Of all the physical states, the gaseous state is the easiest to study, as there are minimum physical interactions between the molecules (or atoms). The parameters are limited only to temperature, pressure and the number of moles of the molecules.
Robert Boyle in 1661 showed that the volume of a fixed amount of gas in a container is inversely proportional to the pressure at a fixed temperature.
1/P  Robert Boyle |
 Jacques Charles |
 Gay-Lussac |
Around 1787 Jacques Charles had found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 degree interval, but did not publish it. In 1802, Joseph LouisGay-Lussac published that for a fixed amount of gas the volume is directly proportional to the temperature at a fixed pressure.
T This eventually came to be known as Charles' Law.
In the early 1800s Amedo Avogadro proposed that the number of mole of gaseous molecules in a container is directly proportional to the volume when pressure and temperature are held constant.
n Putting all these together, this meant that;
nT/P A proportional relationship can always be made an equation by including a constant. So,
It is only to be expected that the constant R be known as the gas constant. This equation is now known as the ideal gas equation, or just gas equation.
Avogradro has found that one mole of gas at 760 torr pressure and 0�C, occupied a volume of 22.41 litres. In SI units it will be 1 mole, 101.325 x 10� J m‾� pressure, 273.15 K temperature, and 22.414 x 10‾� m� volume. So the value of R would be;
| R | = | PV / nT |
| = | 101.325 x 10� J m‾� x 22.414 x 10‾� m� / (1 mol x 273.15 K) | |
| = | 8.314 JK‾� mol‾� |
Tutorial 1
- What is the density of nitrogen gas at STP?
- Given that the the density of oxygen is 1.429 kg m‾�, what will be the density of argon at the same condition of temperature and pressure (Atomic mass: oxygen=16, argon=40)?
- Assuming that air is 78% by volume of nitrogen, 21% by volume of oxygen and 1% by volume of carbon dioxide, what is the density of air at 25�C, at 760 torr pressure?
- The density of the following gases were found to be:
Compute the relative mass of the molecule or atom given that the mass of oxygen gas (O2) is 32. Answer
Gas Oxygen Nitrogen Flourine Neon Chlorine Density / kg m-3 1.429 1.251 1.697 0.9000 3.160
 John Dalton |
Tutorial 2
Dalton's Law of Partial Pressures: The pressure exerted by a mixture of perfect gases is the sum of the pressures exerted by the individual gases if they have occupied the same volume individually. Let us put n(A) moles of a gas in a container of volume V and at temperature T. Let the pressure measured be p(A). Let us repeat this with gas B, then gas C, etc; and the respective pressure measured be p(B), p(C), etc. Derive Dalton's Law of Partial Pressure. Answer
INDUSTRIAL USES of GASES
The beauty of the gas equation is because the molecules are free from all intermolecular interactions. However at high pressure (forcing the molecules closer to each other) or low temperature (the molecules are moving slowly passed each other) such a freedom is lost, and the gas equation fails.
So at low pressure (like 760 torr pressure) and high temperature (like 25�C) we refer to it as ideal gas or perfect gas; a sentiment for its freedom rather then from a scientific perspective.
In the industry gases are normally used at high pressure (like 20 bar pressure or more). So from an application perspective gases at such high pressure is referred to as real gas, like in real action. Again an emotional choice rather then from a scientific perspective. After all gases are real.
To quantify when a gas becomes a real gas we define a compression factor Z;
When the values of (p, V, T, n) T gives a value of Z = 1 we say it is a perfect gas. When Z > 1, then it is a real gas and the molecules are experiencing repulsive intermolecular interaction. When Z < 1, they are experiencing attractive intermolecular interaction.
When Z ≠ 1 the gas equation is no longer applicable. However, to consider each and every interaction between the molecules to derive a suitable equation is too complex. The best approach is by trial and error. A very well used method is to write a series expansion.
Then using the data compiled for the system we will attempt to compute for A1, A2, A3 ..... Depending on the magnitude of intermolecular interactions the value of A2 and above might be so small that they contribute negligibly to the value of pV and the equation for the particular system can assume to be;
So this becomes a tedious experimental approach where the equation for the gas must be derived on a case−by−case basis and the appropriate Ais values determined. Such equations are known as the virial equation of state.
Even among the virial equations of state there are several forms. For example there is also an attempt to express it as;
The coefficients, Ais and are not unique. They need to be determined for each and every system.
The van der Waals Equation
 van der Waal |
where nb is the volume taken up by the molecules. Since the pressure is affected by the intermolecular interaction he proposed that the pressure be corrected by the square of the concentration of the gas (n/V)�. That is;
This became known as the van der Waals equation of state. The "a and b" are obtained from experiments and must be determined for each system. It was found that within expectation it is fairly constant for the same gas, regardless of the temperature and pressure.
This is the best attempt up to date. This equation is not derived from first principles. It is just trying to do the best under a difficult situation. So it must taken as a fair correlation.
Tutorial 3
Comment on the trends of a and b? Answer
Gas He Ar N2 CO2 a / atm dm6mol‾� 0.034 1.345 1.390 3.592 b / dm� mol‾� 0.0237 0.0322 0.0391 0.0427
Even so the equation is not easy to use. For example if you want to compute the volume of one mole of gas at certain (T, p, a and b) values, you will end up having to solve the equation;
Try this out for one mole of helium at 100�C and 100 atm. (Answer: 0.329 dm� mol‾�)
