Question

Modeling a population inversion. Population inversion, in which more particles of a system are in an excited state than a ground state, is used to produce laser action. Consider a system of atoms at \(300 \mathrm{~K}\) with three energy levels: \(0 \mathrm{kcal} \mathrm{mol}^{-1}, 0.5 \mathrm{kcal} \mathrm{mol}^{-1}\), and \(1.0 \mathrm{kcal} \mathrm{mol}^{-1}\). (a) Compute the probabilities \(p_{1}^{*}, p_{2}^{*}\), and \(p_{3}^{*}\) that an atom is in each energy level. (b) What is the average energy \(\langle\varepsilon\rangle\) of the system? What is the entropy (in cal \(\mathrm{mol}^{-1} \mathrm{~K}^{-1}\) )? Now suppose you excite the system to cause a population inversion, resulting in a new population distribution \(p^{* *}\) as follows: \(p_{1}^{* *}=p_{3}^{*}, p_{2}^{* *}=p_{2}^{*}, p_{3}^{* *}=p_{1}^{*}\). (c) What is the average energy \(\langle\varepsilon\rangle\) of the system after the population inversion? What is the entropy (in cal \(\mathrm{mol}^{-1} \mathrm{~K}^{-1}\) ) of the system after the population inversion? (d) What is the temperature of the system after the population inversion?

Short answer

Partition function Z is needed to calculate probabilities. With new probabilities, average energy, entropy, and temperature re-evaluations are performed.

Step-by-step solution

Compute Partition Function Z

The partition function Z is computed using the formula: \[Z = \text{sum}_{i} e^{-E_i / k_B T}\] Here, we have three energy states: 0, 0.5 and 1.0 kcal/mol. We need to convert kcal/mol to J/mol by multiplying with 4184 J/kcal and divide by the Boltzmann constant (8.314 J/mol K) to get energy in the appropriate unit. Next, we compute Z at 300K.

Calculate Temperature-Converted Energy Levels

First, convert each energy level from kcal/mol to J and then to dimensionless form by dividing by \( k_B \times T \), where \( k_B = 8.314 \text{ J/mol K} \) and \( T = 300 \text{ K} \). Energy levels: 0, 0.5 kcal/mol (0.5 \times 4184 J), and 1 kcal/mol (1 \times 4184 J).

Calculate Probabilities

Using the partition function Z, compute the probabilities for each state:\[ p_i = \frac{e^{-E_i / k_B T}}{Z} \]. Calculate \( p_1^*, p_2^*, \) and \( p_3^*\) using their respective energy levels.

Compute Average Energy of the System

The average energy \( \langle \varepsilon \rangle \) is computed by: \[ \langle \varepsilon \rangle = \sum_{i} p_i \varepsilon_i \].

Compute Entropy of the System

Entropy S is computed using Boltzmann's entropy formula: \[ S = -R \sum_{i} p_i \, \ln(p_i) \].

Excite System and Get New Probabilities

The new population distribution is given: \( p_{1}^{**} = p_{3}^* \), \( p_{2}^{**} = p_{2}^* \), \( p_{3}^{**} = p_{1}^* \).

Recompute Average Energy After Inversion

Use the new probabilities \( p_{1}^{**}, p_{2}^{**}, \) and \( p_{3}^{**} \) to recompute the average energy \( \langle \varepsilon \rangle \).

Recompute Entropy After Inversion

Again, use Boltzmann's entropy formula with the new probabilities to calculate: \[ S = -R \sum_{i} p_{i}^{**} \, \ln(p_{i}^{**}) \].

Calculate Effective Temperature After Inversion

The effective temperature T' after population inversion can be estimated using the Boltzmann distribution and average energy:

Key concepts

Partition Function

The partition function, denoted as Z, is a key concept in thermodynamics and statistical mechanics. It is a measure of the statistical distribution of particles among various energy states in a system. The partition function is given by the sum of the Boltzmann factors for each energy level.

Mathematically, it is represented as follows:

\[ Z = \text{sum}_{i} e^{-E_i / k_B T} \]

Here,

• \(E_i\) is the energy of the state \(i\),
• \(k_B\) is the Boltzmann constant,
• \(T\) is the temperature in Kelvin.

The partition function helps determine the probabilities of the system being in different energy states, which is crucial for calculating other thermodynamic properties such as average energy and entropy.

Boltzmann Constant

The Boltzmann constant, denoted by \(k_B\), connects macroscopic and microscopic physics. It plays a critical role in statistical mechanics and thermodynamics. The Boltzmann constant has a value of approximately 8.314 J/(mol K).

This constant appears in the expressions for the partition function, the distribution of particles among energy levels, and helps in converting energy units into temperature units and vice versa. It provides a bridge between the energy of individual particles and the temperature of the system.

Entropy Calculation

Entropy is a measure of randomness or disorder in a system. In statistical mechanics, it quantifies how spread out the energy of a system is among its possible states.

The entropy \(S\) can be calculated using Boltzmann's entropy formula:

\[ S = -R \, \text{sum}_{i} p_i \, \text{ln}(p_i) \]

Here,

• \(R\) is the universal gas constant,
• \(p_i\) is the probability of the system being in state \(i\).

The formula sums up for all possible states, each weighted by its probability of occurrence. Entropy increases with the number of accessible microstates and provides vital insights into the spontaneity and equilibrium of chemical processes.

Average Energy

The average energy \( \langle \, \text{E} \, \rangle \) of a system indicates the expected energy of particles in the system at a given temperature. This is essential for understanding the behavior and properties of the system.

The average energy is computed using the formula:

\[ \langle \, \text{E} \, \rangle = \text{sum}_{i} p_i \, \text{E}_i \]

Here,

• \(p_i\) is the probability of being in state \(i\),
• \(\text{E}_i\) is the energy of state \(i\).

This weighted sum provides an insight into the energy distribution in the system, reflecting the influence of temperature and energy states.

Effective Temperature

The effective temperature after a population inversion can be an abstract but highly informative concept. This temperature describes how the average energy of a system corresponds to a thermal equilibrium state.

Once you've computed the new average energy \( \langle \, \text{E} \, \rangle \) after population inversion, the effective temperature \(T'\) can be estimated using the Boltzmann distribution principles and the partition function.

By comparing the new energy levels and the shifted population probabilities, the effective temperature elucidates the new thermal dynamics of the system post-inversion, bridging theoretical models with observable phenomena in thermodynamics.