11. Ideal Gas Laws

IA2 PHYSICS Ideal gas laws

TEACHING NOTES

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 = p∆ V ..... 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: i)

the specific (or molar) heat capacity at constant volume, cv

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IA2 PHYSICS Ideal gas laws ii)

TEACHING NOTES

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

cp - cv = p∆ V 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.

Therefore

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IA2 PHYSICS Ideal gas laws From B to C

TEACHING NOTES

Compress the gas slowly to change V and p at constant T. We can apply the Boyle/Marriotte law to this change. Therefore

p1V' = p2V2

It 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. The equation of state is therefore usually written as

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IA2 PHYSICS Ideal gas laws

TEACHING NOTES

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. II.

there must be negligible forces of attraction between its molecules 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

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IA2 PHYSICS Ideal gas laws

TEACHING NOTES

helium

-272

-269

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)

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IA2 PHYSICS Ideal gas laws •

TEACHING NOTES

the speed of the molecules (this has 2 effects, one is the change in the momentum and the other is the more often the molecules collide with container) so the total effect depends upon the speed squared. P α 2 -------------- (2)

•

The number of molecules in the container PαN

----------------- (3)

So, by combining the above three equations, we get P α (N m 2) / V P = 1/3 (N m 2) / V V is the volume of the gas (m3)

Where : P is the pressure of the gas (Pa), 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 :

P = 1/3 ρ 2

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

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IA2 PHYSICS Ideal gas laws

TEACHING NOTES

The more familiar form expresses the average molecular kinetic energy:

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