Question

If work is given by $W=\alpha {\beta }^2{e}^{\left[\frac{-x^2}{\alpha kT}\right]}$ where, $k=$ Boltzmann constant, $T$ = temperature, $x$ is displacement, then the dimensional formula for $\beta$ is

• A. $\left[M^1{L}^1{T}^{-2}\right]$
• B. $\left[M^{-1}{L}^{-1}{T}^2\right]$
• C. $\left[M^1{L}^2{T}^{-2}\right]$
• D. $\left[M^2{L}^1{T}^2\right]$

Answer 1

The correct option is A $\left[M^1{L}^1{T}^{-2}\right]$

Given, work $W=\alpha {\beta }^2{e}^{\left[\frac{-x^2}{\alpha kT}\right]}$

We know that the power term is dimensionless. So,

$\left[\frac{x^2}{\alpha kT}\right]=$ dimensionless

$\Rightarrow \left[\alpha \right]=\left[\frac{x^2}{kT}\right]$

As we know, $v_{rms}=\sqrt{\frac{3kT}{M}}$
Hence, $\left[kT\right]=\left[v^{2}M\right]$ where $M$ is molecular mass (in $\text{kg}$)

$\Rightarrow \left[\alpha \right]=\left[\frac{x^2}{v^{2}M}\right]$

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

$\Rightarrow \left[\alpha \right]=\left[M^{-1}{T}^2\right]..........\left(1\right)$

Now, as the exponential function is also dimensionless, so we will have for work $\text{W}$,

[$W$] $=\left[\alpha {\beta }^2\right]\times$ Dimensionless

$\Rightarrow \left[M{L}^2{T}^{-2}\right]=\left[M^{-1}{T}^2\right]\left[{\beta }^2\right]$

$\Rightarrow \left[{\beta }^2\right]=\left[M^2{L}^2{T}^{-4}\right]$

$\Rightarrow \left[\beta \right]=\left[M^1{L}^1{T}^{-2}\right]$

$\begin{array}{c}\text{Why this question}?\\ \text{Concept - Exponential function and power of an exponential}\\ \text{function is a dimensionless quantity.}\\ \text{If}A=e^x,\\ x\to \text{dimensionless}\\ A\to \text{dimensionless}\end{array}$