Two rigid boxes containing different ideal gases are placed on a table. Box A contains one mole of nitrogen at temperature T 0 , while box B contains one mole of helium at temperature 7 3 T 0 . The bo

Question

Two rigid boxes containing different ideal gases are placed on a table. Box A contains one mole of nitrogen at temperature T 0 , while box B contains one mole of helium at temperature 7 3 T 0 . The boxes are then put into thermal contact with each other, and heat flows between them until the gases reach a common final temperature (Ignore the heat capacity of boxes). Then , the final temperature of the gases, T f , in terms of T 0 is

TestHub · Accepted Answer

Here, change in internal energy of the system is zero, ie, increase in internal energy of one is equal to decrease in internal energy of other.

$∆ U_{A} = 1 \times \frac{5 R}{2} (T_{f} - T_{o})$

$∆ U_{B} = 1 \times \frac{3 R}{2} (T_{f} - \frac{7}{3} T_{0})$

Now, $∆ U_{A} + ∆ U_{B} = 0$

$\frac{5 R}{2} (T_{f} - T_{0}) + \frac{3 R}{2} (T_{f} - \frac{7 T_{0}}{3}) = 0$

$5 T_{f} - 5 T_{0} + 3 T_{f} - 7 T_{0} = 0$

$\Rightarrow 8 T_{f} = 12 T_{0}$

$\Rightarrow T_{f} = \frac{12}{8} T_{0} = \frac{3}{2} T_{0}$

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