According to Avogadro's law, why do equal volumes of molecular hydrogen and nitrogen contain the same number of molecules when they are at the same temperature and pressure?

This is a very interesting question. Avogadro Law gives the relationship between volume and amount when pressure and temperature are held constant. Remember amount is measured in moles. Also, since volume is one of the variables, that means the container holding the gas is flexible in some way and can expand or contract.

If the amount of gas in a container is increased, the volume increases.

If the amount of gas in a container is decreased, the volume decreases.

Suppose the amount is increased. This means there are more gas molecules and this will increase the number of impacts on the container walls. This means the gas pressure inside the container will increase (for an instant), becoming greater than the pressure on the outside of the walls. This causes the walls to move outward. In this process the volume of gas has increased.

Suppose the amount is decreased. This means there are less gas molecules and this will decrease the number of impacts on the container walls. This means the gas pressure inside the container will decrease (for an instant), becoming lesser than the pressure on the outside of the walls. This causes the walls to move inward. In this process the volume of gas will decrease.

The mathematical form of Avogadro's Law is: V ÷ n = k

This means that the volume-amount fraction will always be the same value if the pressure and temperature remain constant.

Let #V_1# and #n_1# be a volume-amount pair of data at the start of an experiment. If the amount is changed to a new value called #n_2#, then the volume will change to #V_2#.

We know this: #V_1# ÷ #n_1# = k

And we know this: #V_2# ÷ #n_2# = k

Since k = k, we can conclude that #V_1# ÷ #n_1# = #V_2# ÷ #n_2#

Let us suppose that we have a container whose Volume is #V_1# Liters , It is filled with n moles of Hydrogen gas, the temperature of the gas is T Kelvin, and its Pressure is P atmosphere.

we have a container whose Volume is #V_1# Liters , It is filled with x moles of Hydrogen gas, the temperature of the gas is T Kelvin, and its Pressure is P atmosphere.

as per the law
#V_1# / #n_1# = #V_2# /#n_2#

#V_1# / n = #V_1# / x

#V_1# . x = #V_1# . n

on solving

x = v , the two gases if they take up same volume and has pressure , at the same temperature then they must have similar # of moles.