P-V plots for two gases during an adiabatic p | vedclass

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(B) For an adiabatic process, the equation is $PV^{\gamma} = 	ext{constant}$.Taking the derivative with respect to $V$, we get $P(\gamma V^{\gamma-1}

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$O_2$ and $He$

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(B) For an adiabatic process, the equation is $PV^{\gamma} = 	ext{constant}$.Taking the derivative with respect to $V$, we get $P(\gamma V^{\gamma-1}) + V^{\gamma} \frac{dP}{dV} = 0$.This simplifies to $\frac{dP}{dV} = -\gamma \frac{P}{V}$.The magnitude of the slope of the $P-V$ curve is proportional to the adiabatic index $\gamma$, where $\gamma = \frac{C_p}{C_v}$.For a monoatomic gas (like $He$), $\gamma = \frac{5}{3} \approx 1.67$.For a diatomic gas (like $O_2$), $\gamma = \frac{7}{5} = 1.4$.Since the slope of curve $(2)$ is greater than the slope of curve $(1)$, curve $(2)$ corresponds to the gas with a higher $\gamma$ value (monoatomic gas, $He$) and curve $(1)$ corresponds to the gas with a lower $\gamma$ value (diatomic gas, $O_2$).Therefore, plots $(1)$ and $(2)$ correspond to $O_2$ and $He$ respectively.

$P-V$ plots for two gases during an adiabatic process are shown. Plots $(1)$ and $(2)$ correspond to

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