Question

In the relation $P=\frac{\alpha }{\beta }{e}^{-\frac{\alpha Z}{k\theta }}$, $P$ is pressure, $Z$ is 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. $\left[M{L}^{2}T\right]$
• C. $\left[M{L}^0{T}^{-1}\right]$
• D. $\left[M^0{L}^2{T}^{-1}\right]$

Answer 1

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

Since exponential term is dimensionless,

Dimension of $\alpha$, $\left[\alpha \right]=\frac{\left[k\right]\left[\theta \right]}{\left[Z\right]}=\frac{\left(M{L}^2{T}^{-2}{K}^{-1}\right)\ast \left(K\right)}{L}=ML{T}^{-2}$

Since $\frac{\alpha }{\beta }$ must have the dimensions of pressure($M{L}^{-1}{T}^{-2}$)

Therefore, dimensions of $\beta$, $\left[\beta \right]=\frac{ML{T}^{-2}}{M{L}^{-1}{T}^{-2}}=L^2$