question_answer 15)

                  Consider two containers A and B containing identical gases at the same pressure, volume and temperature. The gas in container A is compressed to half of its original volume isothermally while the gas in container B is compressed to half of its original value adiabatically. The ratio of final pressure of gas in B to that of gas in A is                 (a) \[{{2}^{r-1}}\]                                  (b) \[{{\left( \frac{1}{2} \right)}^{r-1}}\]                 (c) \[{{\left( \frac{1}{1-\gamma } \right)}^{2}}\]             (d) \[{{\left( \frac{1}{\gamma -1} \right)}^{2}}\]                

Answer:

                  (a) For isothermal process, \[{{P}_{1}}{{V}_{1}}={{P}_{2}}{{V}_{2}}\]                 For adiabatic process, \[{{P}_{1}}V_{1}^{\gamma }\,=\,{{P}_{2}}V_{2}^{\gamma }\]                 For isothermal process,                 For adiabatic process, \[{{P}_{2a}}\,={{P}_{1}}\,{{\left( \frac{{{V}_{1}}}{{{V}_{2}}} \right)}^{\gamma }}\]\[={{2}^{\gamma }}\,{{P}_{1}}\]                 \[\therefore \] \[\frac{{{P}_{2a}}}{{{P}_{2i}}}\,={{2}^{\gamma -1}}\]