Question

Imagine dividing a box into two equal parts, part A on the left and part \(\mathrm{B}\) on the right. Four identical gas atoms, numbered 1 through \(4,\) are placed in the box. What are most probable and second most probable distributions (for example, 3 atoms in \(\mathrm{A}, 1\) atom in \(\mathrm{B}\) ) of gas atoms in the box? Calculate the entropy, \(S\), for these two distributions. Note that the configuration with 3 atoms in \(A\) and 1 atom in \(B\) and that with 1 atom in \(A\) and three atoms in B count as different configurations.

Short answer

Answer: The most probable distribution has 2 atoms in each part, with an entropy of \(S_{1} = \ln{6}\). The second most probable distribution has 3 atoms in one part and 1 atom in the other part, with an entropy of \(S_{2} = \ln{4}\).

Step-by-step solution

List all possible distributions

Since there are 2 parts and 4 atoms, there are a total of \(2^4 = 16\) possible distributions for 4 atoms in 2 parts. We list them all: - A: (1, 2, 3, 4), B: () - 1 possibility - A: (), B: (1, 2, 3, 4) - 1 possibility - A: (1, 2, 3), B: (4) - 4 possibilities - A: (1, 2, 4), B: (3) - 4 possibilities - A: (1, 3, 4), B: (2) - 4 possibilities - A: (2, 3, 4), B: (1) - 4 possibilities - A: (1, 2), B: (3, 4) - 6 possibilities - A: (1, 3), B: (2, 4) - 6 possibilities - A: (1, 4), B: (2, 3) - 6 possibilities - A: (2, 3), B: (1, 4) - 6 possibilities - A: (2, 4), B: (1, 3) - 6 possibilities - A: (3, 4), B: (1, 2) - 6 possibilities - A: (1, 2, 4), B: (3, 4) - 12 possibilities - A: (1, 3), B: (2, 3, 4) - 12 possibilities - A: (1, 4), B: (2, 3, 4) - 12 possibilities - A: (2, 3), B: (1, 2, 3, 4) - 12 possibilities

Identify the most probable and second most probable distributions

We notice that the distributions with the largest number of possibilities are the distributions with 2 atoms in each part (A: (1, 2), B: (3, 4) and so on). There are 6 possibilities for each distribution, making it the most probable distribution. The second most probable distribution is the one with 3 atoms in one part and 1 atom in the other part (A: (1, 2, 3), B: (4) and so on). There are 4 possibilities for each such distribution.

Calculate entropy for the most probable distribution

Entropy, \(S\), is defined as \(S = k_B \ln{W}\), where \(k_B\) is Boltzmann's constant, and \(W\) is the number of possible configurations. For the most probable distribution, we have \(W = 6\). Since \(S\) is calculated in units of \(k_B\), we can write \(S_{1} = \ln{6}\) for the most probable distribution.

Calculate entropy for the second most probable distribution

For the second most probable distribution, we have \(W = 4\). The entropy for the second most probable distribution is calculated as \(S_{2} = \ln{4}\).

State results

We have found that the most probable distribution has 2 atoms in each part, with an entropy of \(S_{1} = \ln{6}\). The second most probable distribution has 3 atoms in one part and 1 atom in the other part, with an entropy of \(S_{2} = \ln{4}\).

Key concepts

Statistical Mechanics

Statistical mechanics is a foundational framework in physics that helps us understand and predict the properties of complex systems consisting of a vast number of particles. At its core, it connects the microscopic behavior of individual atoms and molecules with the macroscopic observable properties of materials, such as temperature, pressure, and, indeed, entropy.

This fascinating discipline points out that while we might not track every particle in a system due to their sheer numbers and the quantum uncertainty inherent in their positions and velocities, we can still make accurate predictions using statistical methods. These methods rely on probability distributions to describe how particles, like the gas atoms in our textbook exercise, are likely to be distributed across different states.

Understanding statistical mechanics allows us to infer the most probable arrangement of atoms in our box, as well as any other possible arrangements and their respective likelihoods. This marriage of quantum mechanics, classical mechanics, and statistics gives rise to the power of prediction in thermodynamics and many other fields.

Boltzmann's Constant

When you dive into the mysteries of thermodynamics, you'll inevitably encounter Boltzmann's constant, symbolized as \(k_B\). This fundamental constant connects the micro-world to the macro-world, playing a pivotal role in the relationship between the temperature of a system and the kinetic energy of its particles.

It's essential to grasp that \(k_B\) is not just a constant; it's a bridge that allows us to transition from an understanding of individual particles' behavior to the thermodynamic properties of the entire system. With a value of approximately \(1.38 \times 10^{-23} J/K\), Boltzmann's constant is crucial for calculating entropy, as seen in the step-by-step solution provided for the exercise.

When we talk about entropy calculation in a quantitative manner, \(k_B\) is an indispensable part of the equation, ensuring that the units of measurement are coherent and meaningful across the vast planes of physics.

Probability Distribution

Probability distribution is a cornerstone concept in statistics, and it seamlessly weaves into the fabric of statistical mechanics. In the context of our textbook problem, a probability distribution would tell us how likely it is to find the gas atoms in any given arrangement within the divided box.

The specific type of probability distribution relevant to the exercise is the binomial distribution, which applies to scenarios with two possible outcomes – for example, an atom being in part A or part B of the box. The binomial distribution formula would allow us to calculate the exact probability of having, say, 3 atoms in part A and 1 in part B, by considering the number of atoms and the number of available states for each atom.

This concept is vital when predicting the behavior of the atoms and, by extension, the gas. The more probable a distribution is, the higher is its statistical weight, which directly influences the entropy of the system.

Entropy Calculation

Entropy is often introduced as a measure of disorder within a system, but more accurately, it quantifies the number of ways a system can be arranged without changing its total energy – essentially, it's a measure of uncertainty or randomness. To calculate entropy, we often use the formula \(S = k_B \ln{W}\), where \(S\) represents entropy, \(k_B\) is Boltzmann's constant, and \(W\) is the number of microstates consistent with the given macrostate of the system.

In the exercise presented, the calculation of entropy involves identifying the different distributions of gas atoms and their respective probabilities. The most probable distribution is where there is an equal number of atoms in each part of the box, and the distribution with all atoms on one side or the other is the least probable. We can gain a deeper understanding of thermodynamic processes and the driving forces behind them by parsing out these entropy calculations for different atomic arrangements within the box.