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Q. In the relation $P =\frac{a}{\beta} e ^{-\frac{a z}{ k \theta}}, p$ is the pressure, $z$ the distance, $k$ is Boltzmann constant and $\theta$ is the temperature, the dimensional formula of $\beta$ will be

BITSATBITSAT 2007

Solution:

In given equation, $\frac{ az }{ k \theta}$ should be dimensionless
$\therefore a =\frac{ k \theta}{ z }$
$\Rightarrow a =\frac{ ML ^{2} T ^{-2} K ^{-1} \times K }{ L }= MLT ^{-2}$
And $ p=\frac{a}{\beta} $
$\Rightarrow \beta=\frac{a}{p}=\frac{M L T^{-2}}{M L^{-1} T^{-2}}$
$=M^{0} \,L^{2}\, T^{0}$