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

Answer 1

Answer: (A)

In given equation, $\frac{a z}{k \theta }$ should be dimensionless
$ \, \, \alpha =\frac{k \theta }{z}$
$\Rightarrow \, \, \left[\alpha \right]=\frac{\left[M L^{2} T^{- 2} K^{- 1} \times K\right]}{\left[L\right]}=\left[M L T^{- 2}\right]$
And $ \, p=\frac{\alpha }{\beta }$
$\Rightarrow \, \left[\beta \right]=\left[\frac{\alpha }{p}\right]=\frac{\left[M L T^{- 2}\right]}{\left[M L^{- 1} T^{- 2}\right]}=\left[M^{0} L^{2} T^{0}\right]$