The slopes of isothermal and adiabatic curves are related as
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(c) For Isothermal process $PV = $constant
$ \Rightarrow \left( {\frac{{dP}}{{dV}}} \right) = \frac{{ - P}}{V} = $ Slope of Isothermal curve
For adiabatic$P{V^\gamma } = $constant
$ \Rightarrow \frac{{dP}}{{dV}} = \frac{{ - \gamma P}}{V} = $ Slop of adiabatic curve slope
$ \Rightarrow \left( {\frac{{dP}}{{dV}}} \right) = \frac{{ - P}}{V} = $ Slope of Isothermal curve
For adiabatic$P{V^\gamma } = $constant
$ \Rightarrow \frac{{dP}}{{dV}} = \frac{{ - \gamma P}}{V} = $ Slop of adiabatic curve slope
Clearly, ${\left( {\frac{{dP}}{{dV}}} \right)_{{\rm{adiabatic}}}} = \gamma {\left( {\frac{{dP}}{{dV}}} \right)_{{\rm{Isothermal }}}}$
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