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

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Adiabatic 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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