Derive an expression for the entropy of one mole of an ideal gas

Entropy:the measure of a system's thermal energy per unit temperature that is unavailable for doing useful work. Because work is obtained from ordered molecular motion, the amount of entropy is also a measure of the molecular disorder, or randomness, of a system.

Entropy for one mole of gas:

\[\Delta S=\ \ \frac{\Delta Q}{T}\ \rightarrow \ \Delta Q = \Delta E+Pdv \\\ \Delta Q=C_{v}dt+pdv\left ( \Delta E= C_{v}dt \right ) \\\ \Delta S=\frac{ C_{v}dt+pdv }{T} \ = \frac{ C_{v}dt}{T}+\frac{p}{T}dv.\]

For pressure

\[\Delta S=\frac{ C_{v}dt}{T}+\frac{R}{V}dv \ \ \sqsubset PV=nRT \ \ \rightarrow P=\frac{RT}{V} \sqsupset \\\ \Delta S= C_{v}\frac{dt}{t}+R\frac{dv}{v} \ \Delta S= C_{v} \int_{ t_{1}}^{ t_{2}}dt+R\int_{ V_{1}}^{ V_{2}}\frac{dv}{v}\]

\[\Delta S= C_{v}ln\frac{ t_{2}}{ t_{1}}+R \ ln\frac{ v_{2}}{ v_{1}} \rightarrow [R= C{p}- C_{v}] \\\ \Delta S= C_{v}ln\frac{ t_{2}} { t_{1}}+ (C_{p}- C_{v}) \ ln\frac{ v_{2}}{ v_{1}} \\\ \Delta S= C_{v}ln\frac{ P_{2} V_{2}}{ V_{1} P_{1}}+ C_{p}ln\frac{ V_{2}}{ V_{1}}- C_{v}ln\frac{ V_{2}}{ V_{1}}\]

\[C_{v}ln\frac{ P_{2}}{ P_{1}}+ C_{v}ln\frac{ V_{2}}{ V_{1}}+C_{p}ln\frac{ V_{2}}{ V_{1}} - C_{v}ln\frac{ V_{2}}{ V_{1}} \ \ \ \ [\frac{T2}{T1}=\frac{P2V2}{P1V1}] \\\ answer= C_{v}ln\frac{P2}{P1}+ C_{p}\frac{V2}{V1} .\]