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 $P$ will be

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

Answer 1

Answer: (A)

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