A diatomic ideal gas is heated at constant volume until its pressure becomes three times. It is again heated at constant pressure until its volume is doubled. Find the molar heat capacity for the whole process.

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Solution

Let the pressure, temperature and volume of the gas at A be $P, T$ and $V$ respectively.
For diatomic gas, $C_{V} = \frac{5}{2} R$ and $C_{P} = \frac{7}{2} R$
Process AB : Isochoric Process $⟹ \frac{P_{A}}{T_{A}} = \frac{P_{B}}{T_{B}}$
$∴$ $\frac{P}{T} = \frac{3 P}{T_{B}}$ $⟹ T_{B} = 3 T$
Heat supplied at constant volume $Q_{V} = Δ U = n C_{V} (T_{B} - T_{A})$
$∴$ $Q_{V} = \frac{5}{2} n R (2 T)$ ...............(1)

Process BC : Isobaric Process $⟹ \frac{V_{B}}{T_{B}} = \frac{V_{C}}{T_{C}}$
$∴$ $\frac{V}{3 T} = \frac{2 V}{T_{C}}$ $⟹ T_{C} = 6 T$
Heat supplied at constant pressure $Q_{P} = n C_{P} (T_{C} - T_{B})$
$∴$ $Q_{P} = \frac{7}{2} n R (3 T)$ ...............(2)
Let the molar heat capacity for the whole process be $C$
$∴$ $Q_{T o t a l} = Q_{V} + Q_{P}$
OR $n C (T_{C} - T_{A}) = Q_{V} + Q_{P}$

OR $n C (5 T) =$ $\frac{31}{2} n R T$ $⟹ C = 3.1 R$

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