Work Done During a Change of Volume of a Gas
Consider a quantity of gas in a container which has a frictionless piston as shown in the diagram below. The volume occupied by the gas is changed by V.
The pressure, p, acting on the surface of the piston produces a force, F, equal to pA. During the change in volume, this force does some work.
The work done by this force is w = Fs = pAs
but As is the change in the volume occupied by the gas, V. Therefore
w = pV
...
When the volume of a gas increases, work is done by the gas.
When the volume of a gas decreases, work is done on the gas by an external force.
The above result shows that if the temperature of a gas is increased at constant volume, no work is done.
However, if the temperature is increased and the gas is allowed to expand, work will be done. In this case, extra energy will have to be supplied to do this work.
For this reason, gases are said to have two principal specific (or molar) heat capacities:
ii) the specific (or molar) heat capacity at constant pressure, cp
It should be clear that cp > cv and that the difference between them is given by
c
p- c
v= pV
The Equation of State of an Ideal Gas
The three experiments referred to above can be easily carried out (in a school laboratory) for a range of temperatures and pressures not too far from "normal"; for example, between 0°C and 100°C and between about ½ to 2 atmospheres of pressure. Within these limits real gases give the results described above. However, if gases are compressed to extreme pressures at low temperature, their behaviour is very different.
An (imaginary) gas which obeys the gas laws perfectly for any temperature and pressure is called an ideal (or perfect) gas.
Consider a quantity of gas which experiences the following changes.
...A B ...C
From A to B
Energy supplied to increase T and V at constant p. We can apply Charles’ law to this change.
From B to C
Compress the gas slowly to change V and p at constant T. We can apply the Boyle/Marriotte law to this change.
Therefore
p
1V
'
= p
2V
2It is clear that V’ can be eliminated from these two equations giving
This is called the equation of state of an ideal gas and is usually written in the following form
pV = (a constant)×T
The Universal Gas Constant
The equation of state for an ideal gas can be applied to real gases as long as we limit the range of temperatures and pressures.
The "constant" in the equation obviously depends on the quantity of gas in the container. It also depends on the type of gas; oxygen, hydrogen etc., because, for a given mass of gas we have a different number of particles for different gases.
Avogadro suggested that at a given temperature and pressure, equal volumes of any gas (behaving as an ideal gas) contain equal numbers of particles. This is called Avogadro’s law and has been confirmed by experiment.
Therefore, if we consider a given number of particles of any gas in our cylinder we can find a really constant constant! This is called the universal gas constant, R.
The number of particles we chose to define this constant is (approximately) 6×1023. This number is called Avogadro’s number, NA. If we have this number of particles of a substance, we say we have 1mol of that substance.
pV = nRT
where, n is the number of mols of gas.
The units of R are JK-1mol-1
It is often found useful to consider individual molecules of a gas. Fo this reason, we define the gas constant per molecule (or Boltzmann’s constant), k, as follows
Ideal Gas and Real Gases
Ideal Gas
An (imaginary) gas which obeys the gas laws perfectly for all temperatures and pressures is called an ideal (or perfect) gas.
In order for a gas to be considered ideal
I. there must be negligible forces of attraction between its molecules
II. the total volume of its molecules must be negligible compared with the volume occupied by the gas.
Real Gases
Real gases near s.t.p. (760mmHg and 0°C) behave like an ideal gas.
Real gas molecules attract each other and do not occupy negligible volume when the gas is at high pressure. If we decrease the temperature and increase the pressure of a real gas it will eventually change its state. At this stage the gas laws no longer apply (obvious really…since you don’t have a gas any more!). Some melting and boiling points of elements which are gaseous at room temperature are shown in the table below.
Gas Melting Point /°C Boiling Point /°C
hydrogen -259 -253
nitrogen -210 -196
oxygen -219 -183
These temperatures are for normal atmospheric pressure.
Kinetic Theory
The kinetic theory of gases is the study of the microscopic behaviour of molecules and the interactions which lead to macroscopic relationships like the ideal gas law.
The study of the molecules of a gas is a good example of a physical situation where statistical methods give precise and dependable results for
macroscopic manifestations of microscopic phenomena. For example, the pressure, volume and temperature calculations from the ideal gas law are very precise. The average energy associated with the molecular motion has its foundation in the Boltzmann distribution, a statistical distribution
function. Yet the temperature and energy of a gas can be measured precisely.
The pressure of a gas in a given volume depends on 3 factors:
the mass of the molecules P α m --- (1)
the speed of the molecules (this has 2 effects, one is the change in the
P α <c>2 --- (2)
The number of molecules in the container P α N --- (3)
So, by combining the above three equations, we get
P α (N m <c>2) / V
P = 1/3 (N m <c>2) / V
Where : P is the pressure of the gas (Pa), V is the volume of the gas (m3)
N is the number of molecules, m is the mass of one molecule (kg)
c is an average molecular speed (m s-1)
The above equation can be rearranged as follows:
(N m) / V = (number of molecules x mass of one molecule) / volume of the gas
= total mass of the gas / volume of the gas
= density of the gas, ρ
So we can re-write this ideal gas equation as :
The relation between microscopic (average kinetic energy) quantity and macroscopic quantity (Absolute temperature):
The expression for gas pressure developed from kinetic theory relates pressure and volume to the average molecular kinetic energy. Comparison with the ideal gas law leads to an expression for temperature sometimes referred to as the kinetic temperature.
This leads to the expression
The more familiar form expresses the average molecular kinetic energy:
