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Chapter 15: Problem 64

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}$.

Short Answer

Expert verified
Question: Arrange the given scenarios in increasing order of 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, and (d) a mixture of 1 mol of water at \(30^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(30^{\circ} \mathrm{C}\). Answer: The order of increasing entropy is (a) 鈫� (c) 鈫� (b) 鈫� (d).

Step by step solution

01

Analyze the entropy in different scenarios

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.
02

Compare the entropy of the given scenarios

(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.
03

Arrange the scenarios in increasing order of entropy

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.

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Most popular questions from this chapter

Consider a heat engine that is not reversible. The engine uses $1.000 \mathrm{mol}\( of a diatomic ideal gas. In the first step \)(\mathrm{A})$ there is a constant temperature expansion while in contact with a warm reservoir at \(373 \mathrm{K}\) from \(P_{1}=1.55 \times 10^{5} \mathrm{Pa}\) and $V_{1}=2.00 \times 10^{-2} \mathrm{m}^{3}$ to \(P_{2}=1.24 \times 10^{5} \mathrm{Pa}\) and $V_{2}=2.50 \times 10^{-2} \mathrm{m}^{3} .$ Then (B) a heat reservoir at the cooler temperature of \(273 \mathrm{K}\) is used to cool the gas at constant volume to \(273 \mathrm{K}\) from \(P_{2}\) to $P_{3}=0.91 \times 10^{5} \mathrm{Pa} .$ This is followed by (C) a constant temperature compression while still in contact with the cold reservoir at \(273 \mathrm{K}\) from \(P_{3}, V_{2}\) to \(P_{4}=1.01 \times 10^{5} \mathrm{Pa}, V_{1} .\) The final step (D) is heating the gas at constant volume from \(273 \mathrm{K}\) to \(373 \mathrm{K}\) by being in contact with the warm reservoir again, to return from \(P_{4}, V_{1}\) to \(P_{1}, V_{1} .\) Find the change in entropy of the cold reservoir in step \(\mathrm{B}\). Remember that the gas is always in contact with the cold reservoir. (b) What is the change in entropy of the hot reservoir in step D? (c) Using this information, find the change in entropy of the total system of gas plus reservoirs during the whole cycle.
(a) How much heat does an engine with an efficiency of \(33.3 \%\) absorb in order to deliver \(1.00 \mathrm{kJ}\) of work? (b) How much heat is exhausted by the engine?
A \(0.500-\mathrm{kg}\) block of iron at \(60.0^{\circ} \mathrm{C}\) is placed in contact with a \(0.500-\mathrm{kg}\) block of iron at $20.0^{\circ} \mathrm{C} .\( (a) The blocks soon come to a common temperature of \)40.0^{\circ} \mathrm{C} .$ Estimate the entropy change of the universe when this occurs. [Hint: Assume that all the heat flow occurs at an average temperature for each block.] (b) Estimate the entropy change of the universe if, instead, the temperature of the hotter block increased to \(80.0^{\circ} \mathrm{C}\) while the temperature of the colder block decreased to \(0.0^{\circ} \mathrm{C}\) [Hint: The answer is negative, indicating that the process is impossible. \(]\)
List these in order of increasing entropy: (a) 0.01 mol of \(\mathrm{N}_{2}\) gas in a \(1-\mathrm{L}\) container at \(0^{\circ} \mathrm{C} ;\) (b) 0.01 mol of \(\mathrm{N}_{2}\) gas in a 2 -L container at \(0^{\circ} \mathrm{C} ;\) (c) 0.01 mol of liquid \(\mathrm{N}_{2}\).

A large, cold \(\left(0.0^{\circ} \mathrm{C}\right)\) block of iron is immersed in a tub of hot \(\left(100.0^{\circ} \mathrm{C}\right)\) water. In the first \(10.0 \mathrm{s}, 41.86 \mathrm{kJ}\) of heat are transferred, although the temperatures of the water and the iron do not change much in this time. Ignoring heat flow between the system (iron + water) and its surroundings, calculate the change in entropy of the system (iron + water) during this time.
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