Grand Canonical ensemble

In summary, the problem involves a surface with N sites that can have 0, 1, or 2 atoms without any energy cost. It is in contact with a particle reservoir at chemical potential \mu and temperature T. The probability for a site to be empty, have 1 atom, or have 2 atoms is equal at 1/3 for each state. The average number of atoms in the gas can be calculated using the partition function, which can be written as (1+e^{\frac{\mu }{kT}}+ e^{\frac{2\mu }{kT}})^N.
  • #1

j93

189
2

Homework Statement


Grand Canonical ensemble problem- A surface of N sites that can have 0,1,2 atoms. It costs no energy to adsorb an atom. Grand canonical problem therefore in contact with particle reservoir. Assume \mu chem potentiol and temp T.
What is probability for site to be empty,1,or 2 atoms.
average number of atoms in gas.

Homework Equations


The Attempt at a Solution


I believe the answer is that they have equal probability because there is no energy cost to adsorb. ie 1/3 for all 3.

for second question I believe the partition function is [tex]Z = \Sigma e^{mu/kT}e^{E/kT}[/tex] where E = 0 for all 3 different states ie [tex]Z= 3^Ne^{mu/KT}[/tex]. I don't think this is right though
 
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  • #2
I think Z the partition function is
1[tex] (1 + e^{\frac{\mu N}{kT}}+ e^{\frac{2\mu N}{kT}}) [/tex]
or
[tex](1+e^{\frac{\mu }{kT}}+ e^{\frac{2\mu }{kT}})^N[/tex]
 
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1. What is the Grand Canonical ensemble?

The Grand Canonical ensemble is a statistical mechanical ensemble used to describe systems in thermodynamic equilibrium where both the number of particles and the energy can fluctuate. It is often used to describe systems with a fixed temperature, volume, and chemical potential.

2. How is the Grand Canonical ensemble different from other ensembles?

The Grand Canonical ensemble differs from other ensembles, such as the Canonical and Microcanonical ensembles, in that it allows for fluctuations in both the number of particles and the energy. This makes it more suitable for systems that exchange particles with a reservoir, such as a gas in contact with a heat bath.

3. What is the significance of the chemical potential in the Grand Canonical ensemble?

The chemical potential in the Grand Canonical ensemble represents the energy required to add or remove a particle from the system. It is a measure of how strongly the system is coupled to its reservoir and plays a crucial role in determining the equilibrium behavior of the system.

4. How is the Grand Canonical ensemble related to the other thermodynamic potentials?

The Grand Canonical ensemble is related to the other thermodynamic potentials through the Legendre transformation. The Helmholtz free energy, Gibbs free energy, and grand potential are all related to the partition function of the Grand Canonical ensemble in different ways and can be used to calculate thermodynamic properties of the system.

5. What are the applications of the Grand Canonical ensemble?

The Grand Canonical ensemble is commonly used in the study of systems with a variable number of particles, such as gases, liquids, and plasmas. It is also used in the study of phase transitions and critical phenomena. Additionally, it has applications in fields such as astrophysics, where systems may exchange both particles and energy with their surroundings.

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