JEE Main & Advanced Physics Thermodynamical Processes Entropy

Entropy is a measure of disorder of molecular motion of a system. Greater is the disorder, greater is the entropy.

The change in entropy i.e.

\[dS=\frac{\text{Heat absorbed by system}}{\text{Absolute temperature}}\] or \[dS=\frac{dQ}{T}\]

The relation is called the mathematical form of Second Law of Thermodynamics.

(1) For solids and liquids

(i) When heat given to a substance changes its state at constant temperature, then change in entropy \[dS=\frac{dQ}{T}=\pm \frac{mL}{T}\]

where positive sign refers to heat absorption and negative sign to heat evolution.

(ii) When heat given to a substance raises its temperature from \[{{T}_{1}}\] to \[{{T}_{2}},\] then change in entropy

\[dS=\int{\frac{dQ}{T}}=\int_{\,{{T}_{1}}}^{\,{{T}_{2}}}{mc}\frac{dT}{T}=mc{{\log }_{e}}\left( \frac{{{T}_{2}}}{{{T}_{1}}} \right)\]

\[\Rightarrow \] \[\Delta S=2.303\,mc{{\log }_{10}}\left( \frac{{{T}_{2}}}{{{T}_{1}}} \right)\].

(2) For a perfect gas : Perfect gas equation for n moles is \[PV=nRT\] \[\Delta S=\int{\frac{dQ}{T}}=\int{\frac{\mu {{C}_{V}}dT+P\,dV}{T}}\]              

 [As \[dQ=dU+dW\]]

\[\Rightarrow \] \[\Delta S=\int{\frac{\mu {{C}_{V}}dT+\frac{\mu RT}{V}dV}{T}}\] \[=\mu {{C}_{V}}\int_{\,{{T}_{1}}}^{\,{{T}_{2}}}{\frac{dT}{T}}+\mu R\int_{\,{{V}_{1}}}^{\,{{V}_{2}}}{\frac{dV}{V}}\]                                          [As \[PV=\mu RT\]]

\[\therefore \]\[\Delta S=\mu {{C}_{V}}{{\log }_{e}}\left( \frac{{{T}_{2}}}{{{T}_{1}}} \right)+\mu R{{\log }_{e}}\left( \frac{{{V}_{2}}}{{{V}_{1}}} \right)\]

In terms of T and P,  \[\Delta S=\mu {{C}_{P}}{{\log }_{e}}\left( \frac{{{T}_{2}}}{{{T}_{1}}} \right)-\mu R{{\log }_{e}}\left( \frac{{{P}_{2}}}{{{P}_{1}}} \right)\] and in terms of P and V \[\Delta S=\mu {{C}_{V}}{{\log }_{e}}\left( \frac{{{P}_{2}}}{{{P}_{1}}} \right)+\mu {{C}_{P}}{{\log }_{e}}\left( \frac{{{V}_{2}}}{{{V}_{1}}} \right)\]