Question

The entropy of any system is given by ${\displaystyle S={\alpha }^2\beta In[\frac{\mu kR}{J{\beta }^2}+3]}$ Where α and β are the constants µ J, k, and R are no. of moles, the mechanical equivalent of heat, Boltzmann constant, and gas constant respectively. ${\displaystyle [\text{take S}=\frac{dQ}{T}]}$

Choose the incorrect option from the following.

पर्याय

• S, β, k, and µR have the same dimensions.

• α and J have the same dimensions.

• S and α have different dimensions.

• α and k have the same dimensions.

Answer 1

α and k have the same dimensions.

Explanation:

${\displaystyle S={\alpha }^2\beta In[\frac{\mu kR}{J{\beta }^2}+3]}$ ......(i)

${\displaystyle \frac{\mu kR}{J{\beta }^2}}$ → dimensionless

${\displaystyle S=\frac{Q}{T}=\frac{\text{Joule}}{\text{Kelvin}}}$

⇒ [S] = ${\displaystyle \frac{M{L}^2{T}^{-2}}{\theta }=M{L}^2{T}^{-2}{\theta }^{-1}}$

Ideal gas equation, PV = µRT

µR = ${\displaystyle \frac{\text{PV}}{T}}$

[µR] = ML²T^-2θ^-1

k → Boltzmann constant [k] = ML²T^-2θ^-1

J → dimensionless constant