An expression for a dimensionless quantity P is given by $\mathrm{P}=\frac{\alpha}{\beta} \log _{\mathrm{e}}\left(\frac{\mathrm{kt}}{\beta \mathrm{x}}\right)$; where $\alpha$ and $\beta$ are constants, x is distance; k is Boltzmann constant and $t$ is the temperature. Then the dimensions of $\alpha$ will be :

(C) $\left[\mathrm{MLT}^{-2}\right]$

(A) $\left[\mathrm{M}^{0} \mathrm{~L}^{-1} \mathrm{~T}^{0}\right]$

(B) $\left[\mathrm{ML}^{0} \mathrm{~T}^{-2}\right]$

(D) $\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]$

JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Units & Measurement

An expression for a dimensionless quantity P is given by

P = \frac{α}{β} lo g_{e} (\frac{kt}{β x})

; where

α

and

β

are constants, x is distance; k is Boltzmann constant and

t

is the temperature. Then the dimensions of

α

will be :

Answer: (C)

Since P is dimensionless, the argument of the logarithm must be dimensionless. From

\frac{k t}{β x} = 1

, we get

[β] = \frac{[ k ] [ t ]}{[ x ]}

. Since Boltzmann constant k has dimensions of energy (

k = E / T

), we have

[k] [t] = [ML^{2} T^{- 2}]

. Therefore

[β] = \frac{[ ML ^{2} T ^{- 2} ]}{[ L ]} = [MLT^{- 2}]

. Since P is dimensionless and

P = \frac{α}{β} \times (dimensionless)

, we must have

[α] = [β] = [MLT^{- 2}]

JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Single correct option