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Step-by-Step Solution
Step 1: Identify the Processes
The system moves from state A to B along a straight (linear) path on the PāV diagram:
Then, it goes from B to C by an isobaric (constant-pressure) process.
Step 2: Work Done from A to B
ā¢ Pressure changes linearly from
$P_A = 8000\,\mathrm{dyn\,cm}^{-2}$ to $P_B = 4000\,\mathrm{dyn\,cm}^{-2}$.
ā¢ Volume changes from $V_A = 3\,\mathrm{m}^3$ to $V_B = 7\,\mathrm{m}^3$.
To find the work done under a linearly varying pressure, we use the average pressure
$P_{\text{avg}}$ times the change in volume $\Delta V$.
$P_{\text{avg}} = \frac{P_A + P_B}{2}
= \frac{8000 + 4000}{2}
= 6000\,\mathrm{dyn\,cm}^{-2}.$
$\Delta V = V_B - V_A = 7 - 3 = 4\,\mathrm{m}^3.$
Converting $4\,\mathrm{m}^3$ to $\mathrm{cm}^3$:
$4\,\mathrm{m}^3 = 4 \times 10^6\,\mathrm{cm}^3.$
So, the work done from A to B in cgs units (erg) is:
$$
W_{AB} = P_{\text{avg}} \times \Delta V
= 6000\,\mathrm{dyn\,cm}^{-2} \times 4 \times 10^6\,\mathrm{cm}^3
= 2.4 \times 10^{10}\,\mathrm{erg}.
$$
Since $1\,\mathrm{erg} = 10^{-7}\,\mathrm{J}$, converting to Joules:
$$
W_{AB} = 2.4 \times 10^{10} \times 10^{-7}\,\mathrm{J}
= 2400\,\mathrm{J}.
$$
Step 3: Work Done from B to C
ā¢ This process is isobaric at
$P = 4000\,\mathrm{dyn\,cm}^{-2}$.
ā¢ The volume decreases from $V_B = 7\,\mathrm{m}^3$ to $V_C = 3\,\mathrm{m}^3$, so
$$
\Delta V = V_C - V_B = 3 - 7 = -4\,\mathrm{m}^3.
$$
The negative sign indicates a decrease in volume.
Converting $4\,\mathrm{m}^3$ to $\mathrm{cm}^3$ again gives
$4 \times 10^6\,\mathrm{cm}^3.$
Hence, the work done by the gas from B to C is:
$$
W_{BC}
= P \times \Delta V
= 4000\,\mathrm{dyn\,cm}^{-2} \times \bigl(-4 \times 10^6\,\mathrm{cm}^3\bigr)
= -1.6 \times 10^{10}\,\mathrm{erg}
= -1600\,\mathrm{J}.
$$
(Negative sign indicates work is done on the gas.)
Step 4: Total Work Done
Summing the work done in each process:
$$
W_{\text{total}} = W_{AB} + W_{BC} = 2400\,\mathrm{J} + (-1600\,\mathrm{J}) = 800\,\mathrm{J}.
$$
Final Answer
The total work done by the gas from A to B and then B to C is $800\,\mathrm{J}$.
