JEE Main · 2021 · 24 Feb 2021 · Shift 1 · Units & Measurement

The work done by a gas molecule in an isolated system is given by

W = α β^{2} e^{- \frac{x ^{2}}{α k T}}

, where

x

is the displacement,

k

is the Boltzmann constant and

T

is the temperature.

α

and

β

are constants. Then the dimensions of

β

will be:

Answer: (A)

Solution

For the exponent

\frac{x ^{2}}{α k T}

to be dimensionless, we need

[α] = \frac{L ^{2}}{M L ^{2} T ^{- 2}} = M^{- 1} T^{2}

.

Since

W

has dimensions

[M L^{2} T^{- 2}]

, and

W = α β^{2}

, we have:

[α β^{2}] = M L^{2} T^{- 2}

[β^{2}] = \frac{M L ^{2} T ^{- 2}}{M ^{- 1} T ^{2}} = M^{2} L^{2} T^{- 4}

[β] = M L T^{- 2}

JEE Main · 2021 · 24 Feb 2021 · Shift 1 · Single correct option