(Grand) Canonical Ensemble?

In summary, when considering systems where the number of particles is not constant, such as in the adsorption of He atoms onto a solid surface, the grand partition function should be used. This is because the number of particles on the surface is not constant, and the grand canonical ensemble allows for fluctuations in the number of particles. The normal partition function is only appropriate for systems where the number of particles remains constant.
  • #1
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I am taking a statictical mechanics course, and one thing bothers me. I am not sure when we should use the normal partition function (Z) and when the grand partition function (twisty Z). In particular, why can we not use grand partition function when we are considering the following system:

He atoms may be adsorbed from vapor phase onto a solid surface, forming a 2-dimensional gas. If the adsorption energy is e, then by treating the vapor as a reservoir, find the density of He atom on the surface.

Shouldn't we always use grand partition function whenever the number of particle N is not constant? Or is there no limitation on which one we Have to use, just which is more convenient?
 
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  • #2
For the adsorption problem, you should use the grand canonical ensemble (which uses the grand partition function), since the number of particles on the surface is clearly not constant.
 
  • #3


The (Grand) Canonical Ensemble is a statistical mechanical model used to describe the behavior of a system in equilibrium with a reservoir, where the number of particles can fluctuate. In this ensemble, the system is allowed to exchange both energy and particles with the reservoir, while the volume and temperature are kept constant.

The main difference between the normal partition function (Z) and the grand partition function (Ξ) is that Z only takes into account the energy of the system, while Ξ also considers the number of particles. So, in cases where the number of particles is not constant, it is more appropriate to use the grand partition function.

In the example you mentioned, where He atoms are adsorbed from a vapor onto a solid surface, the number of particles on the surface can fluctuate as they are adsorbed or desorbed. Therefore, it would be more accurate to use the grand partition function in this case.

However, there may be cases where using the normal partition function is more convenient, even if the number of particles is not constant. It ultimately depends on the specific system and the information that is needed to be calculated.

In summary, the grand partition function should be used when the number of particles is not constant, but there may be situations where using the normal partition function is also acceptable. It is important to carefully consider the system and the information needed before deciding which one to use.
 

1. What is the (Grand) Canonical Ensemble?

The (Grand) Canonical Ensemble is a statistical mechanics ensemble used to describe the behavior of a system in equilibrium with a reservoir of particles and energy. It is commonly used in the study of systems with varying particle numbers and chemical potentials.

2. How does the (Grand) Canonical Ensemble differ from other ensembles?

The (Grand) Canonical Ensemble differs from other ensembles, such as the Microcanonical and Canonical ensembles, in that it allows for fluctuations in both the number of particles and the energy of the system. This makes it more suitable for studying systems in equilibrium with a larger reservoir.

3. What are the main assumptions of the (Grand) Canonical Ensemble?

The main assumptions of the (Grand) Canonical Ensemble include: 1) the system is in equilibrium with a reservoir, 2) the reservoir has a fixed temperature and chemical potential, 3) the system can exchange particles and energy with the reservoir, and 4) the system and reservoir are weakly coupled so that their energies are not significantly affected by the exchange.

4. What is the role of the chemical potential in the (Grand) Canonical Ensemble?

The chemical potential, denoted by μ, plays a crucial role in the (Grand) Canonical Ensemble as it determines the average number of particles in the system. It is a measure of the energy required to add or remove a particle from the reservoir, and it balances the exchange of particles between the system and reservoir to maintain equilibrium.

5. How is the (Grand) Canonical Ensemble used in practical applications?

The (Grand) Canonical Ensemble is used in many practical applications, such as in the study of gases, fluids, and phase transitions. It allows for a more realistic description of systems with varying particle numbers and chemical potentials, making it a valuable tool in understanding the behavior of these complex systems at equilibrium.

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