1. Tardigrade
  2. Question
  3. Physics
  4. Two rigid boxes containing different ideal gases are placed on a table. Box A contains one mole of nitrogen at temperature T0 , while box B contains one mole of helium at temperature (7/3)T0 . 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, Tf, in terms of T0 is

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Q. 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 $\frac{7}{3}T_{0}$ . 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

NTA AbhyasNTA Abhyas 2020Thermodynamics

Solution:

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.
Solution
$\Delta U_{A}=1 \times \frac{5 R}{2}\left(T_{f}-T_{o}\right)$
$\Delta U_{B}=1 \times \frac{3 R}{2}\left(T_{f}-\frac{7}{3} T_{0}\right)$
Now, $\Delta U_{A}+\Delta U_{B}=0$
$\frac{5 R}{2}\left(T_{f}-T_{0}\right)+\frac{3 R}{2}\left(T_{f}-\frac{7 T_{0}}{3}\right)=0$
$5 T_{f}-5 T_{0}+3 T_{f}-7 T_{0}=0$
$\Rightarrow 8 T_{f}=12 T_{0}$
$\Rightarrow T_{f}=\frac{12}{8} T_{0}=\frac{3}{2} T_{0}$