The workdone by a gas molecule in an isolated system is given by, W =α β2 e -( x 2/α kT ), where x is the displacement, k is the Boltzmann constant and T is the temperature, α and β are constants. Then the dimension of β will be :

Question

The workdone by a gas molecule in an isolated system is given by, W =α β2 e -( x 2/α kT ), where x is the displacement, k is the Boltzmann constant and T is the temperature, α and β are constants. Then the dimension of β will be : (A) [ M L 2 T -2] (B) [ M L T -2] (C) [ M 2 L T 2] (D) [ M 0 L T 0]

Tardigrade Admin · Accepted Answer

( x 2/α kT ) arrow dimensionless ⇒ [α]=([ x 2]/[ kT ]) =( L 2/ ML 2 T -2)= M -1 T 2 Now [ W ]=[α][β]2 [β]=√( ML 2 T -2/ M -1 T 2)= M 1 L 1 T -2

Q. The workdone 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 dimension of $β$ will be :

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

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

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

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

$\frac{x ^{2}}{α k T} \to$ dimensionless
$\Rightarrow [α] = \frac{[ x ^{2} ]}{[ k T ]}$
$= \frac{L ^{2}}{M L ^{2} T ^{- 2}} = M^{- 1} T^{2}$
Now $[W] = [α] [β]^{2}$
$[β] = \frac{M L ^{2} T ^{- 2}}{M ^{- 1} T ^{2}} = M^{1} L^{1} T^{- 2}$

Q. The workdone by a gas molecule in an isolated system is given by, W=αβ2e−αkTx2​, where x is the displacement, k is the Boltzmann constant and T is the temperature, α and β are constants. Then the dimension of β will be :

[ML2T−2]

[MLT−2]

[M2LT2]

[M0LT0]