In a typical combustion engine the work done by a gas molecule is given W =α2 β e (-β x 2/ kT ), where x is the displacement, k is the Boltzmann constant and T is the temperature. If α and β are constants, dimensions of α will be :

Question

In a typical combustion engine the work done by a gas molecule is given W =α2 β e (-β x 2/ kT ), where x is the displacement, k is the Boltzmann constant and T is the temperature. If α and β are constants, dimensions of α will be : (A) [ MLT -2] (B) [ M 0 LT 0] (C) [ M 2 LT -2] (D) [MLT-1]

Tardigrade Admin · Accepted Answer

kT has dimension of energy (β x 2/ kT ) is dimensionless [β][ L 2]=[ ML 2 T -2] [β]=[ MT -2] α2 β has dimensions of work [α2][ MT -2]=[ ML 2 T -2] [α]=[ M 0 LT 0]

Q. In a typical combustion engine the work done by a gas molecule is given $W = α^{2} β e^{\frac{- β x ^{2}}{k T}}$ , where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature. If $α$ and $β$ are constants, dimensions of $α$ will be :

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

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

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

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

$k T$ has dimension of energy
$\frac{β x ^{2}}{k T}$ is dimensionless
$[β] [L^{2}] = [M L^{2} T^{- 2}]$
$[β] = [M T^{- 2}]$
$α^{2} β$ has dimensions of work
$[α^{2}] [M T^{- 2}] = [M L^{2} T^{- 2}]$
$[α] = [M^{0} L T^{0}]$

Q. In a typical combustion engine the work done by a gas molecule is given W=α2βekT−βx2​, where x is the displacement, k is the Boltzmann constant and T is the temperature. If α and β are constants, dimensions of α will be :

[MLT−2]

[M0LT0]

[M2LT−2]

[MLT−1]