Question

In a typical combustion engine, the work done by a gas molecule is given by,

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

Where $x$ is the displacement, $k$ is the Boltzmann constant, $T$ is the temperature, $\alpha$ and $\beta$ are constants, then the dimension of $\alpha$ will be :

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

Answer 1

The correct option is A $\left[M^{0}L{T}^0\right]$

Given:
Work done, $W={\alpha }^2\beta e^{-\beta x^2/kT}$

Where,
$x=$ Displacement
$k=$ Boltzmann constant
$T=$ Temperature
$\alpha ,\beta =$ Constants

We know that,
$-\frac{\beta x^2}{kT}$ $=$ dimensionless

$\Rightarrow \left[\frac{\beta x^2}{kT}\right]=\left[M^0{L}^0{T}^0\right]$

$\left[kT\right]=\left[PV\right]=\left[M{L}^2{T}^{-2}\right]$
$\left[x^2\right]=\left[L^2\right]$

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

Also,
$\left[W\right]=\left[{\alpha }^2\beta \right]$
$\Rightarrow \left[M{L}^2{T}^{-2}\right]=\left[M^1{T}^{-2}\right]\left[{\alpha }^2\right]$
$\Rightarrow \left[{\alpha }^2\right]=\left[L^2\right]$
$\Rightarrow \left[\alpha \right]=\left[M^{0}L{T}^0\right]$