The entropy of any system is given by

Question:

The entropy of any system is given by

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

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.

$\left[\operatorname{Take} S=\frac{\mathrm{dQ}}{\mathrm{T}}\right]$

Choose the incorrect option from the following :

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

  2. $\mathrm{S}, \beta, \mathrm{k}$ and $\mu \mathrm{R}$ have the same dimensions.

  3. $\mathrm{S}$ and $\alpha$ have different dimensions.

  4. $\alpha$ and $k$ have the same dimensions.

Correct Option: , 4

Solution:

$\mathrm{S}=\alpha^{2} \beta \ln \left(\frac{\mu \mathrm{KR}}{\mathrm{J} \beta^{2}}+3\right)$

$\mathrm{S}=\frac{\mathrm{Q}}{\mathrm{T}}=$ joulek $/ \mathrm{k}$

$\left[\alpha^{2} \beta\right]=$ Joule $/ \mathrm{k}$

$\mathrm{PV}=\mathrm{nRT} \quad\left[\frac{\mu \mathrm{KR}}{\mathrm{J} \beta^{2}}\right]=1$

$R=\frac{\text { Joule }}{K}$

$\Rightarrow \mathrm{R}=\frac{\text { Joule }}{\mathrm{K}}, \mathrm{K}=\frac{\text { Joule }}{\mathrm{R}}$

$\Rightarrow \beta=\left(\frac{\text { Joule }}{K}\right)$

$\alpha^{2} \beta=\left(\frac{\text { Joule }}{\mathrm{K}}\right)$

$\Rightarrow \alpha=$ dimensionless

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