First slide

Kinetic theory of ideal gases

Question

3 moles of an ideal monoatomic gas at a temperature of 27^oC are mixed with 2 moles of an ideal monoatomic gas at a temperature 227^oC. Determine the equilibrium temperature of the mixture, assuming no loss of energy.

Moderate

A

280 K
B

320 K
C

350 K
D

380 K

Solution

Energy possessed by the ideal gas which is at 27^oC is
$E_{1} = 3 \times internal energy in the form of KE$
$E_{1} = 3 \times (\frac{3}{2} R T) = 3 (\frac{3}{2} \times R (273 + 27)) = \frac{2700 R}{2}$
Energy possessed by the ideal gas which is at 227^oC is
$E_{2} = 2 (\frac{3 R}{2} \times (273 + 227)) = 1500 R$
If T be the equilibrium temperature of the mixture, then its energy will be $E_{m} = 5 (\frac{3 R T}{2})$
Since, energy remains conserved, $E_{m} = E_{1} + E_{2}$
$5 (\frac{3 R T}{2}) = \frac{2700 R}{2} + 1500 R$
$\Rightarrow T = 380 K or T = 107 ° C$

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