Question

In the relation $p=\frac{\alpha }{\beta }{e}^{-\frac{\alpha z}{k\theta }},p$ is the pressure, $z$ the distance, $k$ is Boltzmann constant and $\theta$ is the temperature, the dimensional formula of $\beta$ will be

• A. $\left[M^0{L}^2{T}^0\right]$
• B. $\left[M{L}^2{T}^0\right]$
• C. $\left[M{L}^0{T}^{-1}\right]$
• D. $\left[M{L}^2{T}^{-1}\right]$

Answer 1

The correct option is A $\left[M^0{L}^2{T}^0\right]$

In the given equation, $\frac{\alpha z}{k\theta }$ should be dimensionless
$\therefore \alpha =\frac{k\theta }{z}$
$\Rightarrow \left[a\right]=\frac{\left[M{L}^2{T}^{-2}{K}^{-1}\right]\times \left[K\right]}{\left[L\right]}=\left[ML{T}^{-2}\right]$
and $P=\frac{\alpha }{\beta }$
$\Rightarrow \left[a\right]=\frac{\left[a\right]}{\left[r\right]}=\frac{\left[ML{T}^2\right]}{\left[M{L}^{-1}{T}^{-2}\right]}=\left[M^0{L}^2{T}^0\right]$.