# Boltzmann Distribution: Calculate Probability of Particle in 4 States

• shayan825
In summary, the probability that the particle is in the ground state is 0.032, the probability that it is in a particular excited state is 0.7725, and the probability that it is in a state of energy -3.0 eV is 0.1666.
shayan825
A certain particle is interacting with a reservoir at 500 k and can be in any four possible states. The ground state has energy 3.1 eV and three excited states all have the same energy. what is the probability that the particle is in ground state? what is the probability that the particle is in a particular excited state? what is the probability that it is in a state of energy -3.0 eV? using Boltzmann distribution

I use the formula (e^E/kT)/sum of e^E/kT but get a wrong answer.

There is a factor of -1 missing from the exponent. Are you also including the degeneracy in the sum?

tman12321 said:
There is a factor of -1 missing from the exponent. Are you also including the degeneracy in the sum?

yes, for the first question(the probability that the particle is in the ground state) I get 0.032. e^(-3.1/500k) / e^(-3.1/500k) + 3e^(-3.0/500k). The answer should be 0.7725

Why would 3.1 eV be the energy of the ground state, and -3.0 eV be the energy of the excited state? Is it supposed to be the other way around?

tman12321 said:
Why would 3.1 eV be the energy of the ground state, and -3.0 eV be the energy of the excited state? Is it supposed to be the other way around?[/QUOTwell,

that's what the book says..ya

There seem to be a couple of problems here:
(1) The energy of the ground state should not be greater than the energy of an excited state. The ground state should have the lowest energy. I guess you meant that the ground state is -3.1 eV and the excited state is -3.0 eV, i.e. a 0.1 eV difference. (You were missing this minus sign before).
(2) The Boltzmann factor is Exp[-ΔE/kT], not Exp[ΔE/kT], where ΔE > 0.
(3) You seem to be mixing the usage of k and K. K denotes a temperature in Kelvin and is a unit, and k is the Boltzmann constant, a number. You say that the reservoir is at 500 k, when I think you meant 500 K. You then divide the energies in eV by 500k, but it is unclear whether you think this means 500*k or 500 K. (500*k would be right assuming you're using the value of k in eV/K).

It will be easiest, from a computational standpoint, if you redefine -3.1 eV to be the zero of the energy . Then the ground state has energy 0 eV, and the excited states have energy 0.1 eV.

## 1. What is the Boltzmann Distribution?

The Boltzmann Distribution is a concept in statistical mechanics that describes the probability of a particle being in a particular energy state at a given temperature. It is named after Austrian physicist Ludwig Boltzmann.

## 2. How is the probability of a particle in a specific energy state calculated using the Boltzmann Distribution?

The probability of a particle being in a specific energy state is calculated using the Boltzmann Distribution formula, which is P(E) = (1/Z) * e^(-E/kT), where P(E) is the probability, Z is the partition function, E is the energy of the state, k is the Boltzmann constant, and T is the temperature in Kelvin.

## 3. What is the significance of the Boltzmann Distribution in thermodynamics?

The Boltzmann Distribution is significant in thermodynamics because it provides a way to calculate the probability of particles being in different energy states at a given temperature. This information can then be used to determine the overall properties of a system, such as its entropy and free energy.

## 4. Can the Boltzmann Distribution be applied to all types of particles?

Yes, the Boltzmann Distribution can be applied to all types of particles, including atoms, molecules, and ions. It is a fundamental concept in statistical mechanics and is used to describe the behavior of particles in various systems.

## 5. How does temperature affect the probability of particles in different energy states according to the Boltzmann Distribution?

Temperature has a direct effect on the probability of particles being in different energy states according to the Boltzmann Distribution. As temperature increases, the probability of particles being in higher energy states also increases, while the probability of particles being in lower energy states decreases.

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