Question

A cubical box of side of $1 \,m$ contains helium gas (atomic weight $4$) at a pressure of $100 \,N / m ^{2}$. During an observation time of $1 \,s$, an atom traveling with the root-mean-square speed parallel to one of the edges of the cube, was found to make $500$ hits with a particular wall, without any collision with other atoms. Evaluate the temperature (in $ K$) of the gas. (Take $R=$ $25 / 3 \,J \,mol ^{-1} \,K ^{-1}$ and $\left.k=1.38 \times 10^{-23} \,J / K \right)$

Answer 1

$v_{ rms }=\sqrt{\frac{3 R T}{M}}$
Since gas molecule makes $500$ collisions in $1\, s$,
$\therefore $ Distance travelled in $1 \,s =1000 \,m$
$\therefore $ From the given relation,
$\sqrt{\frac{3 R T}{M}}=1000$
$ \Rightarrow 3 \times \frac{25}{3} \times T \times \frac{1}{4 \times 10^{-3}}=10^{6}$
$\Rightarrow T=160 \,K$