Question

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. $\left[ M L ^{2} T ^{-2}\right]$
• B. $\left[ M L T ^{-2}\right]$
• C. $\left[ M ^{2} L T ^{2}\right]$
• D. $\left[ M ^{0} L T ^{0}\right]$

Answer 1

Answer: (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 $[ W ]=[\alpha][\beta]^{2}$
$[\beta]=\sqrt{\frac{ ML ^{2} T ^{-2}}{ M ^{-1} T ^{2}}}= M ^{1} L ^{1} T ^{-2}$