An expression for a dimensionless quantity P is given by $P = \frac{α}{β} \log_{e} (\frac{k t}{β x})$ ;where $α$ and $β$ are constants, x is distance; k is Boltzmann constant and t is the temperature. Then the dimensions of $α$ will be

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

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

$[M L^{0} T^{- 2}]$

$[M^{0} L^{- 1} T^{0}]$

answer is C.

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Detailed Solution

$P = \frac{α}{β} \log_{e} (\frac{k t}{β x}), \frac{k t}{β x} = 1 \Rightarrow β = \frac{k t}{x} = \frac{M L^{2} T^{- 2}}{L}$

$(∵ E = \frac{1}{2} k t)$

As P is dimensionless $[α] = [β] = [M L T^{- 2}]$

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