Empirical laws

There are three main empirically derived laws that combine into the Ideal Gas Law. This article then gives a worked example of how to apply them.

Avogadro’s Law

Avogadro’s Law states that the volume a gas occupies is proportional to the number of moles in it. If we increase the number of moles, the volume increases, too. This can be expressed as (V propto n), but it’s more convenient when looking at the ideal gas law to think of:

[frac{V_1}{n_1} = frac{V_2}{n_2}]

Gay-Lussac’s law

The Gay-Lussac’s law shows that pressure and temperature are related if the volume is fixed. If we increase pressure in a fixed volume, temperature increases. If we increase temperature, pressure increases. This allows us to calculate a new temperature or pressure if we know the starting point, and what changes:

[frac{p_1}{T_1} = frac{p_2}{T_2}]

Boyle’s Law

Boyle’s Law shows that pressure multiplied by volume is a constant. If we change pressure or volume (at a constant temperature) this allows us to calculate a new volume or pressure. It’s expressed conveniently as:

[p_1 V_1 = p_2 V_2]

Charles’ Law

Finally, Charles’ Law relates volume and temperature the same way. If we fix pressure, then we can calculate volume and temperature changes. This is particularly useful for systems that are open to the atmosphere, because pressure is constant.

[frac{V_1}{T_1} = frac{V_2}{T_2}]

Combining these together

The above laws were established as far back as the 18th century. Benoît Paul Émile Clapeyron combined them together in 1834, proposing the Ideal Gas Law:

[pV=nRT]

Where R was a universal gas constant of 8.314 J K^-1 mol^-1, that we can obtain from experiment. If you notice carefully, we can rearrange the ideal gas law to obtain all four of the above laws – if we know what’s constant.

We rearrange pV=nRT to move all of the constant variables to one side. Most typically, “moles” is constant, because we work with changes to a single gas. Then you find that the variables on the other side look like the empirical laws above.

You probably don’t need to memorise all of these, or who they’re named after, as you can derive them from the ideal gas law. You can also use the ideal gas law to do a calculation in two steps. 1) work out the factors that are constant, and then 2) recalculate your unknown variable. Or, you can do it in one step by rearranging one of the empirical laws.

Worked example

Boyle’s Law

Boyle’s Law is (p_1 V_1 = p_2 V_2), which is true for constant temperature, as (pV=nRT) – the right hand side is constant for constant temperature.

If we begin at 1 atm and 1 m³, and then double the pressure, the volume should half.

So, by Boyle’s law, (p_1 V_1 = p_2 V_2), so:

[frac{p_1 V_1}{p_2} = V_2]

Adding in the numbers:

[frac{1 atm times 1 m^3}{2 atm} = frac{1}{2} m^3]

One useful thing about this approach is that you need to be less careful with your units. In this example, there is simply a ratio of two pressures involved; and that ratio is independent of the unit. It can be atmospheres or Pascals, the ratio will be the same. So whatever your input volume unit is (in this case, m³), will be what your output unit is.