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(C) For an ideal gas,the equation of state is $PV = nRT$. Thus,the temperature $T$ is proportional to the product $PV$. In the given $P-V$ diagram,the

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Heated in the beginning and cooled towards the end

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(C) For an ideal gas,the equation of state is $PV = nRT$. Thus,the temperature $T$ is proportional to the product $PV$. In the given $P-V$ diagram,the process is a straight line from state $1$ to state $2$. Let the equation of the line be $P = -mV + c$,where $m$ is the slope and $c$ is the intercept. The product $PV = (-mV + c)V = -mV^2 + cV$. To find the variation of $T$ with $V$,we look at the function $f(V) = -mV^2 + cV$. This is a downward-opening parabola. The value of $PV$ (and thus $T$) increases initially as $V$ increases,reaches a maximum at $V = c/(2m)$,and then decreases as $V$ increases further. Therefore,the gas is heated in the beginning and cooled towards the end.

$A$ volume $V$ and pressure $P$ diagram was obtained from state $1$ to state $2$ when a given mass of a gas is subjected to temperature changes. During this process the gas is

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