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1.Units, Dimensions and Measurement
hard
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
$\left[ MLT ^{-1}\right]$
(JEE MAIN-2021)
Solution
$kT$ 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
$\left[\alpha^{2}\right]\left[ MT ^{-2}\right]=\left[ ML ^{2} T ^{-2}\right]$
$[\alpha]=\left[ M ^{0} LT ^{0}\right]$
Standard 11
Physics
