Ch. 20 - Second Law of Thermodynamics

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 20 - Second Law of Thermodynamics

Problem 89a

Chapter 20, Problem 89a

A bowl contains many red, orange, and green jelly beans, in equal numbers. You are to make a line of 3 jelly beans by randomly taking 3 beans from the bowl. Construct a table showing the number of microstates that correspond to each macrostate.

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Understand the problem: A macrostate is defined by the number of jelly beans of each color in the line (e.g., 1 red, 1 orange, 1 green). A microstate is a specific arrangement of jelly beans in the line (e.g., red first, orange second, green third). The goal is to count the microstates for each macrostate.

Identify all possible macrostates: Since there are three colors (red, orange, green) and three jelly beans in total, the macrostates are determined by the number of each color in the line. Possible macrostates include: (3 red, 0 orange, 0 green), (2 red, 1 orange, 0 green), (1 red, 1 orange, 1 green), etc.

Use combinatorics to calculate the number of microstates for each macrostate: For example, if the macrostate is (2 red, 1 orange, 0 green), calculate the number of ways to arrange 2 red and 1 orange jelly bean in a line using the formula for permutations: \( \frac{n!}{n_1! \cdot n_2! \cdot n_3!} \), where \( n \) is the total number of jelly beans, and \( n_1, n_2, n_3 \) are the counts of each color.

Construct the table: For each macrostate, list the number of jelly beans of each color and the corresponding number of microstates. For example, the row for (1 red, 1 orange, 1 green) would include the number of microstates calculated as \( 3! = 6 \), since all three colors are distinct and can be arranged in any order.

Verify the total number of microstates: Add up the microstates for all macrostates to ensure the total matches the number of ways to choose and arrange 3 jelly beans from the bowl. This total should be \( 3^3 = 27 \), as there are 3 choices for each of the 3 positions in the line.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Microstates and Macrostates

In statistical mechanics, a microstate refers to a specific detailed configuration of a system, while a macrostate is defined by macroscopic properties like temperature or pressure. For example, in the jelly bean scenario, each unique arrangement of the jelly beans represents a microstate, while the overall color distribution (e.g., 1 red, 1 orange, 1 green) represents a macrostate.

Combinatorial Counting

Combinatorial counting is a mathematical technique used to determine the number of ways to choose or arrange items. In this context, it helps calculate the number of different combinations of jelly beans that can be selected, which is essential for constructing the table of microstates corresponding to each macrostate.

Probability Distribution

A probability distribution describes how the probabilities of different outcomes are distributed in a system. In the jelly bean example, understanding the probability of selecting different combinations of colors helps in analyzing the likelihood of each macrostate occurring, which is crucial for interpreting the results of the constructed table.

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Related Practice

Textbook Question

A dehumidifier removes water vapor from air and has been referred to as a “refrigerator with an open door.” The humid air is pulled in by a fan and passes over a cold coil, whose temperature is less than the dew point, and some of the air’s water condenses. After this water is extracted, the air is warmed back to its original temperature and sent into the room. In a well-designed dehumidifier, the heat that is removed by the cooling coil mostly comes from the condensation of water vapor to liquid, and this heat is used to re-warm the air. Estimate how much water is removed in 1.0 h by an ideal dehumidifier, if the temperature of the room is 25°C, the water condenses at 8°C, and the dehumidifier does work at the rate of 550 W of electrical power.

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Textbook Question

Trees can help a bit to offset the buildup of CO₂ due to burning coal and other fossil fuels. CO₂ can be absorbed by tree foliage. Trees use the carbon to grow, and release O₂ into the atmosphere. Suppose a refrigerator uses 600 kWh of electricity per year (about 2 x 10⁹ J) from a 33% efficient coal-fired power plant. Burning 1 kg of coal releases about 2 x 10⁷ J of energy. Assume coal is all carbon, which when burned in air becomes CO₂.

(a) How much coal is burned per year to run this refrigerator?

(b) Assuming a forest can capture 1700 kg of carbon per hectare ( = 10, 000 m²) per year, estimate how many square meters of forest are needed to capture the carbon (in the form now of CO₂) emitted by the refrigerator in (a).

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Textbook Question

Suppose a heat pump has a stationary bicycle attachment that allows you to provide the work instead of using an electrical wall outlet. If your heat pump has a coefficient of performance of 2.0 and you can cycle at a racing pace output of about 200 W for a half hour, how much heat can you provide?

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Textbook Question

Two 1100-kg cars are traveling 75 km/h in opposite directions when they collide and are brought to rest. Estimate the change in entropy of the universe as a result of this collision. Assume T = 20°C.

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Textbook Question

A bowl contains many red, orange, and green jelly beans, in equal numbers. You are to make a line of 3 jelly beans by randomly taking 3 beans from the bowl. Construct a table showing the number of microstates that correspond to each macrostate. Then, determine the probability of all 3 beans red.

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Textbook Question

Rank the following five-card hands in order of increasing probability: (a) four aces and a king; (b) six of hearts, eight of diamonds, queen of clubs, three of hearts, jack of spades; (c) two jacks, two queens, and an ace; and (d) any hand having no two equal-value cards (no pairs, etc.). Discuss your ranking in terms of microstates and macrostates.

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