Q.Deduce Avogadro's law based on kinetic theory.

The historical law. Avogadro's law (or Avogadro's hypothesis) states that equal volumes of any two gases, at the same temperature and pressure, contain exactly the same number of molecules -- regardless of what the gases actually are, or how different their individual molecular masses might be. Historically this was proposed purely as a hypothesis to explain gas-combination volume ratios; kinetic theory derives it as a direct, unavoidable consequence.

The derivation. For two gases at the same $T$ and $P$, applying the kinetic-theory pressure formula to each separately gives $P=\tfrac13\tfrac{N_1}{V}m_1\overline{v_1^2}=\tfrac13\tfrac{N_2}{V}m_2\overline{v_2^2}$. Since both gases are at the same temperature, they share the same average kinetic energy per molecule (temperature depends only on $\overline{KE}$, never on molecular identity): $\tfrac12m_1\overline{v_1^2}=\tfrac12m_2\overline{v_2^2}$. Dividing the pressure equation by this kinetic-energy equation cancels every mass-and-speed factor entirely, leaving simply $N_1=N_2$. …