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Gas Pressure and Temperature

P-V diagram o...

Question

$P - V$ diagram of an ideal gas for a process $A B C$ is as shown in the figure. Change in internal energy of the gas during the process $A B C$ .

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Solution

Let the moles of an ideal gas be $1$ and the temperature at B be $T_{o}$ .
Thus $P_{o} V_{o} = R T_{o}$ $⟹ T_{o} = \frac{P_{o} V_{o}}{R}$ .......................(1)

Using ideal gas equation at A : $P_{A} V_{A} = R T_{A}$
$∴$ $P_{o} (3 V_{o}) = R T_{A}$ $⟹ T_{A} = 3 T_{o}$ (using 1)

Using ideal gas equation at C : $P_{C} V_{C} = R T_{C}$
$∴$ $(3 P_{o}) V_{o} = R T_{C}$ $⟹ T_{C} = 3 T_{o}$ (using 1)
Change in internal energy of gas during process ABC $Δ U_{A B C} = C_{v} (T_{C} - T_{A})$
$∴$ $Δ U_{A B C} = C_{v} (3 T_{o} - 3 T_{o}) = 0$

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