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A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.?
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A system has two energy levels with energies ε and 2ε. The lower level...
Introduction:
In this problem, we are given a system with two energy levels, ε and 2ε. The lower energy level is 4-fold degenerate, meaning there are four possible states with energy ε. The upper energy level is doubly degenerate, meaning there are two possible states with energy 2ε. We need to calculate the partition function and the fraction of particles in the upper energy level.
Partition function:
The partition function, denoted by Z, is a fundamental quantity in statistical mechanics that describes the distribution of particles in different energy states. It is given by the sum of the Boltzmann factors for each energy state:
Z = e^(-βε) + e^(-βε) + e^(-βε) + e^(-βε) + e^(-2βε) + e^(-2βε)
where β = 1/(kT) is the inverse temperature and k is the Boltzmann constant.
Simplifying the expression, we get:
Z = 4e^(-βε) + 2e^(-2βε)
Fraction of particles in the upper energy level:
To calculate the fraction of particles in the upper energy level, we need to find the probability of a particle being in the upper energy level. The probability P_i of a particle being in a particular energy state i is given by the Boltzmann factor for that energy state divided by the partition function:
P_i = e^(-βE_i) / Z
where E_i is the energy of state i.
For the upper energy level with energy 2ε, the probability P_upper is:
P_upper = 2e^(-2βε) / Z
To find the fraction of particles in the upper energy level, we need to sum the probabilities for each particle and divide by the total number of particles N:
Fraction_upper = N * P_upper
Summary:
In this problem, we calculated the partition function for a system with two energy levels and determined the fraction of particles in the upper energy level. The partition function was found by summing the Boltzmann factors for each energy state, and the fraction of particles in the upper energy level was calculated by dividing the probability of a particle being in the upper energy level by the partition function and multiplying by the total number of particles.
In this problem, we are given a system with two energy levels, ε and 2ε. The lower energy level is 4-fold degenerate, meaning there are four possible states with energy ε. The upper energy level is doubly degenerate, meaning there are two possible states with energy 2ε. We need to calculate the partition function and the fraction of particles in the upper energy level.
Partition function:
The partition function, denoted by Z, is a fundamental quantity in statistical mechanics that describes the distribution of particles in different energy states. It is given by the sum of the Boltzmann factors for each energy state:
Z = e^(-βε) + e^(-βε) + e^(-βε) + e^(-βε) + e^(-2βε) + e^(-2βε)
where β = 1/(kT) is the inverse temperature and k is the Boltzmann constant.
Simplifying the expression, we get:
Z = 4e^(-βε) + 2e^(-2βε)
Fraction of particles in the upper energy level:
To calculate the fraction of particles in the upper energy level, we need to find the probability of a particle being in the upper energy level. The probability P_i of a particle being in a particular energy state i is given by the Boltzmann factor for that energy state divided by the partition function:
P_i = e^(-βE_i) / Z
where E_i is the energy of state i.
For the upper energy level with energy 2ε, the probability P_upper is:
P_upper = 2e^(-2βε) / Z
To find the fraction of particles in the upper energy level, we need to sum the probabilities for each particle and divide by the total number of particles N:
Fraction_upper = N * P_upper
Summary:
In this problem, we calculated the partition function for a system with two energy levels and determined the fraction of particles in the upper energy level. The partition function was found by summing the Boltzmann factors for each energy state, and the fraction of particles in the upper energy level was calculated by dividing the probability of a particle being in the upper energy level by the partition function and multiplying by the total number of particles.
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A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.?
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A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.?.
A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.?.
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A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.?, a detailed solution for A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.? has been provided alongside types of A system has two energy levels with energies ε and 2ε. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non- interacting classical particles in the system in equilibrium at temperature T, calculate the partition function and the fraction of particles in the upper energy level.? theory, EduRev gives you an
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