Question

For a set of nondegenerate levels with energy \(\varepsilon / k=0,100 .,\) and \(200 .\) K, calculate the probability of occupying each state when \(T=50 ., 500 .,\) and \(5000 .\) K. As the temperature continues to increase, the probabilities will reach a limiting value. What is this limiting value?

Short answer

At temperatures T = 50, 500, and 5000 K, the occupation probabilities for the energy states with \(\varepsilon /k = 0, 100, 200\) K can be calculated using the Boltzmann distribution and are as follows: T = 50 K: \( P_{0,50} = \frac{1}{Q_{50}} \) \( P_{100,50} = \frac{e^{-100/50}}{Q_{50}} \) \( P_{200,50} = \frac{e^{-200/50}}{Q_{50}} \) T = 500 K: \( P_{0,500} = \frac{1}{Q_{500}} \) \( P_{100,500} = \frac{e^{-100/500}}{Q_{500}} \) \( P_{200,500} = \frac{e^{-200/500}}{Q_{500}} \) T = 5000 K: \( P_{0,5000} = \frac{1}{Q_{5000}} \) \( P_{100,5000} = \frac{e^{-100/5000}}{Q_{5000}} \) \( P_{200,5000} = \frac{e^{-200/5000}}{Q_{5000}} \) As the temperature continues to increase, the probabilities will reach a limiting value of 1/3 for each energy state, meaning that as the temperature approaches infinity, the probability of finding a molecule in any of the three states becomes equal.

Step-by-step solution

Understanding the Boltzmann distribution

The Boltzmann distribution states that the probability of a molecule having energy E in a system at temperature T is given by: \( P(E) = \frac{e^{-E/(k_BT)}}{Q} \) where \(k_B\) is the Boltzmann constant and Q is the partition function, defined as: \( Q = \sum_{i} e^{-E_i/(k_BT)} \) In this exercise, we have three energy levels, with energies \(\varepsilon/ k = 0, 100, 200\) K, and we need to calculate the probabilities at temperatures T = 50, 500, and 5000 K.

Calculate the partition function Q

For each temperature, we need to calculate the partition function Q. Using the given energy levels and the temperature, we can plug into the expression for Q: \( Q = e^{-0/(k_BT)} + e^{-100/(k_BT)} + e^{-200/(k_BT)} \) Since \(\varepsilon/k\) is already given, \(Q = 1 + e^{-100/T} + e^{-200/T}\) Calculating Q for each temperature: At T = 50 K: \( Q_{50} = 1 + e^{-100/50} + e^{-200/50} \) At T = 500 K: \( Q_{500} = 1 + e^{-100/500} + e^{-200/500} \) At T = 5000 K: \( Q_{5000} = 1 + e^{-100/5000} + e^{-200/5000} \)

Calculate the probabilities for each energy level

Using the calculated Q values and the Boltzmann distribution expression, we can find probabilities for each energy state at the respective temperatures: At T = 50 K: \( P_{0,50} = \frac{1}{Q_{50}} \) \( P_{100,50} = \frac{e^{-100/50}}{Q_{50}} \) \( P_{200,50} = \frac{e^{-200/50}}{Q_{50}} \) At T = 500 K: \( P_{0,500} = \frac{1}{Q_{500}} \) \( P_{100,500} = \frac{e^{-100/500}}{Q_{500}} \) \( P_{200,500} = \frac{e^{-200/500}}{Q_{500}} \) At T = 5000 K: \( P_{0,5000} = \frac{1}{Q_{5000}} \) \( P_{100,5000} = \frac{e^{-100/5000}}{Q_{5000}} \) \( P_{200,5000} = \frac{e^{-200/5000}}{Q_{5000}} \)

Find the limiting value of probabilities as temperature increases

As the temperature approaches infinity, the probabilities of occupying each state will converge to a limiting value. We can find this limiting value by taking the limit of the probability expressions as T goes to infinity: Limiting value for state 0: \( \lim_{T \to \infty} P_{0,T} = \lim_{T \to \infty} \frac{1}{1 + e^{-100/T} + e^{-200/T}} = \frac{1}{3} \) Limiting value for state 100: \( \lim_{T \to \infty} P_{100,T} = \lim_{T \to \infty} \frac{e^{-100/T}}{1 + e^{-100/T} + e^{-200/T}} = \frac{1}{3} \) Limiting value for state 200: \( \lim_{T \to \infty} P_{200,T} = \lim_{T \to \infty} \frac{e^{-200/T}}{1 + e^{-100/T} + e^{-200/T}} = \frac{1}{3} \) As temperature approaches infinity, the occupation probabilities for each energy state converge to the limiting value of 1/3, meaning that the probability of finding a molecule in any of the three states becomes equal.

Key concepts

Statistical Mechanics

Statistical mechanics is a branch of physics that uses statistical methods to explain the properties of matter and the behavior of particles based on their microscopic details. At the core, it provides a bridge between the microscopic laws of quantum mechanics and the macroscopic observations of thermodynamics. One pivotal application of statistical mechanics is understanding how particles, like atoms or molecules, distribute themselves among available energy states.

The behavior of these particles is random and unpredictable when looked at individually. However, when observing large numbers of particles, clear patterns emerge which can be described statistically. Tools such as the Boltzmann distribution, which we have used in the exercise, allow scientists to predict how particles will distribute among different energy states at various temperatures. By doing so, statistical mechanics gives us valuable insights into the properties of gases, solids, liquids, and plasmas.

Partition Function

The partition function, denoted as Q, plays a central role in statistical mechanics by encapsulating all the possible states that a system can occupy. It is essentially a sum over all the exponential terms that involve the energy levels of the system, inversely weighted by the temperature and Boltzmann's constant.

In mathematical terms, the partition function is expressed as:
\[ Q = \sum_{i} e^{-E_i/(k_BT)} \]
Calculating the partition function allows us to make various probability calculations about the system. For example, it helps to determine the likelihood of the system being in any given energy state at a certain temperature. Importantly, as the partition function changes with temperature, so does the distribution of probabilities across energy levels, which was calculated in the exercise.

Energy Levels

Energy levels refer to the discrete energies that particles in a system can possess. Quantum mechanics dictates that particles such as electrons in atoms or molecules can only occupy certain energy levels. A primary feature is that not all energy levels are equally probable for occupation; lower energy levels tend to be more populated than higher ones, particularly at lower temperatures.

In the provided exercise, the nondegenerate energy levels were given as 0, 100, and 200 times the Boltzmann constant times temperature (K). These levels are key to determining the partition function, which in turn influences the probability of particles occupying these states. As temperature changes, so does the relative occupation of these levels, which leads to different macroscopic properties observable in the physical world.

Probability Calculations

In statistical mechanics, probability calculations are essential for predicting the behavior of systems. They provide the likelihood of finding a particle in a particular energy state. The Boltzmann distribution gives a formula for calculating these probabilities, which is reliant on both the energy of the state and the temperature of the system.

To compute the probability of an energy state, the expression:
\[ P(E) = \frac{e^{-E/(k_BT)}}{Q} \]
is used, where Q is the partition function. The exercise illustrated these calculations at different temperatures, offering a clear example of how probabilities change as temperature varies. Ultimately, such calculations can predict outcomes in a variety of physical processes, from chemical reactions to the behavior of stars.