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

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(B) In an adiabatic process, the equation is $PV^{\gamma} = 	ext{constant}$.Taking the derivative with respect to $V$, we get $\frac{dP}{dV} V^{\gamm

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${O_2}$ and $He$

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(B) In an adiabatic process, the equation is $PV^{\gamma} = 	ext{constant}$.Taking the derivative with respect to $V$, we get $\frac{dP}{dV} V^{\gamma} + P \gamma V^{\gamma-1} = 0$.Thus, the slope of the $P-V$ graph is given by $\frac{dP}{dV} = - \gamma \frac{P}{V}$.Therefore, the magnitude of the slope is directly proportional to the adiabatic index $\gamma$ (i.e., $|	ext{Slope}| \propto \gamma$).From the given graph, the slope of curve $2$ is greater than the slope of curve $1$ (i.e., $|	ext{Slope}|_2 > |	ext{Slope}|_1$).This implies that $\gamma_2 > \gamma_1$.For $O_2$ (a diatomic gas), $\gamma = 1.4$, and for $He$ (a monatomic gas), $\gamma = 1.66$.Since $\gamma_{He} > \gamma_{O_2}$, curve $2$ corresponds to $He$ and curve $1$ corresponds to $O_2$.Thus, plots $1$ and $2$ correspond to $O_2$ and $He$ respectively.

$P-V$ plots for two gases during an adiabatic process are shown in the figure. Plots $1$ and $2$ should correspond respectively to

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