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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 repectively. [Take ${S}=\frac{{dQ}}{{T}}$ ]
Choose the incorrect option from the following:
${S}, \beta, {k}$ and $\mu {R}$ have the same dimensions.
$\alpha$ and ${J}$ have the same dimensions.
${S}$ and $\alpha$ have different dimensions.
$\alpha$ and ${k}$ have the same dimensions.
Solution
$S=\alpha^{2} \beta \ell {n}\left(\frac{\mu K R}{j \beta^{2}}+3\right)$
$S=\frac{Q}{T}=\text { Joule } / k$
$P V=n R T \quad\left[\frac{\mu K R}{J \beta^{2}}\right]=1$
${R=\frac{\text { Joule }}{K}}$
${\Rightarrow \beta=\left(\frac{\text { Joule }}{K}\right)}$
${\Rightarrow \alpha=\text { dimensionless }}$
