Correct Answer - Option 3 : Polytropic process
Concept:
For a polytropic process,
\(\frac{{{P_1}}}{{{P_2}}} = {\left( {\frac{{{V_2}}}{{{V_1}}}} \right)^n} = {\left( {\frac{{{T_1}}}{{{T_2}}}} \right)^{\frac{n}{{n - 1}}}}\)
\({s_2} - {s_1} = {c_v}\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}} + R\ln \frac{{{{\rm{V}}_2}}}{{{{\rm{V}}_1}}} = {c_v}\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}} + \;R\ln {\left( {\frac{{{T_1}}}{{{T_2}}}} \right)^{\frac{1}{{n - 1}}}} = {c_v}\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}} + \frac{R}{{n - 1}}\ln \frac{{{T_1}}}{{{T_2}}}\)
\({s_2} - {s_1} = {c_v}\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}} - \frac{R}{{n - 1}}\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}} = {c_v}\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}} - \frac{{{c_v}\left( {\gamma - 1} \right)}}{{n - 1}}\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}} = {c_v}\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}}\left( {1 - \frac{{\left( {\gamma - 1} \right)}}{{n - 1}}} \right)\)
\({s_2} - {s_1} = {c_v}\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}}\left( {\frac{{n - \gamma }}{{n - 1}}} \right)\)
Entropy change for m kg of gas
\(S_2-S_1=\frac{\gamma-n}{n-1}.m.c_v.ln\frac{T_1}{T_2}\)
The entropy change is given by:
\(\Delta {\rm{S}} = {{\rm{c}}_{\rm{v}}}{\rm{\;}}\left[ {\frac{{{\rm{n}} - {\rm{\gamma }}}}{{{\rm{n}} - 1}}} \right]\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}}\)
\(\Delta {\rm{S}} = \frac{{{\rm{n}} - {\rm{\gamma }}}}{{\left( {{\rm{n}} - 1} \right)\left( {\gamma - 1} \right)}}R\ln \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}}\)
