Graphs below show the entropy versus energy $U$ of two systems $1$ and $2$ at constant volume. The initial energies of the systems are indicated by $U_{1, i}$ and $U_{2, i}$, respectively. Graphs are drawn to the same scale. The systems are then brought into thermal contact with each other. Assume that, at all times the combined energy of the two systems remains constant. Choose the most appropriate option indicating the energies of the two systems and the total entropy after they achieve the equilibrium.

Q: Graphs below show the entropy versus energy $U$ of two systems $1$ and $2$ at constant volume. The initial energies of the systems are indicated by $U_{1, i}$ and $U_{2, i}$, respectively. Graphs are drawn to the same scale. The systems are then brought into thermal contact with each other. Assume that, at all times the combined energy of the two systems remains constant. Choose the most appropriate option indicating the energies of the two systems and the total entropy after they achieve the equilibrium.

c $(c)$ By second law of thermodynamics, entropy of a system either increases or remains constant (only in reversible quasistatic processes). Also, from graphs given $S_{2} > S_{1} \Rightarrow \frac{Q}{T_{2}} > \frac{Q}{T_{1}}$ where, $S$ is entropy, $Q$ is heat and $T$ is the temperature. or $\quad T_{1} < T_{2}$ Hence, temperature of system $1$ is less than system $2$. So, heat flows from system $2$ to system $1$ . Hence, $U_{1}$ increases and $U_{2}$ decreases.

Question

A$U_{1}$ increases and $U_{2}$ decreases and the total entropy remains the same
B$U_{1}$ decreases and $U_{2}$ increases and the total entropy remains the same
C$U_{1}$ increases and $U_{2}$ decreases and the total entropy increases
D$U_{1}$ decreases and $U_{2}$ increases and the total entropy increases

KVPY 2018, Advanced

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