Question

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

• A. $\left[ MLT ^{-2}\right]$
• B. $\left[ M ^{0} LT ^{0}\right]$
• C. $\left[ M ^{2} LT ^{-2}\right]$
• D. $[MLT^{-1}]$

Answer 1

Answer: (B)

$kT$ has dimension of energy
$\frac{\beta x ^{2}}{ kT }$ is dimensionless
$[\beta]\left[ L ^{2}\right]=\left[ ML ^{2} T ^{-2}\right]$
$[\beta]=\left[ MT ^{-2}\right]$
$\alpha^{2} \beta$ has dimensions of work
$\left[\alpha^{2}\right]\left[ MT ^{-2}\right]=\left[ ML ^{2} T ^{-2}\right]$
$[\alpha]=\left[ M ^{0} LT ^{0}\right]$