A gas within a closed chamber undergoes the cycle shown in the $p - V$ diagram of figure. The horizontal scale is set by $V_{s} = 4.0 m^{3}$ . Calculate the net energy added to the system as heat during one complete cycle.

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Solution

Over a cycle, the internal energy is the same at the beginning and end, so the heat $Q$ absorbed equals the work done: $Q = W$ . Over the portion of the cycle from $A$ to $B$ the pressure $p$ is a linear function of the volume $V$ , and we may write $p = a + b V$ . The work done over this portion of the cycle is
$W_{A B} = \int_{V_{A}}^{V_{B}} p d V = \int_{V_{A}}^{V_{B}} (a + b V) d V = a (V_{B} - V_{A}) + \frac{1}{2} b (V_{B}^{2} - V_{A}^{2})$
The $B C$ portion of the cycle is at constant pressure, and the work done by the gas is
$W_{B C} = p_{B} Δ V_{B C} = p_{B} (V_{C} - V_{B})$
The $C A$ portion of the cycle is at constant volume, so no work is done. The total work done by the gas is
$W = W_{A B} + W_{B C} + E_{C A}$
The pressure function can be written as
$p = \frac{10}{3} P a + (\frac{20}{3} P a / m^{3}) V$ .
where the coefficient $a$ and $b$ were chosen so the $p = 10 P a$ when $V = 1.0 m^{3}$ and $p = 30 P a$ when $V = 4.0 m^{3}$ . Therefore, the work done going from $A$ to $B$ is
$W_{A B} = a (V_{B} - V_{A}) + \frac{1}{2} b (V_{B}^{2} - V_{A}^{2})$
$= (\frac{10}{3} P a) (4.0 m^{3} - 1.0 m^{3}) + \frac{1}{2} (\frac{20}{3} P a / m^{3}) [(4.0 m^{3})^{2} - (1.0 m^{3})^{2}]$
$= 10 J + 50 J = 6 J$
Similarly, with $p_{B} = p_{C} = 30 P a, V_{C} = 1.0 m^{3}$ , and $V_{B} = 4.0 m^{3}$ , we have
$W_{B C} = p_{B} (V_{C} - V_{B}) = (30 P a) (1.0 m^{3} - 4.0 m^{3}) = 90 J$ .
Adding up all contributions, we find the total work done by the gas to be

$W = W_{A B} + W_{B C} + W_{C A} = 60 J - 90 J + 0 = - 30 J$ .

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A gas within a closed chamber undergoes the cycle shown in the p−V diagram of figure. The horizontal scale is set by Vs=4.0 m3. Calculate the net energy added to the system as heat during one complete cycle.

A gas within a closed chamber undergoes the cycle shown in the $p - V$ diagram of figure. The horizontal scale is set by $V_{s} = 4.0 m^{3}$ . Calculate the net energy added to the system as heat during one complete cycle.