Question

# Assertion :Isothermal and adiabatic, two processes are shown on $$P-V$$ diagram. Process-1 is adiabatic and process-2 is isothermal. Reason: At a given point, slope of adiabatic process $$=\, \gamma\, \times$$ slope of isothermal process.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C
Assertion is correct but Reason is incorrect
D
Both Assertion and Reason are incorrect

Solution

## The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for AssertionAdiabatic Process$$\Delta Q= 0\quad hence \ \Delta W=-\Delta U$$Thermodynamic equation $$PV^{\gamma}=constant$$ Or $$TV^{\gamma-1}=constant$$   $$\Delta W=\dfrac {P_2V_2-P_1V_1}{1-\gamma}=\dfrac {nR\Delta T}{1-\gamma}$$Slope of P-V curve  $$\dfrac {dP}{dV}=\gamma \frac {P}{V}$$For an adiabatic process:$$dP V^{\gamma} + P \gamma V^{\gamma -1} dV= 0$$$$\Rightarrow (\dfrac{dP}{dV})_{adiabatic} = -\gamma\dfrac{P}{V}$$For an isothermal process  $$PV = constant$$$$( \dfrac{dP}{dV})_{isothermal} = -\dfrac{P}{V}$$Combining both,$$(\dfrac{dP}{dV})_{adiabatic} = \gamma ( \dfrac{dP}{dV})_{isothermal}$$Since $$\gamma \gt 1$$, $$(\dfrac{dP}{dV})_{adiabatic} \gt ( \dfrac{dP}{dV})_{isothermal}$$Physics

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