Question

A system of noninteracting atoms has a positional multiplicity of 120 and an energetic multiplicity of \(5 .\) What is the total multiplicity? What are the positional, energetic, and total entropies of the system?

Short answer

Total multiplicity is 600, with positional, energetic, and total entropies calculated using their respective multiplicities.

Step-by-step solution

Understanding Multiplicities

Multiplicity refers to the number of different ways in which a system can be arranged. In this problem, you are given a positional multiplicity of 120 and an energetic multiplicity of 5.

Calculate the Total Multiplicity

Total multiplicity is the product of positional and energetic multiplicities. Multiply the positional multiplicity (120) by the energetic multiplicity (5) to find the total multiplicity.\[ \text{Total Multiplicity} = 120 \times 5 = 600 \]

Calculate Positional Entropy

Use the formula for entropy, \( S = k \ln(W) \), where \( k \) is Boltzmann's constant (approximately \( 1.38 \times 10^{-23} \text{ J/K} \)) and \( W \) is the multiplicity. For positional entropy, \( W = 120 \).\[ S_{\text{positional}} = k \ln(120) \approx 1.38 \times 10^{-23} \ln(120) \]

Calculate Energetic Entropy

Similarly, calculate the energetic entropy using the energetic multiplicity. Here, \( W = 5 \).\[ S_{\text{energetic}} = k \ln(5) \approx 1.38 \times 10^{-23} \ln(5) \]

Calculate Total Entropy

The total entropy of the system is calculated using the total multiplicity.\[ S_{\text{total}} = k \ln(600) \approx 1.38 \times 10^{-23} \ln(600) \]

Summarize the Entropy Values

Compute the actual numerical values for each entropy and summarize. Use a calculator for the natural logarithms:- \( \ln(120) \approx 4.787 \)- \( \ln(5) \approx 1.609 \)- \( \ln(600) \approx 6.396 \)Substitute these into the entropy formula to find:- \( S_{\text{positional}} = 1.38 \times 10^{-23} \times 4.787 \)- \( S_{\text{energetic}} = 1.38 \times 10^{-23} \times 1.609 \)- \( S_{\text{total}} = 1.38 \times 10^{-23} \times 6.396 \)

Key concepts

Multiplicity

In the context of thermodynamics, multiplicity refers to the number of possible arrangements available to a system. In simpler terms, it is a way to count how many different configurations can occur for a given set of energy levels or positions of particles. This is crucial because the number of ways a system can be configured often determines its entropy. Entropy, in this sense, is linked to the disorder or randomness within the system.
For example, given a system with two types of multiplicities—positional and energetic—you can calculate the total multiplicity by multiplying these values. For instance, if a system has a positional multiplicity of 120 and an energetic multiplicity of 5, the total multiplicity is calculated as:

• 120 (positional) × 5 (energetic) = 600

This total multiplicity gives you a measure of the overall randomness of the system, combining both position and energy levels into one comprehensive value.

Boltzmann's Constant

Boltzmann's constant, often symbolized as \( k \), is a key factor in statistical mechanics and thermodynamics and plays a crucial role in connecting the macroscopic and microscopic worlds of physics. It is named after Ludwig Boltzmann, an Austrian physicist noted for his foundational contributions to statistical mechanics.
Boltzmann's constant is approximately \( 1.38 \times 10^{-23} \text{ J/K} \) and it's used in the formula for entropy:

• \( S = k \ln(W) \)

where \( S \) is the entropy and \( W \) is the multiplicity of the system. Boltzmann's constant essentially helps scale the otherwise dimensionless measure of multiplicity to an appropriate physical value of entropy measured in joules per kelvin (J/K). This scale conversion is crucial because it allows for the quantification of the disorder or randomness in a thermal system.

Positional Entropy

Positional entropy is a component of the system's total entropy that reflects the number of ways particles can be spatially arranged. It deals with the distribution of particles in different locations and configurations, irrespective of their energy states. This type of entropy is particularly significant when considering gases, as different spatial arrangements can occur even at the same energy level.
To calculate the positional entropy, use the formula:

• \( S_{\text{positional}} = k \ln(W_{\text{positional}}) \)

where \( W_{\text{positional}} \) is the positional multiplicity. In the given example, if the positional multiplicity is 120, plug this value into the formula to find positional entropy, which measures how the particles fill the space available to them.

Energetic Entropy

Energetic entropy quantifies the number of ways a system's energy can be distributed among its particles. Unlike positional entropy, which considers spatial distribution, energetic entropy focuses on the energy levels themselves. It plays a critical role in understanding how energy is shared among particles in given energy states.
The energetic entropy is calculated similarly to positional entropy, using the energetic multiplicity:

• \( S_{\text{energetic}} = k \ln(W_{\text{energetic}}) \)

where \( W_{\text{energetic}} \) is the energy state multiplicity. For example, with an energetic multiplicity of 5, the calculation reflects the different ways energy can be partitioned among particles and contributes to understanding the overall system behavior. Together with positional entropy, energetic entropy helps grasp the complete picture of a system's randomness and disorder in both spatial and energetic dimensions.