Ch. 20 - Second Law of Thermodynamics

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 20 - Second Law of Thermodynamics

문제 60b

20장, 문제 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?

검증된 단계별 안내

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}}

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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

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