List these in order of increasing entropy: (a) 1 mol of water at $20^{\circ} \mathrm{C}\( and 1 mol of ethanol at \)20^{\circ} \mathrm{C}$ in separate containers; (b) a mixture of 1 mol of water at \(20^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(20^{\circ} \mathrm{C} ;\) (c) 0.5 mol of water at \(20^{\circ} \mathrm{C}\) and 0.5 mol of ethanol at \(20^{\circ} \mathrm{C}\) in separate containers; (d) a mixture of 1 mol of water at $30^{\circ} \mathrm{C}\( and 1 mol of ethanol at \)30^{\circ} \mathrm{C}$.

Entropies are extensive properties; therefore, entropy increases when the amount of substance increases. Mixing also increases the entropy of a system because the substance molecules become dispersed, creating more randomness. Changing the temperature of a substance also affects its entropy.

(a) 1 mol of water at \(20^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(20^{\circ} \mathrm{C}\) in separate containers - Here, the entropy is relatively lower than in the other scenarios due to the substances being in separate containers. (b) a mixture of 1 mol of water at \(20^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(20^{\circ} \mathrm{C}\) - In this scenario, the mixed substances create more randomness, resulting in higher entropy than the first scenario. (c) 0.5 mol of water at \(20^{\circ} \mathrm{C}\) and 0.5 mol of ethanol at \(20^{\circ} \mathrm{C}\) in separate containers - Even though the substances are in separate containers, the entropy is still higher than the first scenario due to the presence of more amount of substance. (d) a mixture of 1 mol of water at \(30^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(30^{\circ} \mathrm{C}\) - Here, we have a mixture of the substances at a higher temperature than the other scenarios, which further increases the entropy due to increased randomness at higher temperatures.

After analyzing the entropy of the scenarios, we can arrange them in increasing order of entropy: (a) \(\rightarrow\) (c) \(\rightarrow\) (b) \(\rightarrow\) (d) The order of increasing entropy is 1 mol of water at \(20^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(20^{\circ} \mathrm{C}\) in separate containers (a), followed by 0.5 mol of water at \(20^{\circ} \mathrm{C}\) and 0.5 mol of ethanol at \(20^{\circ} \mathrm{C}\) in separate containers (c), followed by a mixture of 1 mol of water at \(20^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(20^{\circ} \mathrm{C}\) (b), and finally, a mixture of 1 mol of water at \(30^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(30^{\circ} \mathrm{C}\) (d) has the highest entropy.