Boyle’s law requires that the temperature of a gas is fixed. What happens if the temperature of the gas is allowed to change? Figure 20.6 shows the results of an experiment in which a fixed mass of gas is cooled at constant pressure. The gas contracts; its volume decreases.

This graph does not show that the volume of a gas is proportional to its temperature on the Celsius scale.
If a gas contracted to zero volume at ${0^ \circ }C$, the atmosphere would condense on a cold day and we would have a great deal of difficulty in breathing! However, the graph does show that there is a temperature at which the volume of a gas does, in principle, shrink to zero. Looking at the lower temperature scale on the graph, where temperatures are shown in kelvin (K), we can see that this temperature is $0 K$, or absolute zero. (Historically, this is how the idea of absolute zero first arose.)
We can represent the relationship between volume V and thermodynamic temperature T as:

$V \propto T$

or simply:

$\frac{V}{T} = constant$

Note that this relationship only applies to a fixed mass of gas and to constant pressure.
This relationship is an expression of Charles’s law, named after the French physicist Jacques Charles, who in 1787 experimented with different gases kept at constant pressure.
If we combine Boyle’s law and Charles’s law, we can arrive at a single equation for a fixed mass of gas:

$\frac{{pV}}{T} = constant$

Shortly, we will look at the constant quantity that appears in this equation, but first we will consider the extent to which this equation applies to real gases.

Real and ideal gases

The relationships between p, V and T that we have considered are based on experimental observations of gases such as air, helium, nitrogen and so on, at temperatures and pressures around room temperature and pressure. In practice, if we change to more extreme conditions, such as low temperatures or high pressures, gases start to deviate from these laws as the gas atoms exert significant electrical forces on each other. For example, Figure 20.7 shows what happens when nitrogen is cooled down towards absolute zero. At first, the graph of volume against temperature follows a good straight line. However, as it approaches the temperature at which it condenses, it deviates from ideal behaviour and at $77 K$ it condenses to become liquid nitrogen.

Thus, we have to attach a condition to the relationships discussed earlier. We say that they apply to an ideal gas.
When we are dealing with real gases, we have to be aware that their behaviour may be significantly different from the ideal gas.
An ideal gas is thus one for which we can apply the equation:

$\frac{{pV}}{T} = constant\,for\,a\,fixed\,mass\,of\,gas$