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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

From Table \(14.4,\) we know that approximately \(2256 \mathrm{kJ}\) are needed to transform \(1.00 \mathrm{kg}\) of water at \(100^{\circ} \mathrm{C}\) to steam at \(100^{\circ} \mathrm{C}\). What is the change in entropy of \(1.00 \mathrm{kg}\) of water evaporating at \(100.0^{\circ} \mathrm{C} ?\) (Specify whether the change in entropy is an increase, \(+\), or a decrease, \(-.\) )
A reversible refrigerator has a coefficient of performance of \(3.0 .\) How much work must be done to freeze \(1.0 \mathrm{kg}\) of liquid water initially at \(0^{\circ} \mathrm{C} ?\)
Suppose 1.00 mol of oxygen is heated at constant pressure of 1.00 atm from \(10.0^{\circ} \mathrm{C}\) to \(25.0^{\circ} \mathrm{C} .\) (a) How much heat is absorbed by the gas? (b) Using the ideal gas law, calculate the change of volume of the gas in this process. (c) What is the work done by the gas during this expansion? (d) From the first law, calculate the change of internal energy of the gas in this process.
An inventor proposes a heat engine to propel a ship, using the temperature difference between the water at the surface and the water \(10 \mathrm{m}\) below the surface as the two reservoirs. If these temperatures are \(15.0^{\circ} \mathrm{C}\) and \(10.0^{\circ} \mathrm{C},\) respectively, what is the maximum possible efficiency of the engine?
Show that the coefficient of performance for a reversible heat pump is $1 /\left(1-T_{\mathrm{C}} / T_{\mathrm{H}}\right)$.
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