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

We are given that $$P = \frac{\alpha}{\beta} \log_e\left(\frac{kT}{\beta x}\right)$$ is dimensionless. The argument of the logarithm must therefore be dimensionless, which implies $$[kT] = [\beta x]$$. Since $$k$$ is the Boltzmann constant with dimensions $$[ML^2T^{-2}K^{-1}]$$ and $$T$$ is temperature $$[K]$$, it follows that $$[kT] = [ML^2T^{-2}]$$. Moreover, as $$x$$ is a distance with dimensions $$[L]$$, we obtain $$[\beta] = \frac{[kT]}{[x]} = \frac{[ML^2T^{-2}]}{[L]} = [MLT^{-2}]$$.

Furthermore, because $$P$$ is dimensionless and $$\log_e(\cdot)$$ is also dimensionless, we have $$[P] = \frac{[\alpha]}{[\beta]} \Rightarrow [M^0L^0T^0] = \frac{[\alpha]}{[MLT^{-2}]}$$. This gives $$[\alpha] = [MLT^{-2}]$$.

The dimensions of $$\alpha$$ are $$[MLT^{-2}]$$. The correct answer is Option C.