A mixture of ideal gas containing moles of monoatomic gas and 1 mole of rigid diatomic gas is initially at pressure , volume , and temperature . If the gas mixture is adiabatically compressed to a volume then the correct statement(s) is/are, (Given R is

Question

A mixture of ideal gas containing 5 moles of monoatomic gas and 1 mole of rigid diatomic gas is initially at pressure $P_{0}$ , volume $V_{0}$ , and temperature $T_{0}$ . If the gas mixture is adiabatically compressed to a volume $\frac{V_{0}}{4},$ then the correct statement(s) is/are,

(Given $2^{1.2} = 2.3, 2^{3.2} = 9.2,$ R is gas constant) [2019]

Smart Achievers · Accepted Answer

(1, 2, 4)

$Adiabatic constant of the gas mixture,$

$γ_{m} = \frac{n_{1} C_{p 1} + n_{2} C_{p 2}}{n_{1} C_{v 1} + n_{2} C_{v 2}} = \frac{5 \times \frac{5 R}{2} + 1 \times \frac{7 R}{2}}{5 \times \frac{3 R}{2} + 1 \times \frac{5 R}{2}} = 1.6$

$For an adiabatic process, P V^{γ} = Constant$

$∴$ $P = P_{0} {(\frac{V_{0}}{V})}^{1.6} = P_{0} {(4)}^{1.6}$

$= P_{0} {(2^{2})}^{1.6} = P_{0} 2^{3.2} = 9.2 P_{0}$

$Work done during the process,$

$W = \frac{P_{2} V_{2} - P_{1} V_{1}}{1 - γ} = \frac{9.2 P_{0} \times (\frac{V_{0}}{4}) - P_{0} V_{0}}{1 - 1.6} = \frac{- 13 P_{0} V_{0}}{6}$

$But P_{0} V_{0} = 6 R T_{0}$ $(as n = 5 + 1 = 6)$

$∴$ $W = \frac{- 13 (6 R T_{0})}{6} = - 13 R T_{0}$

$∴$ $| W |$ $= 13 R T_{0}$

$The average K.E. of the gas mixture,$

$\bar{K.E.} = n C V_{mi} \times T_{2}$

$From, T_{1} V_{1}^{γ - 1} = T_{2} V_{2}^{γ - 1}$

$or, T_{2} = T_{1} {(2)}^{6 / 5} = 23 T_{0}$

$∴$ $\bar{K.E.} = n C V_{mi} \times T_{2} = 23 R T_{0}$

Solution

1 The work $| W |$ done during the process is $13 R T_{0}$ .

2 The final pressure of the gas mixture after compression is in between $9 P_{0}$ and $10 P_{0}$ .

3 The average kinetic energy of the gas mixture after compression is in between $18 R T_{0}$ and $19 R T_{0}$ .

4 Adiabatic constant of the gas mixture is 1.6.

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Solution

1 The work |W| done during the process is 13RT0.

2 The final pressure of the gas mixture after compression is in between 9P0 and 10P0.

3 The average kinetic energy of the gas mixture after compression is in between 18RT0 and 19RT0.

4 Adiabatic constant of the gas mixture is 1.6.

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1 The work $| W |$ done during the process is $13 R T_{0}$ .

2 The final pressure of the gas mixture after compression is in between $9 P_{0}$ and $10 P_{0}$ .

3 The average kinetic energy of the gas mixture after compression is in between $18 R T_{0}$ and $19 R T_{0}$ .