First slide

Dimensions of physical quantities

Question

In the relation $p = \frac{α}{β} e^{- \frac{a z}{k θ}}$ , $p$ is pressure, $Z$ is distance, $k$ is Boltzmann constant and $θ$ is the temperature. The dimensional formula of $β$ will be

Moderate

A

$[M^{0} L^{2} T^{0}]$
B

$[{ML}^{2} T]$
C

${[ML T}^{- 1}]$
D

$[M^{0} L^{2} T^{- 1}]$

Solution

$[\frac{α Z}{k θ}] = [M^{0} L^{0} T^{0}] [α] = [\frac{k θ}{Z}]$
Further,
$[p] = [\frac{α}{β}] ∴ [β] = [\frac{α}{p}] = [\frac{k θ}{Z p}]$
Dimensions of $k θ$ are that to energy. Hence,
$[β] = \frac{M L^{2} T^{- 2}}{L M L^{- 1} T^{- 2}} = [M^{0} L^{2} T^{0}]$

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