Q. In the relation $P = \frac{\alpha}{\beta}e^{\frac{\alpha Z}{k\theta}}$ $P$ is pressure, $Z$ is the distance, $k$ is Boltzmann’s constant and $\theta$ is the temperature. The dimensional formula of $\beta$ will be
A) $\lbrack M^{0}L^{2}T^{0}\rbrack$
B) $\lbrack M^{1}L^{2}T^{1}\rbrack$
C) $\lbrack M^{1}L^{0}T^{- 1}\rbrack$
D) $\lbrack M^{0}L^{2}T^{- 1}\rbrack$
Explanation
In given equation, $\frac{\alpha z}{k\theta}$ should be dimensionless $\therefore\ \alpha = \frac{k\theta}{z}$ $\Rightarrow \lbrack\alpha\rbrack = \frac{\lbrack ML^{2}T^{- 2}K^{- 1} \times K\rbrack}{\lbrack L\rbrack} = \lbrack MLT^{- 2}\rbrack$ And $P = \frac{\alpha}{\beta}$ $\Rightarrow \lbrack\beta\rbrack = \left\lbrack \frac{\alpha}{P} \right\rbrack = \frac{\left\lbrack MLT^{- 2} \right\rbrack}{\lbrack ML^{- 1}T^{- 2}\rbrack}$ $\Rightarrow \lbrack\beta\rbrack = \lbrack M^{0}L^{2}T^{0}\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.