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A gas is in e...

Question

A gas is in equilibrium at $T$ kelvin. Mass of one molecule is $m$ and, its component of velocity in $x$ direction is $v_{x}$ . The mean of its $v_{x}^{2}$ is represented by $\frac{n k T}{m}$ . Find $n$ .

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Solution

$v = v_{x}^i + v_{y}^j + v_{z}^k$
$v = \sqrt{v_{x}^{2} + v_{y}^{2} + v_{z}^{2}}$ for random motion $v_{x} = v_{y} = v_{z}$
$v = \sqrt{3} v_{x}$
$v = \sqrt{\frac{3 k T}{m}} \Rightarrow \sqrt{3} v_{x} = \sqrt{\frac{3 k T}{m}}$
$v_{x}^{2} = \frac{k T}{m}$

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