The entropy of any system is given by,
$$S = \alpha^2\beta\ln\left[\frac{\mu kR}{J\beta^2} + 3\right]$$
where $\alpha$ and $\beta$ are the constants. $\mu$, $J$, $k$ and $R$ are number of moles, mechanical equivalent of heat, Boltzmann's constant and gas constant, respectively. [Take $S = \dfrac{dQ}{T}$]
Choose the incorrect option from the following:
Answer: (D) $\alpha$ and $k$ have the same dimensions.
$J$ is a pure number (a conversion factor), and the argument of the log is dimensionless, so $[\beta^2] = [\mu kR]$. Here $[k] = [\text{J K}^{-1}]$ and $[\mu R] = [\text{J K}^{-1}]$, so $[\beta] = [\text{J K}^{-1}]$, the same as entropy.
Then $S = \alpha^2\beta\times(\text{number})$ gives $\alpha$ dimensionless.
So (1) is true (both dimensionless), (2) is true, (3) is true, and (4) is false: $\alpha$ is dimensionless but $k$ is not.
Solution by Sreeraj P, M.Sc Physics