Question

Pressure depends on distance as, $P=\frac{\alpha }{\beta }exp(-\frac{\alpha z}{k\theta })$, where $\alpha ,\beta$ are constants, $z$ is distance, $k$ is Boltzman's constant and $\theta$ is temperature. The dimensions of $\beta$ are

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

Answer 1

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

The term $\frac{\alpha z}{k\theta }$ should be dimensionless as it as an exponent over $e$.
$\frac{\alpha z}{k\theta }=\frac{\alpha m}{\frac{J}{K}.K}=\frac{\alpha m}{J}$

For the fraction to be dimensionless $\alpha$ should have units of $J{m}^{-1}$.
And $\frac{\alpha }{\beta }$ should have the dimension of Pressure that is $N{m}^{-2}$.

$\frac{J{m}^{-1}}{\beta }=N{m}^{-2}$
$\beta =\frac{J{m}^{-1}}{N{m}^{-2}}=\frac{Nm.m^{-1}}{N{m}^{-2}}=m^2=L^2$