇

=𝑛𝑅

Universal gas constant

Standard pressure Standard volume

Standard temperatureNumber of moles

Calculating the number n of moles

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑜𝑙𝑒𝑠=𝑚𝑎𝑠𝑠

𝑚𝑜𝑙𝑎𝑟𝑚𝑎𝑠𝑠

When dealing with individual molecules of gas, a different gas constant is needed because the molar gas constant, R, is applicable for n number of moles of gas.

pV = NkT

Where N is the total number of molecules of the gas, and k is the Boltzmann constant: the gas constant for a single molecule. What is the value of k?k = R / NA

k = (8.31 J mol-1 K-1) / (6.02 × 1023

mol-1)

k = 1.38 × 10-23 J K-1

Modelling gases – the kinetic model

The kinetic theory of gases links the macroscopic behaviour of a gas with its microscopic behaviour.

In macroscopic terms, a gas is a phase of matter that has a fixed mass and whose volume is equal to the volume of its container. The gas exerts pressure on the walls of the container

In microscopic terms, a gas is a collection of many particles that collide with each other and with the container walls.

The kinetic theory relates the pressure exerted by a gas to the motion of its particles.

Boyle’s law and real gases

Under conditions of low pressure and high temperature, real gases will obey Boyle’s law and are said to be behaving in an ideal manner.

At high pressures/low volumes or at temperatures near the gas’s boiling point, real gases stop behaving ideally and no longer follow Boyle’s law.

This is because the gas’s molecules/atoms are occupying a significant proportion of the volume.

An ideal gas is one that obeys Boyle’s law under all conditions. In reality, however, no real gas is an ideal gas under all conditions.

Ideal gas assumptions

When describing an ideal gas, the following assumptions are made:

all molecules of a particular gas are identical the internal energy of the gas is entirely kinetic all collisions between molecules and the walls of the

container are completely elastic Newton’s laws of motion apply molecules take up negligible volume gravitational and electrostatic forces can be ignored the motion of all molecules is random all molecules travel in straight lines.

Assumption Explanation/commentA gas contains a very large number of spherical particles (atoms or molecules).

A small ‘cube’ of air can have as many as 1020 molecules.

The forces between particles are negligible,except during collisions.

If the particles attracted each other strongly over long distances, they would all tend to clump together in the middle of the container. The particles travel in straightlines between collisions.

Assumption Explanation/commentThe volume of the particlesis negligible compared tothe volume occupied bythe gas.

When a liquid boils tobecome a gas, its particlesbecome much farther apart.

Most of the time, a particlemoves in a straight line at aconstant velocity. The timeof collision with anotherparticle or with thecontainer walls is negligiblecompared with the timebetween collisions.

The particles collide with the walls of the container andwith each other, but for most of the time they are moving with constant velocity.

Assumption Explanation/commentThe collisions of particleswith each other and withthe container are perfectlyelastic, so that no kineticenergy is lost.

Kinetic energy cannot be lost. The internal energy of the gas is the total kinetic energy of the particles.

PressureThe pressure that a gas exerts is caused by the collisions of its molecules with the walls of the container.

A gas particle is shown colliding elastically with the right wall of the container and rebounding from it.

∆ 𝑝=−𝑚𝑐−𝑚𝑐=−2𝑚𝑐Time between collisions:

∆ 𝑡=2 𝑙𝑐

Average force on the molecule:

F=∆𝑝∆ 𝑡

=−2𝑚𝑐2 𝑙𝑐

=−2𝑚𝑐2

2 𝑙=−𝑚𝑐2

𝑙

Average force on the wall: F=𝑚𝑐2

𝑙

Pressure:

𝑃=𝐹𝐴

=

𝑚𝑐2

𝑙𝑙2

=𝑚𝑐2

𝑙3

This is for one molecule, but there is a large number N of molecules in the box. Each has a different velocity, and each contributes to the pressure. We write the average value of as < >, and multiply by N to find the total pressure:

𝑃=13𝑁𝑚¿𝑐2> ¿

𝑙3¿

Molecules are moving in all three dimensions equally, so we need to divide by 3 to find the pressure exerted.

𝑃=𝑁𝑚¿𝑐2> ¿𝑙3

¿

𝑃=13𝑁𝑚¿𝑐2> ¿

𝑙3¿

volume

mass

𝑃=13𝑀 ¿𝑐2> ¿

𝑉¿ 𝑀

𝑉=𝜌

𝑃=13𝜌<𝑐2>¿

Kinetic energy of the gas molecules

𝑃𝑉=13𝑁𝑚¿𝑐2>¿

𝑃𝑉=𝑛𝑅𝑇13𝑁𝑚¿ 𝑐2>¿𝑛𝑅𝑇

𝑚¿𝑐2>¿3𝑛𝑅𝑇  𝑁

𝑛𝑁 𝐴=𝑁

𝑛𝑁

=1𝑁 𝐴

𝑚¿𝑐2>¿3𝑅𝑇𝑁 𝐴

𝑅𝑁𝐴

=𝑘𝑚¿𝑐2>¿3𝑘𝑇

𝑚¿𝑐2>¿3𝑘𝑇

𝑚¿𝑐2> ¿2=32𝑘𝑇 ¿

𝐸𝑘=32𝑘𝑇

The mean translational kinetic energy of an atom (or molecule) of an ideal gas is proportional to the thermodynamic temperature.

Root mean square speed

¿𝑐2>¿ Mean square speed

√¿𝑐2>   Root mean square speed ( )

𝑚¿𝑐2>¿3𝑘𝑇

¿𝑐2>¿3𝑘𝑚

𝑇

¿𝑐2>∝𝑇 Mean square speed is proportional to the temperature

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.

Kinetic energy of a real gas

Key equations𝑃1𝑉 1

𝑇1

=𝑃2𝑉 2

𝑇2

𝑃𝑉𝑇

=𝑛𝑅

𝑃𝑉𝑇

=𝑁𝑘

𝑛=𝑁𝑁 𝐴

𝑃=13𝜌<𝑐2>¿

𝐸𝑘=32𝑘𝑇

𝑅𝑁𝐴

=𝑘