Explain Avogadro's hypothesis by relating pressure,temperature,and volume of a gas.

(N/A) The equation relating pressure $(P)$,volume $(V)$,and temperature $(T)$ of a gas is given by:
$PV = KT$
Where $T$ is the absolute temperature in Kelvin.
$K$ is a constant for a given gas,but it varies with the amount of gas.
$K$ is proportional to the number of molecules $(N)$ in the given sample of gas.
$\therefore K \propto N$
$\therefore K = k_{B} N$
Here,$k_{B}$ is the Boltzmann constant.
Its value is the same for all gases.
The unit of $k_{B}$ is $\text{J/K}$ and its dimension is $[M^{1} L^{2} T^{-2} K^{-1}]$.
Substituting $K$ into the first equation:
$PV = k_{B} NT$
$\therefore \frac{PV}{NT} = k_{B} = \text{constant}$
This implies that for two different gas samples:
$\frac{P_{1} V_{1}}{N_{1} T_{1}} = \frac{P_{2} V_{2}}{N_{2} T_{2}}$
If $P, V,$ and $T$ are the same for different gases,then $N$ must also be equal. This represents Avogadro's hypothesis,which states: Equal volumes of all gases at the same temperature and pressure contain the same number of molecules.