Why is the chemical potential negligible at equilibrium?

In summary: Sorry. Ultra relativistic fluid are out of my area of expertise. But, whenever something is negligible, it has to be negligible compared to something else. This is particularly relevant to chemical potential, since chemical potential is not an absolute quantity, but is reckoned relative to some reference state of the material. Maybe they are talking about the effects of changes in chemical potential (or variations in chemical potential) being negligible.
  • #1
center o bass
560
2
I am currently reading cosmology and I read the statement "... chemical
potential which is nearly always negligible when processes are in equilibrium."

So the current understanding I got of chemical potential is through the law:
$$dE = - pdV + TdS + \mu dN.$$

Hence for processes which occur at constant volume and without heat transfer
$$\frac{\partial E}{\partial N} = \mu$$
indicating that $$\mu$$ is the work done on the system in transferring one particle to it.

Now, why is this energy negligible at equilibrium?
 
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  • #2
center o bass said:
I am currently reading cosmology and I read the statement "... chemical
potential which is nearly always negligible when processes are in equilibrium."

So the current understanding I got of chemical potential is through the law:
$$dE = - pdV + TdS + \mu dN.$$

Hence for processes which occur at constant volume and without heat transfer
$$\frac{\partial E}{\partial N} = \mu$$
indicating that $$\mu$$ is the work done on the system in transferring one particle to it.

Now, why is this energy negligible at equilibrium?
The statement is not quite correct. What they are trying to say depends on the context. But, if they are talking about equilibrium between phases, they meant to say that the chemical potential of a species is the same in all physical phases if the system is equilibrium. So the difference in chemical potential of a species between phase A and phase B is zero.

Chet
 
  • #3
Chestermiller said:
The statement is not quite correct. What they are trying to say depends on the context. But, if they are talking about equilibrium between phases, they meant to say that the chemical potential of a species is the same in all physical phases if the system is equilibrium. So the difference in chemical potential of a species between phase A and phase B is zero.

Chet
Thanks. Would you happen to know when -- and the reasons why -- one might be able to neglect the chemical potential? For example: I have read that for ultra relativistic fluids one can neglect it (but I have not found any reasons for this).
 
  • #4
center o bass said:
Thanks. Would you happen to know when -- and the reasons why -- one might be able to neglect the chemical potential? For example: I have read that for ultra relativistic fluids one can neglect it (but I have not found any reasons for this).
Sorry. Ultra relativistic fluid are out of my area of expertise. But, whenever something is negligible, it has to be negligible compared to something else. This is particularly relevant to chemical potential, since chemical potential is not an absolute quantity, but is reckoned relative to some reference state of the material. Maybe they are talking about the effects of changes in chemical potential (or variations in chemical potential) being negligible.

Chet
 
  • #5
In general that's not enirely true. One relativistic fluid we investigate today is the matter created in ultrarelativistic heavy-ion collisions. There two atomic nuclei are smashed together at high energies (e.g., two gold nuclei at the Relativistic Heavy Ion Collider at the Brookhaven National Lab or two lead nuclei at the Large Hadron Collider at CERN, Geneva), and a lot of particles and antiparticles are created. At the highest energies a socalled Quark-Gluon Plasma is created at huge temperatures of about 500 MeV (usually we set the Boltzmann constant to 1 and measure temperatures in units of energy), which rapidly expands and cools down. What's measured are the hadrons which are formed from the quarks and gluons.

This fireball can to a quite high precision be described as the collective fluid-like motion, using relativistic (ideal or to better precision viscous) hydrodynamics. In relativistic equilibrium thermodynamics the chemical potential has a slightly different meaning than in the non-relavivistic case. You must use it as a Lagrange parameter to keep the average of some conserved quantity rather than particle number. In this case this is the net-baryon number (being +1 for baryons and -1 for anti-baryons). The higher the collision energy the more particles are newly created in the collision, and in each process you must produce as many baryons as anti-baryons. The small amount of baryons already present in the colliding nuclei then becomes negligible compared to these many baryon-anti-baryon pairs, and thus the baryo-chemical potential is close to 0 and the higher the initial temperature becomes.

To the contrary, at not so high collision energies, the baryo-chemical potential is larger to keep the surplus of baryons compared to anti-baryons correct in the thermodynamic description. This is a regime being investigated at RHIC in the beam-energy-scan program right now and is the motivation to build a new accelerator here in Germany at the Helmholtz Heavy-Ion Research Center (GSI) in Darmstadt.
 
  • #6
Interesting, Hendrik. I remember that in protons there are also much more sea quark-antiquark pairs than the three constituent quarks. So is there a mu=0 approximation possible also for elementary particles?
 

Related to Why is the chemical potential negligible at equilibrium?

1. Why is the chemical potential considered negligible at equilibrium?

The chemical potential is considered negligible at equilibrium because it is the state of a system where there is no net change in the composition or properties of the system over time. This means that the system has reached a state of balance, with no driving force for further change. As a result, the chemical potential of each component in the system becomes equal, and any slight differences in chemical potential between components become insignificant.

2. How is the chemical potential defined?

The chemical potential is a thermodynamic property that measures the change in energy of a system when the number of particles of a particular component is altered while keeping the temperature, volume, and number of particles of other components constant. It is represented by the symbol μ and is expressed in units of energy per particle (e.g. J/mol).

3. What factors affect the chemical potential?

The chemical potential is affected by several factors, including temperature, pressure, concentration, and the nature of the chemical species involved. It is also affected by external factors such as electric or magnetic fields. Additionally, the chemical potential can change depending on the phase of the system (e.g. gas, liquid, solid) and the interactions between particles.

4. Can the chemical potential ever be completely zero?

In theory, the chemical potential can be zero, but it is highly unlikely in a real-world system. This would require all the particles in the system to have the same energy, which is highly improbable due to thermal fluctuations. In most cases, the chemical potential is very close to zero at equilibrium, but never exactly zero.

5. How does the concept of chemical potential relate to chemical reactions?

The concept of chemical potential is crucial in understanding and predicting the direction of chemical reactions. In a chemical reaction, the reactants and products have different chemical potentials, and the reaction will proceed in the direction that minimizes the overall chemical potential of the system. This is why reactions tend to move towards equilibrium, where the chemical potentials of the reactants and products are equal.

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