Thermodynamics

JEE Main level, single correct, medium

186 of 190

Question 186 of 190

The entropy of any system is given by,
S=α2βln⁡[μkRJβ2+3]S={\alpha }^{2}\beta \ln \left[\dfrac{\mu kR}{J{\beta }^{2}}+3\right]
where α\alpha and β\beta are the constants. μ,J,k\mu ,J,k and RR are number of moles, mechanical equivalent of heat, Boltzmann's constant and gas constant, respectively.
[TakeS=dQT]\left[\text{Take}S=\dfrac{dQ}{T}\right]
Choose the incorrect option from the following:

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Answer: (D) α\alpha and kk have the same dimensions.

S=α2βℓn(μkRJβ2+3)S={\alpha }^{2}\beta \ell n\left(\dfrac{\mu kR}{J{\beta }^{2}}+3\right)
S=QT=Joulek−1S=\dfrac{Q}{T}=Joule{k}^{-1}
[α2β]=Joulek−1\left[{\alpha }^{2}\beta \right]=Joule{k}^{-1}
PV=nRT[μkRJβ2]=1PV=nRT\left[\dfrac{\mu kR}{J{\beta }^{2}}\right]=1
R=JoulekR=\dfrac{\text{Joule}}{k}
⇒R=Joulek,k=JouleR\Rightarrow R=\dfrac{\text{Joule}}{k},k=\dfrac{\text{Joule}}{R}
⇒β=(Joulek)\Rightarrow \beta =\left(\dfrac{\text{Joule}}{k}\right)
α2β=(Joulek){\alpha }^{2}\beta =\left(\dfrac{\text{Joule}}{k}\right)
⇒α=\Rightarrow \alpha = dimensionless

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