Derivation of Ideal Gas Equation

Gases are defined by 4 important parameters

These are pressure, volume, temperature and number of molecules

Several scientists conducted experiments to find relationships between these paramters

From those experiments, they gave laws called gas laws

Let us begin with Boyle’s law

Boyle’s law states that at a given temperature, Pressure exerted by a fixed amount of gas is inversely proportional to the volume occupied by the gas

Therefore according to Boyle's law, at a constant $T$ and $n$ - $V\propto \frac{1}{P}$

Now lets see Charles' law

Charles’ law states that at a given pressure, the volume of a fixed amount of gas is directly proportional to its temperature

Therefore according to Charles' law, at a constant $P$ and $n$ - $V\propto T$

Finally lets see Avogadro’s law

Avogadro’s law states that for gases having equal volume, at the same Temperature and Pressure, the gases will have equal no of molecules as well

Therefore at constant $T$ and $P$, $V\propto n$

A gas that follows Boyle’s law, Charles’ law and Avogadro's law strictly is called an ideal gas

So these three gas laws can be combined to form an equation which is followed by all ideal gases

This equation is called the Ideal Gas equation

From the three laws, we have $V\propto \frac{1}{P}$, at constant T and n $V\propto T$, at constant P and n $V\propto n$, at constant T and P

Combining the three laws we get,$$\begin{gathered} V\propto \frac{Tn}{P}, \\ PV\propto nT \end{gathered}$$

We can re-write this as $PV=nRT$ This is the Ideal Gas Equation

Here R is a constant called the universal gas constant. R is equal to $0.082057Latmmo{l}^{-1}{K}^{-1}$

The ideal gas equation is very important because if we know 3 gas parameters, we can find the 4ᵗʰ one using it

Revision

The Ideal Gas equation is obtained from the Ideal Gas model formulated by scientists like Charles, Boyle, Avogadro etc to explain the behaviour of gases

The Ideal Gas Equation is given as - $$\begin{gathered} PV=nRT,\text{where,} \\ P\text{=Pressure of the Gas} \\ V\text{=Volume of the Gas} \\ n\text{=No of moles of the Gas} \\ R\text{=Universal Gas Constant} \\ T\text{=Temperature of the Gas} \end{gathered}$$

R is equal to $0.082057Latmmo{l}^{-1}{K}^{-1}$

The End