An expression for a dimensionless quantity P is given by P= _ e (… - Vidyadip

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

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Answer

$P=\frac{\alpha}{\beta} \log _{ e }\left(\frac{ kt }{\beta x }\right)$

$\frac{ kt }{\beta x }=1 \Rightarrow \beta=\frac{ kt }{ x }=\frac{ ML ^{2} T ^{-2}}{ L }$

$\left(\because E =\frac{1}{2} kt \right)$

$As \; P$ is dimensionless

$\Rightarrow[\alpha]=[\beta]=\left[ MLT ^{-2}\right]$

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