The entropy of any system is given by S = α²βln [(μ kR)/(Jβ²) + 3] where α and β are the constants. μ, J, k and R are no. of moles, mechanical equivalent of heat, Boltzmann constant and gas constant respectively. [Take S = dQ/T ] Choose the incorrect option from the following: (a) α and J have the same dimensions. (b) S and α have different dimensions (c) S, β, k and μ R have the same dimensions (d) α and k have the same dimensions

Answer: (d). The log argument is dimensionless and J (J/cal) is dimensionless, so [β²] = [μ kR] = (J/K)², giving [β] = J/K = [S]. S = α²β then makes α dimensionless. (a) α and J both dimensionless: correct. (b) S has dimensions, α does not: correct. (c) S, β, k, μ R are all J/K: correct. (d) k has dimension J/K but α is dimensionless, so (d) is the incorrect option.

The entropy of any system is given by S = α²βln [(μ kR)/(Jβ²) + 3]… | JEE Main 2021 (20 Jul, Shift 1)

Q46JEE Main 2021 · 20 JUL · SHIFT 1Medium

Applications of Dimensional Analysis

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 no. of moles, mechanical equivalent of heat, Boltzmann constant and gas constant respectively.

[Take $S = \frac{dQ}{T}$ ]

Choose the incorrect option from the following:

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Answer: (d)

The log argument is dimensionless and $J$ (J/cal) is dimensionless, so $[\beta^2] = [\mu kR] = (\text{J/K})^2$ , giving $[\beta] =$ J/K $= [S]$ .
$S = \alpha^2\beta$ then makes $\alpha$ dimensionless.
(a) $\alpha$ and $J$ both dimensionless: correct. (b) $S$ has dimensions, $\alpha$ does not: correct. (c) $S, \beta, k, \mu R$ are all J/K: correct.
(d) $k$ has dimension J/K but $\alpha$ is dimensionless, so (d) is the incorrect option.

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Applications of Dimensional Analysis question from JEE Main 2021 (20 Jul, Shift 1)

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