Given expression

$W = \alpha \beta^2 e^{-\frac{x^2}{\alpha kT}}$

where:

• $W$ = work done
• $x$ = displacement
• $k$ = Boltzmann constant
• $T$ = temperature

Dimension of work

Work has dimensions:

$[W] = [ML^2T^{-2}]$

Use the exponential condition

The argument of an exponential must be dimensionless:

$\frac{x^2}{\alpha kT} \text{ is dimensionless}$

Therefore,

$[x^2] = [\alpha kT]$

Find dimension of $kT$

Boltzmann constant $k$ has dimension:

$[k] = \frac{\text{energy}}{\text{temperature}} = [ML^2T^{-2}\Theta^{-1}]$

Hence,

$[kT] = [ML^2T^{-2}]$

Find dimension of $\alpha$

Since

$[x^2] = [L^2]$

and

$[\alpha kT] = [\alpha][ML^2T^{-2}]$

equating,

$[\alpha][ML^2T^{-2}] = [L^2]$

so

$[\alpha] = [M^{-1}T^2]$

Now use the full equation for work

Since exponential term is dimensionless,

$[W] = [\alpha \beta^2]$

Therefore,

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

$[\beta^2] = \frac{[ML^2T^{-2}]}{[M^{-1}T^2]} = [M^2L^2T^{-4}]$

Taking square root,

$[\beta] = [MLT^{-2}]$

Match with options

$[\beta] = [MLT^{-2}]$

This corresponds to Option C.