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Deriving the Ideal Gas Equation
For a constant amount of gas:
 Boyle's Law tells us that the volume of the gas (V) is inversely proportional to its pressure (P):
 Charles' Law tells us that the volume of the gas (V) is directly proportional to its temperature in Kelvin (T):
 and the Combined Gas Equation tells us that the volume of the gas (V) is directly proportional to its temperature in Kelvin (T) and inversely proportional to its pressure (P):
But what if we do not have a constant amount of gas?
What if we were to add more gas, or, remove some gas?
What would happen to the volume of the gas then?
Avogadro's Principle tells us that, for a gas at constant temperature and pressure, the volume of gas (V) is directly proportional to the number of gas molecules (N):
V ∝ N
and since a mole is equivalent to Avogadro's Number of molecules, we can say that the volume of a gas (V) is directly proportional to the moles of gas (n)
V ∝ n
Remember that we combined Boyle's law and Charles' Law to give the Combined Gas Equation:
So now we can also include V ∝ n to give a new relationship between the amount of gas (n), its volume (V), pressure (P), and temperature (T):
We can turn this relationship into an equation by using a constant of proportionality, R:
If we rearrange this equation, we find that
If we know the values P, V, n and T then we can find the value of the constant, R.
The volume (V) of different amounts of gas (n) was measured at a constant pressure of 101.3 kPa (1 atm) and 298 K (25° C).
These values were then used to calculate the value of the constant, R, as shown in the table below:
Experiment 
Pressure (kPa) 
Volume (L) 
Moles (mol) 
Temperature (K) 

1 
101.3 
24.5 
1.00 
298 
101.3 × 24.5 1 × 298 
R = 8.3 

2 
101.3 
48.9 
2.00 
298 
101.3 × 48.9 2 × 298 
R = 8.3 

3 
101.3 
73.4 
3.00 
298 
101.3 × 73.4 3 × 298 
R = 8.3 

4 
101.3 
97.8 
4.00 
298 
101.3 × 97.8 4 × 298 
R = 8.3 

5 
101.3 
122 
5.00 
298 
101.3 × 122 5 × 298 
R = 8.3 

If we measure pressure in kilopascals (kPa), volume in litres (L), temperature in Kelvin (K) and the amount of gas in moles (mol), then we find that R = 8.314 and it has the units kPa L K^{1} mol^{1}.
The pressure exerted by the gas in this volume is actually a measure of the energy of the gas particles, so the units of this Gas Constant, R, are most often expressed as Joules per Kelvin per mole, J K^{1} mol^{1}.
R = 8.314 J K^{1} mol^{1}
The gas constant R is very special because its value does not depend on the nature of the gas used.
If the experiment described above is done using:
 carbon dioxide gas, CO_{2(g)}, then R = 8.314 J K^{1} mol^{1}
 nitrogen gas, N_{2(g)}, then R = 8.314 J K^{1} mol^{1}
 helium gas, He_{(g)}, then R = 8.314 J K^{1} mol^{1}
So R is called the Gas Constant, and the equation PV = nRT is known as the Ideal Gas Equation, or, as the Ideal Gas Law.
Since R depends only on the amount of gas present, and does not depend on what the gas molecules are made up of, then the molecules of gas must not interact with each other, for, if they did, the value of R would be expected to change and reflect the impact of these interactions on the value of pressure (for instance).
By rearranging the Ideal Gas Law (Ideal Gas Equation) PV=nRT, it can be used to calculate the pressure (P), volume (V), temperature (T) or amount (n) of gas:
To calculate gas pressure: 
P = 
nRT V 
To calculate gas volume: 
V = 
nRT P 
To calculate gas temperature: 
T = 
PV nR 
To calculate amount of gas: 
n = 
PV RT 
Note that the Ideal Gas Law is supported by the Kinetic Theory of Gases:
 Ideal Gas Law says that at constant temperature (T) and volume (V), the pressure of a gas (P) is directly proportional to the amount of gas (n)
P ∝ n
That is, increase the moles of gas molecules and the pressure will increase.
Reduce the moles of gas molecules and the pressure will decrease.
Kinetic Theory of Gases tells us that pressure is caused by gas molecules colliding with the walls of the container, in other words, for a given volume of gas (at constant pressure):
P ∝ number of gas molecules
So, since the number of gas molecules is related to the amount of gas in moles, the Ideal Gas Law is supported by the Kinetic Theory of Gases.
 Ideal Gas Law says that the volume of a gas (V) is dependent on the amount of gas (n), its temperature (T) and its volume (V).
Note that the volume of a gas is not dependent on the size, or volume, of the gas molecules.
The volume of a gas depends on the amount of gas and NOT on any other properties of the gas.
Kinetic Theory of Gases tells us that a gas is made up of a huge number of gas particles which are so small that their sizes are negligible compared to the average distances between them, that is, the volume of gas molecules is negligible compared to the volume of space occupied by the gas.
In other words, the volume of space occupied by a gas depends on the number of the gas particles, but does not depend on the nature of the gas particles.
V ∝ number of gas molecules
So, since the number of gas molecules is related to the amount of gas in moles, the Ideal Gas Law is supported by the Kinetic Theory of Gases.