Question

Two rigid boxes containing different ideal gas 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 $\left(7/3\right)T_0$. 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

• A. $T_f=\frac{7}{3}{T}_0$
• B. $T_f=\frac{3}{2}{T}_0$
• C. $T_f=\frac{5}{2}{T}_0$
• D. $T_f=\frac{3}{7}{T}_0$

Answer 1

The correct option is B $T_f=\frac{3}{2}{T}_0$

Here volume remains fixed so

For nitrogen $C_v=\frac{5}{2}R$

and for Helium $C_v=\frac{3}{2}R$

Now if no heat is given or taken from the surrounding so internal energy of the system cant change

$\therefore \Delta U$ nitrogen $+\Delta U$ helium$=0$

where $\Delta U=n{C}_v\Delta T$

$\therefore \Rightarrow 1\times \frac{5}{2}R(T_f-T_o)+\frac{3}{2}R(T_f-\frac{7}{3}{T}_o)=0$

$\Rightarrow \frac{5}{2}R{T}_f-\frac{5}{2}R{T}_o+\frac{3}{2}R{T}_f-\frac{7}{2}R{T}_o=0$

$\Rightarrow 4T_f-6T_o=0$

$\Rightarrow T_f=\frac{8}{2}{T}_o$.