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 60b

Chapter 20, Problem 60b

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?

Verified step by step guidance

Understand the problem: A macrostate refers to the observable outcome (e.g., the number of heads), while a microstate refers to the specific arrangement of the coins (e.g., HHTTHT). The goal is to calculate the number of microstates for each macrostate and determine the probability of obtaining six heads.

Step 1: Use the binomial coefficient formula to calculate the number of microstates for each macrostate. The formula is:

(\frac{n!}{k! (n - k)!})

, where n is the total number of coins (6 in this case), and k is the number of heads.

Step 2: Construct a table for all possible macrostates (0 heads, 1 head, ..., 6 heads). For each macrostate, calculate the number of microstates using the binomial coefficient formula. For example, for 0 heads, the number of microstates is

(\frac{6!}{0! (6 - 0)!})

, and for 6 heads, it is

(\frac{6!}{6! (6 - 6)!})

Step 3: Calculate the total number of microstates by summing up the microstates for all macrostates. This total is equal to

2^{6}

, since each coin has two possible outcomes (heads or tails).

Step 4: To find the probability of obtaining six heads, divide the number of microstates corresponding to six heads by the total number of microstates. The probability is given by:

\frac{(\frac{6!}{6! (6 - 6)!})}{2^{6}}

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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, when tossing coins, each unique arrangement of heads and tails represents a microstate, whereas the overall count of heads (e.g., six heads) represents a macrostate. Understanding the relationship between these concepts is crucial for calculating probabilities.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways to arrange or select items from a set. In the context of the coin toss, it helps determine how many different microstates correspond to a given macrostate, such as six heads. The formula for combinations, often denoted as 'n choose k', is essential for calculating these arrangements, where 'n' is the total number of items and 'k' is the number of selected items.

Probability

Probability quantifies the likelihood of an event occurring, expressed as a ratio of favorable outcomes to the total number of possible outcomes. In this scenario, to find the probability of obtaining six heads when tossing six coins, one must divide the number of favorable microstates (which is one, as there is only one way to get all heads) by the total number of microstates (which is 2^6, or 64). This concept is fundamental for understanding outcomes in random experiments.

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Probability Distribution Graph

Related Practice

Textbook Question

Refrigeration units can be rated in “tons.” A 1-ton air conditioning system can remove sufficient energy to freeze 1 ton (2000 pounds = 909 kg) of 0°C water into 0°C ice in one 24-h day. Assume the hot part of a day averages 35°C and the interior of a house is maintained at 22°C by the continuous operation of a 6-ton air conditioning system for 6 hours a day. How much does this cooling cost the homeowner per day, and per month?Assume the work done by the refrigeration unit is powered by electricity that costs \$0.13 per kWh and that the unit’s coefficient of performance is only 18% of an ideal refrigerator. 1 kWh = 3.60 x 10⁶ J .

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Textbook Question

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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Textbook Question

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

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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 three heads and three tails?

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Textbook Question

(II) Calculate the probabilities, when you throw two dice, of obtaining an 11.

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Textbook Question

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.