A) \[{1}/{\gamma }\;\]
B) \[\gamma +1\]
C) \[\gamma \]
D) \[\frac{1}{\gamma +1}\]
Correct Answer: C
Solution :
\[Adiabatic\,\,process\,\,p{{v}^{\gamma }}\,\,=\,\,c\,\,or\,\,p\,\,=\,\,\frac{c}{{{v}^{\gamma }}}\] \[Slope,\,\,\frac{dp}{dv}\,\,=\,\,c\,\left[ \,-\,\,\gamma {{v}^{\,-\,\gamma \,-\,1}} \right]\] \[\,=\,\,p{{v}^{\gamma }}\left[ \,-\,\,r{{v}^{\,-\,\gamma \,-\,1}} \right]\] \[\,=\,\,-\,\,p\gamma {{v}^{\,-\,1}}\] \[Isothermal\,\,process,\,\,pv\,\,=\,\,c\,\,or\,\,p\,\,=\,\,\frac{c}{v}\] \[Slope;\,\,\frac{dp}{dv}=-\,\,c{{v}^{\,-\,2}}=-\,\,pv\times {{v}^{\,-2}}=-\,\,p{{v}^{\,-\,1}}\] \[\frac{{{\left( {dp}/{dv}\; \right)}_{adb}}}{{{\left( {dp}/{dv}\; \right)}_{iso}}}=\frac{-\,\,p\gamma {{v}^{\,-\,1}}}{-\,\,p{{v}^{\,-\,1}}}=\gamma \]
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