The workdone by a gas molecule in an isolated system is given by,… - Vidyadip

MCQ

The workdone by a gas molecule in an isolated system is given by, $W =\alpha \beta^{2} e ^{-\frac{ x ^{2}}{\alpha kT }},$ where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature, $\alpha$ and $\beta$ are constants. Then the dimension of $\beta$ will be

A
$\left[ M L ^{2} T ^{-2}\right]$
✓
$\left[ M L T ^{-2}\right]$
C
$\left[ M ^{2} L T ^{2}\right]$
D
$\left[ M ^{0} L T ^{0}\right]$

✓

Answer

Correct option: B.

$\left[ M L T ^{-2}\right]$

b
$\frac{ x ^{2}}{\alpha kT } \rightarrow$ dimensionless

$\Rightarrow[\alpha]=\frac{\left[ x ^{2}\right]}{[ kT ]}=\frac{ L ^{2}}{ ML ^{2} T ^{-2}}= M ^{-1} T ^{2}$

Now $[ W ]=[\alpha][\beta]^{2}$

$[\beta]=\sqrt{\frac{ ML ^{2} T ^{-2}}{ M ^{-1} T ^{2}}}= M ^{1} L ^{1} T ^{-2}$

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