The entropy of any system is given by ${\rm{S = }}{{\rm{\alpha }}^{\rm{2}}}{\rm{\beta\,\, ln}}\left[ {\frac{{{\rm{\mu kR}}}}{{{\rm{J}}{{\rm{\beta }}^{\rm{2}}}}}\, + \,3} \right]$ where, $\alpha $ and $\beta $ are the constants; $\mu ,\,J,\,k\,\,{\rm{and}}\,\,R$ are number of moles, mechanical equivalent of heat, Boltzmann constant and gas constant, respectively. $\left[ {{\bf{Take}},\,\,{\rm{S = }}\frac{{{\rm{dQ}}}}{{\rm{T}}}\;\;} \right]$ Choose the incorrect option.

$\,\,\alpha \,\,{\rm{and}}\,\,k$ have the same dimension

$\alpha $ and $J$ have the same dimensions.

$S,\,\,\beta ,\,\,k\,\,{\rm{and}}\,\,\mu R$ have the same dimensions

$S\,\,{\rm{and}}\,\,\alpha $ have the different dimensions

JEE PHYSICS MCQ: UNITS & MEASUREMENTS

1 Medium

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Analyzing +4 | -1

The entropy of any system is given by

\mathrm{S = }{\mathrm{\alpha }^\mathrm{2}}\mathrm{\beta\,\, ln}\left[ {\frac{{\mathrm{\mu kR}}}{{\mathrm{J}{\mathrm{\beta }^\mathrm{2}}}}\, + \,3} \right]

where,

\alpha

and

\beta

are the constants;

\mu ,\,J,\,k\,\,\mathrm{and}\,\,R

are number of moles, mechanical equivalent of heat, Boltzmann constant and gas constant, respectively.

\left[ {\mathbf{Take},\,\,\mathrm{S = }\frac{{\mathrm{dQ}}}{\mathrm{T}}\;\;} \right]

Choose the incorrect option.

View answer and explanation

Correct option: D

\,\,\alpha \,\,\mathrm{and}\,\,k

have the same dimension

Option D is correct because

\alpha

and

k

have different dimensions. After dimensional analysis, it's found that

\alpha

is dimensionless while

k

has dimensions of energy per temperature.