Ch.19 - Free Energy & Thermodynamics

Problem 30

Chapter 19, Problem 30

Two systems, each composed of three particles represented by circles, have 30 J of total energy. How many energetically equivalent ways can you distribute the particles in each system? Which system has greater entropy?

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insert step 1> Identify the two systems and the particles within each system. Each system has three particles, and the total energy is 30 J.

insert step 2> Consider the possible energy distributions for each system. For example, you can distribute the energy among the particles in different ways, such as (10 J, 10 J, 10 J), (15 J, 10 J, 5 J), etc.

insert step 3> Calculate the number of energetically equivalent ways to distribute the energy among the particles in each system. This involves finding all possible combinations of energy distributions that sum up to 30 J.

insert step 4> Compare the number of energetically equivalent ways for each system. The system with more ways to distribute the energy has greater entropy.

insert step 5> Conclude which system has greater entropy based on the number of energetically equivalent distributions. The system with more distributions has higher entropy, as entropy is a measure of the number of possible configurations.

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Key Concepts

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

Statistical Mechanics

Statistical mechanics is a branch of physics that uses statistical methods to explain the thermodynamic properties of systems composed of a large number of particles. It provides a framework for understanding how the microscopic behavior of particles relates to macroscopic properties like temperature and energy distribution. In this context, it helps determine the number of ways particles can be arranged, which is crucial for calculating entropy.

Entropy

Entropy is a measure of the disorder or randomness in a system, often interpreted as the number of ways a system can be arranged while maintaining the same energy. In thermodynamics, higher entropy indicates a greater number of possible configurations, which corresponds to a more disordered state. When comparing two systems, the one with higher entropy is considered to have greater disorder and is thermodynamically favored.

Microstates and Macrostates

In statistical mechanics, a macrostate is defined by macroscopic properties like energy and particle number, while microstates are the specific arrangements of particles that correspond to a given macrostate. The number of microstates associated with a macrostate is directly related to the entropy of the system. Understanding the relationship between microstates and macrostates is essential for calculating the entropy and determining which system has greater disorder.

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