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The Equation of State of an Ideal Gas
The experiments which lead us to the three gas laws (the pressure law, Charles' law and the Boyle-Mariotte 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 p1 V1 and T1 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 Boyle-Mariotte 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 (H2) has 16 times as many molecules as the same mass of oxygen (O2).  
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) 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 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-1K-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 1023 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 1023 protons (or neutrons) they have a total mass of about 1gram.   
This is the historical reason for choosing 1023 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 (H2) has a relative molecular mass of 2 so the total mass of 1023 hydrogen molecules is 2g  
Helium (He) has a relative molecular mass of 4 so the total mass of 1023 helium atoms is 4g  
Oxygen (O2) has a relative molecular mass of 32 so the total mass of 1023 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

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