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 59

Chapter 20, Problem 59

Use Eq. 20–14 to determine the entropy of each of the five macrostates listed in Table 20–1 on page 595.

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Identify the equation for entropy given in Eq. 20–14: \( S = k_B \ln W \), where \( S \) is the entropy, \( k_B \) is the Boltzmann constant, and \( W \) is the number of microstates corresponding to a macrostate.

Refer to Table 20–1 on page 595 to extract the values of \( W \) (the number of microstates) for each of the five macrostates.

For each macrostate, substitute the corresponding \( W \) value into the equation \( S = k_B \ln W \). Ensure that the natural logarithm (\( \ln \)) is used.

Multiply the result of \( \ln W \) by the Boltzmann constant \( k_B \) (approximately \( 1.38 \times 10^{-23} \, \text{J/K} \)) to calculate the entropy for each macrostate.

Repeat the calculation for all five macrostates and organize the results in a table or list for clarity.

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Key Concepts

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

Entropy

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it quantifies the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state. Higher entropy indicates greater disorder and a higher number of possible configurations, while lower entropy suggests more order and fewer configurations.

Macrostates and Microstates

In statistical mechanics, a macrostate is defined by macroscopic properties such as temperature, pressure, and volume, while microstates are the specific detailed arrangements of particles that correspond to a macrostate. The relationship between macrostates and microstates is crucial for calculating entropy, as entropy is related to the number of microstates associated with a given macrostate.

Statistical Mechanics

Statistical mechanics is a branch of physics that uses statistical methods to explain and predict the thermodynamic properties of systems composed of a large number of particles. It connects microscopic behavior (individual particles) to macroscopic phenomena (bulk properties), allowing for the calculation of quantities like entropy and temperature based on particle statistics.

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Related Practice

Textbook Question

Suppose that you repeatedly shake six coins in your hand and drop them on the floor. Construct a table showing the number of microstates that correspond to each macrostate. What is the probability of obtaining six heads?

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Textbook Question

Why would you expect the total entropy change in a Carnot cycle to be zero? Do a calculation to show that it is zero.

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Textbook Question

Why would you expect the total entropy change in a Carnot cycle to be zero?

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Textbook Question

(II) Calculate the probabilities, when you throw two dice, of obtaining a 7.

A general theorem states that the amount of energy that becomes unavailable to do useful work in any process is equal to T_L∆S, where T_L is the lowest temperature available and ∆S is the total change in entropy during the process. Show that this is valid in the specific cases of a falling rock that comes to rest when it hits the ground.