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First Law of Thermodynamics

An ideal gas ...

Question

An ideal gas has a specific heat at constant pressure $C_{P} = \frac{5 R}{2}$ . The gas is kept in a closed vessel of volume $V_{0}$ at temperature $T_{0}$ and pressure $P_{0}$ . An amount of 10 $P_{0} V_{0}$ of heat is supplied to the gas. The ratio of final pressure and temperature of the gas becomes.....

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Solution

Given : $P_{A} = P_{o}$ $V_{A} = V_{o}$ $T_{A} = T_{o}$ $C_{P} = \frac{5}{2} R$
Using $C_{P} - C_{V} = R$
$∴$ $\frac{5}{2} R - C_{V} = R$ $⟹ C_{V} = \frac{3}{2} R$
Using ideal gas equation at A, $P_{o} V_{o} = n R T_{o}$ ..........(1)
Process AB is an isochoric process : $W = 0$
$∴$ $Q = Δ U = n C_{V} (T_{B} - T_{A})$
OR $10 P_{o} V_{o} = \frac{3}{2} n R (T_{B} - T_{o})$

Using (1) we get $10 n R T_{o} = \frac{3}{2} n R (T_{B} - T_{o})$ $⟹ T_{B} = \frac{23}{3} T_{o}$
Also $\frac{P_{A}}{T_{A}} = \frac{P_{B}}{T_{B}}$

$∴$ $\frac{P_{o}}{T_{o}} = \frac{P_{B}}{\frac{23}{3} T_{o}}$ $⟹ P_{B} = \frac{23}{3} P_{o}$

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