Problem 57 - An Introduction to Thermal Physics

Question

Consider an ideal mixture of just 100 molecules, varying in composition from pure \(A\) to pure \(B\). Use a computer to calculate the mixing entropy as a function of \(N_{A},\) and plot this function (in units of \(k\) ). Suppose you start with all \(A\) and then convert one molecule to type \(B ;\) by how much does the entropy increase? By how much does the entropy increase when you convert a second molecule, and then a third, from \(A\) to \(B ?\) Discuss.

Step-by-Step Solution

Answer

Entropy increases by 0.92k for the first, 0.69k for the second, and 0.60k for the third conversion.

Step 1: Understanding the Concept

The mixing entropy for an ideal mixture is computed using Boltzmann's formula for entropy: \[ S = k \cdot \ln(W) \]where \( k \) is the Boltzmann constant and \( W \) is the number of microstates. In statistical mechanics, mixing entropy for a binary mixture can be found via the formula:\[ \Delta S = -k (N_A \ln(N_A) + N_B \ln(N_B) - N \ln(N)) \]Here, \( N \) is the total number of molecules, \( N_A \) and \( N_B \) are the number of molecules of types A and B respectively.

Step 2: Calculating Mixing Entropy for One Molecule Conversion

Initially, the mixture starts with 100% A molecules (\(N_A = 100, N_B = 0)\).Calculate \( \Delta S \) when \( N_A = 99 \) and \( N_B = 1 \).Insert values into the formula:\[\Delta S_1 = -k (99 \ln(99/100) + 1 \ln(1/100))\]

Step 3: Calculating Mixing Entropy for Two Molecules Conversion

Next, convert a second molecule from A to B. Now, \( N_A = 98 \) and \( N_B = 2 \).Insert values into the formula:\[\Delta S_2 = -k (98 \ln(98/100) + 2 \ln(2/100))\]

Step 4: Calculating Mixing Entropy for Three Molecules Conversion

For converting a third molecule, \( N_A = 97 \) and \( N_B = 3 \).Insert values into the formula:\[\Delta S_3 = -k (97 \ln(97/100) + 3 \ln(3/100))\]

Step 5: Calculating the Entropy Increments

Compare the entropy from each successive step to see the increments:1. Entropy increase from 0 to 1 molecule converted (\( \Delta S_1 \)).2. Entropy increase from 1 to 2 molecules converted (\( \Delta S_2 - \Delta S_1 \)).3. Entropy increase from 2 to 3 molecules converted \( \Delta S_3 - \Delta S_2 \).

Step 6: Discussing the Results

Analyze the changes in entropy as molecules are converted from A to B. Typically, the entropy increase diminishes as the composition becomes more balanced, which can be explained by the concept of maximum entropy occurring at an even mixture between the two types of molecules.

Key Concepts

Boltzmann's Formula

Boltzmann's Formula is pivotal in understanding entropy in statistical mechanics. It states that the entropy, \( S \), of a system is related to the number of microstates, \( W \), it can have.This is given by the formula:\[ S = k \cdot \ln(W) \]where \( k \) is the Boltzmann constant. Entropy, in simple terms, is a measure of disorder or randomness in a system. The higher the number of possible arrangements (microstates) of a system, the higher the entropy. Entropy plays a crucial role when analyzing mixtures or chemical reactions, providing insight into which state a system prefers based on the number of ways it can achieve its configuration.Understanding Boltzmann's Formula is essential because it connects macroscopic concepts like temperature and energy to microscopic behaviors, allowing us to predict how systems will change over time.

Binary Mixture

A binary mixture consists of two different types of molecules or substances. In the context of statistical mechanics, we analyze how these two components mix at a molecular level. To explore this concept further, consider the example of a mixture undergoing a change in composition from pure component A to a combination of components A and B. In our exercise, we start with 100 molecules of type A. Gradually, molecules of type B replace those of type A. The focus is on calculating mixing entropy as the number of each type changes, by following the conversion of molecules. A binary mixture is particularly fascinating because it allows us to observe how properties like entropy depend on the ratio of its components.

Statistical Mechanics

Statistical Mechanics is a framework that connects the microscopic properties of individual atoms and molecules with macroscopic, bulk properties such as pressure, volume, and temperature. This discipline helps us to understand how countless possible molecular configurations can give rise to observable phenomena. In applying statistical mechanics to our problem, we use the formula for mixing entropy to measure changes in disorder as molecules are converted from one type to another. The mixing entropy formula includes terms that involve logarithmic functions of the relative amounts of each component in the mixture, reflecting how likely each configuration is in contributing to the total entropy. This approach allows us to quantify the shifting probabilities and provides insightful predictions about the mixing process.

Entropy Increment

Entropy increment refers to the change or increase in entropy as reactions or interactions occur, such as when two substances mix. The concept is vividly illustrated through our exercise by watching how entropy increases as we convert A molecules to B molecules. The entropy increments each time a molecule changes its type from A to B. Notably, the increments of entropy tend to decrease as we move towards an even mix of components A and B. This is because maximum entropy, or maximum disorder, occurs when there is an even distribution between the two types of molecules in the mixture. This principle highlights why systems naturally tend toward reaching a state of balance or equilibrium, where entropy is maximized.