Q.Using kinetic theory, show how

From kinetic theory (Section 9.3-9.5), $PV = Nk_BT$.

1. Charles's law. At constant pressure $P$ and for a fixed amount of gas $N$, rearrange to make $V$ the subject:

   $V = \frac{Nk_B}{P}T$

   Since $N$, $k_B$ and $P$ are all held fixed, everything in front of $T$ is a constant, so $V \propto T$ -- Charles's law.
2. Avogadro's law. Consider two DIFFERENT gases occupying the same volume $V$, at the same pressure $P$ and same temperature $T$. Rearranging $PV=Nk_BT$ to make $N$ the subject, $N = \frac{PV}{k_BT}$ …

Rearrange the kinetic-theory relation $PV=Nk_BT$ twice: once holding $P$ fixed (Charles), once compar …

• For Avogadro's law, forgetting to note explicitly that $N=PV/(k_BT)$ contains NO dependence on molecular mass $m$ -- this is exactly why the law holds regardless of which two gases are being compared. …