Question

An expression for a dimensionless quantity $P$ is given by $P =\frac{\alpha}{\beta} \log _{e}\left(\frac{ kt }{\beta x }\right)$; where $\alpha$ and $\beta$ are constants, $x$ is distance ; $k$ is Boltzmann constant and $t$ is the temperature. Then the dimensions of $\alpha$ will be :

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

Answer 1

Answer: (C)

$P =\frac{\alpha}{\beta} \log _{ e }\left(\frac{ kt }{\beta x }\right)$
$\frac{ kt }{\beta x }=1$
$\Rightarrow \beta=\frac{ kt }{ x }=\frac{ ML ^{2} T ^{-2}}{ L }$
$\left(\because E =\frac{1}{2} kt \right)$
As $P$ is dimensionless
$\Rightarrow[\alpha]=[\beta]=\left[ MLT ^{-2}\right]$