The experiments which lead us to the three
gas laws (the pressure law, Charles' law and the BoyleMariotte law)
can easily be 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 ˝ 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, in fact, they cease to be gases!
(See below for more detail.) 



An (imaginary) gas which obeys the gas laws
perfectly for any temperature and pressure is called an ideal
(or perfect) gas. 



Consider a fixed quantity of gas in a cylinder with a movable
piston. 

The gas has initial pressure, volume and temperature of p_{1}
V_{1} and T_{1} as shown here. 



The gas is heated and allowed to
expand at (quasi) constant pressure. 

We therefore now have the situation shown in the next diagram. 



Charles' law
can be applied to this change so we have 





Next, the gas is compressed at (quasi)
constant temperature. 

(In practice, this can be achieved by
compressing slowly in a cylinder made of a good thermal
conductor.) 



The BoyleMariotte law can be applied to
this change, so we can write 





It is clear that V’ can be eliminated
from these two equations giving 



in which we see a sort of combination of the
three gas laws, all in one equation. 



This is called the equation of state of an
ideal gas and is usually written in the slightly more general form: 



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 product of the pressure and the volume of the gas in the
cylinder obviously depends on the mass of gas we put into it. 

If we pump in, say, twice as much gas then the pressure
will be twice as great for a given volume and temperature. 

For this reason, the "constant" in the above equation is not a
universal constant; 

The value of the "constant" also depends on the type of gas
in the cylinder. 

This is not surprising because, for example, a given mass of
hydrogen (H_{2}) has 16 times as many molecules as the same
mass of oxygen (O_{2}). 



The Universal Gas Constant 

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 now 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! 

The number of particles we chose to define this constant is
(approximately) 6×10^{23}. 

This number is called Avogadro’s number, N_{A}. 

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 in the following
form 



where, n is the number of mols of the gas and R is the universal
gas constant. 

The units of R are Jmol^{1}K^{1} 



R therefore represents a quantity of energy possessed by
1mol of particles. 

It is often useful to know the quantity of energy possessed by
individual particles of the gas and so we define the gas
constant per molecule,(or Boltzmann's constant) k as follows 





In case you’re wondering were the number
6×10^{23} came from: 

For comparing the masses of atoms and
molecules, one atomic mass unit, 1u, is defined as follows




In other words 1u is about the mass of a proton or a neutron. 

If you have 6×10^{23}
protons (or neutrons) they have a total mass of about 1gram. 

This is the historical reason for choosing
6×10^{23} as the base for
comparing quantities of substances. 



Relative Molecular Mass 

The relative molecular mass of a substance is the ratio of the mass
of one molecule to the mass 1u. 

This means, in effect, that the r.m.m. of a substance is about equal
to the total number of protons and neutrons in one of its
molecules. 



So, if the relative molecular mass of a substance is μ then the mass
of 1mol of that substance is
μ grams. 

Hydrogen (H_{2}) has a relative molecular mass of 2 so the
total mass of 6×10^{23}
hydrogen molecules is 2g 

Helium (He) has a relative molecular mass of 4 so the total mass of
6×10^{23} helium atoms is
4g 

Oxygen (O_{2}) has a relative molecular mass of 32 so the
total mass of
6×10^{23} oxygen molecules
is 32g and so on… 



Ideal Gas and Real Gases 

As stated above, a gas which obeys the gas laws perfectly for all
temperatures and pressures is called an ideal (or
perfect) gas. 

This gas does not exist! 



However, a gas can behave like an ideal gas if 

 the forces between the molecules are negligible and 

 the total volume of the molecules is negligible
compared to the volume occupied by the gas (because of the
motion of its molecules). 

Both these conditions are very nearly fulfilled at or near s.t.p. 



Real gas molecules attract each other and do not occupy negligible
volume when the gas is compressed into a small volume. 

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 and normal atmospheric pressure are shown in
the table below. 



Gas 
Melting Point, °C 
Boiling Point, °C 
nitrogen 
210 
196 
oxygen 
219 
183 
hydrogen 
259 
253 
helium 
272 
269 

