A system of independent particles (energy levels)

The correct equation to use is e^(-ε1/kT)=0.2. In summary, the conversation is about calculating the temperature at which 20% of independent particles in a system have energy level ε1 using the Boltzmann Distribution formula and the given hint. The correct equation to use is e^(-ε1/kT)=0.2 and z is the partition function.
  • #1
physicisttobe
56
13
Homework Statement
boltzmann distribution
Relevant Equations
…
Hi guys,

Can you give me some feedback on whether my calculation is correct? I applied the formula below (Boltzmann Distribution) but I didn‘t know what to use for the variable z. I don‘t even know if I used the correct equation. Can you help me further?

The task is:
Consider a system of independent particles that can only occupy 2 different energy levels ε0 = 0 (non-degenerate) or ε1 = 50 meV (3-fold degenerate). At what temperature do 20% of the particles have energy ε1?

Hint: k= 1.38E-23 J/K
e= 1.60E-19 As

B3EF1F20-B87A-4A41-AB44-DF8121B70346.jpeg
 
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  • #2
physicisttobe said:
Homework Statement: boltzmann distribution
Relevant Equations: …

Hi guys,

Can you give me some feedback on whether my calculation is correct? I applied the formula below (Boltzmann Distribution) but I didn‘t know what to use for the variable z. I don‘t even know if I used the correct equation. Can you help me further?

The task is:
Consider a system of independent particles that can only occupy 2 different energy levels ε0 = 0 (non-degenerate) or ε1 = 50 meV (3-fold degenerate). At what temperature do 20% of the particles have energy ε1?

Hint: k= 1.38E-23 J/K
e= 1.60E-19 As

View attachment 325293
z is not a variable. It's the partition function.
 

FAQ: A system of independent particles (energy levels)

What is a system of independent particles?

A system of independent particles refers to a collection of particles that do not interact with each other. Each particle in the system occupies discrete energy levels, and the overall properties of the system can be understood by analyzing the individual behaviors of these particles.

How are energy levels determined in a system of independent particles?

Energy levels in a system of independent particles are determined by the quantum mechanical properties of the system. These levels are typically derived from solving the Schrödinger equation for the particles under given potential conditions. The solutions yield quantized energy states that the particles can occupy.

What is the significance of the Fermi-Dirac and Bose-Einstein distributions in such systems?

The Fermi-Dirac and Bose-Einstein distributions describe the statistical behavior of particles in a system of independent particles. The Fermi-Dirac distribution applies to fermions, which obey the Pauli exclusion principle and cannot occupy the same quantum state. The Bose-Einstein distribution applies to bosons, which can occupy the same quantum state. These distributions are crucial for predicting the occupancy of energy levels at different temperatures.

How does temperature affect the energy distribution of particles in the system?

Temperature significantly affects the energy distribution of particles in a system of independent particles. At absolute zero, particles occupy the lowest available energy states. As the temperature increases, particles gain thermal energy and can occupy higher energy levels. The exact distribution of particles across energy levels at a given temperature is described by the Fermi-Dirac or Bose-Einstein statistics, depending on the type of particles involved.

What is the role of degeneracy in the energy levels of independent particles?

Degeneracy refers to the number of distinct quantum states that share the same energy level. In a system of independent particles, degeneracy plays a crucial role in determining the statistical weight of each energy level. Higher degeneracy means more particles can occupy that energy level, influencing the overall distribution of particles and the thermodynamic properties of the system.

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