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 $= γ \times$ slope of isothermal process.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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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
$Δ Q = 0 h e n c e Δ W = - Δ U$
Thermodynamic equation
$P V^{γ} = c o n s t a n t$ Or $T V^{γ - 1} = c o n s t a n t$
$Δ W = \frac{P_{2} V_{2} - P_{1} V_{1}}{1 - γ} = \frac{n R Δ T}{1 - γ}$
Slope of P-V curve $\frac{d P}{d V} = γ \frac{P}{V}$

For an adiabatic process:
$d P V^{γ} + P γ V^{γ - 1} d V = 0$
$\Rightarrow (\frac{d P}{d V})_{a d i a b a t i c} = - γ \frac{P}{V}$
For an isothermal process $P V = c o n s t a n t$
$(\frac{d P}{d V})_{i s o t h e r m a l} = - \frac{P}{V}$
Combining both, $(\frac{d P}{d V})_{a d i a b a t i c} = γ (\frac{d P}{d V})_{i s o t h e r m a l}$
Since $γ > 1$ , $(\frac{d P}{d V})_{a d i a b a t i c} > (\frac{d P}{d V})_{i s o t h e r m a l}$

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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 =γ× slope of isothermal process.

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 $= γ \times$ slope of isothermal process.