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Chapter 15: Problem 47

List these in order of increasing entropy: (a) \(0.5 \mathrm{kg}\) of ice and \(0.5 \mathrm{kg}\) of (liquid) water at \(0^{\circ} \mathrm{C} ;\) (b) $1 \mathrm{kg}\( of ice at \)0^{\circ} \mathrm{C} ;\( (c) \)1 \mathrm{kg}$ of (liquid) water at \(0^{\circ} \mathrm{C} ;\) (d) \(1 \mathrm{kg}\) of water at \(20^{\circ} \mathrm{C}\).

Short Answer

Expert verified
Question: Arrange the following systems in increasing order of entropy: (a) a mixture of 0.5 kg of ice and 0.5 kg of water, both at 0°C; (b) 1 kg of ice at 0°C; (c) 1 kg of water at 0°C; and (d) 1 kg of water at 20°C. Answer: The systems in increasing order of entropy are: (b) 1 kg of ice at 0°C → (c) 1 kg of (liquid) water at 0°C → (a) 0.5 kg of ice and 0.5 kg of (liquid) water at 0°C → (d) 1 kg of water at 20°C.

Step by step solution

01

Compare the phases of matter

We can begin by considering the phases of matter involved in the four systems: - (a) is a mixture of liquid water and solid ice, both at \(0^{\circ} \mathrm{C}\). - (b) is solid ice at \(0^{\circ} \mathrm{C}\). - (c) is liquid water at \(0^{\circ} \mathrm{C}\). - (d) is liquid water at \(20^{\circ} \mathrm{C}\). Entropy increases as a substance changes from solid to liquid to gas, so we can immediately tell that (b) will have the lowest entropy amongst the four as it is the only system that is completely solid.
02

Compare the temperatures of the systems

Entropy increases as the temperature of a system increases. System (d) has the highest temperature so it will have the highest entropy amongst the four. Now we are left with systems (a) and (c), both of which have some liquid water at \(0^{\circ} \mathrm{C}\).
03

Compare the masses of the substances in the systems

Entropy is an extensive property, which means that it increases as the mass of the substance increases. In this case, system (a) has a total mass of \(1 \mathrm{kg}\) (\(0.5 \mathrm{kg}\) ice and \(0.5 \mathrm{kg}\) liquid water), while system (c) has a mass of \(1 \mathrm{kg}\) of liquid water at the same temperature. Since the ice in system (a) adds more disorder compared to the same mass of water in system (c), the entropy of (a) will be higher than (c).
04

Arrange the systems in increasing order of entropy

Based on our analysis above, we can now list the systems in order of increasing entropy: (b) \(1 \mathrm{kg}\) of ice at \(0^{\circ} \mathrm{C}\) → (c) \(1 \mathrm{kg}\) of (liquid) water at \(0^{\circ} \mathrm{C}\) → (a) \(0.5 \mathrm{kg}\) of ice and \(0.5 \mathrm{kg}\) of (liquid) water at \(0^{\circ} \mathrm{C}\) → (d) \(1 \mathrm{kg}\) of water at \(20^{\circ} \mathrm{C}\).

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Most popular questions from this chapter

A coal-fired electrical generating station can use a higher \(T_{\mathrm{H}}\) than a nuclear plant; for safety reasons the core of a nuclear reactor is not allowed to get as hot as coal. Suppose that $T_{\mathrm{H}}=727^{\circ} \mathrm{C}\( for a coal station but \)T_{\mathrm{H}}=527^{\circ} \mathrm{C}$ for a nuclear station. Both power plants exhaust waste heat into a lake at \(T_{\mathrm{C}}=27^{\circ} \mathrm{C} .\) How much waste heat does each plant exhaust into the lake to produce \(1.00 \mathrm{MJ}\) of electricity? Assume both operate as reversible engines.
Suppose you mix 4.0 mol of a monatomic gas at \(20.0^{\circ} \mathrm{C}\) and 3.0 mol of another monatomic gas at \(30.0^{\circ} \mathrm{C} .\) If the mixture is allowed to reach equilibrium, what is the final temperature of the mixture? [Hint: Use energy conservation.]
A town is considering using its lake as a source of power. The average temperature difference from the top to the bottom is \(15^{\circ} \mathrm{C},\) and the average surface temperature is \(22^{\circ} \mathrm{C} .\) (a) Assuming that the town can set up a reversible engine using the surface and bottom of the lake as heat reservoirs, what would be its efficiency? (b) If the town needs about \(1.0 \times 10^{8} \mathrm{W}\) of power to be supplied by the lake, how many \(\mathrm{m}^{3}\) of water does the heat engine use per second? (c) The surface area of the lake is $8.0 \times 10^{7} \mathrm{m}^{2}$ and the average incident intensity (over \(24 \mathrm{h})\) of the sunlight is \(200 \mathrm{W} / \mathrm{m}^{2} .\) Can the lake supply enough heat to meet the town's energy needs with this method?
A reversible heat engine has an efficiency of \(33.3 \%\) removing heat from a hot reservoir and rejecting heat to a cold reservoir at $0^{\circ} \mathrm{C} .$ If the engine now operates in reverse, how long would it take to freeze \(1.0 \mathrm{kg}\) of water at \(0^{\circ} \mathrm{C},\) if it operates on a power of \(186 \mathrm{W} ?\)
Two engines operate between the same two temperatures of \(750 \mathrm{K}\) and \(350 \mathrm{K},\) and have the same rate of heat input. One of the engines is a reversible engine with a power output of \(2.3 \times 10^{4} \mathrm{W} .\) The second engine has an efficiency of \(42 \% .\) What is the power output of the second engine?
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