Chemical Potential: Is It Equal to the Energy of Added Particles?

[Niles]

Homework Statement

Hi all.

The chemical potential is defined as being the amount of energy one has to remove from a system in order to keep its enotrpy and volume constant when one particle is added to the system.

Is this amount of energy the same energy as the energy of the added particle?

Thanks in advance.


Niles.

 

[Mapes]

It would be easy to just tell you the answer, but better in the long run if you thought about a few different systems, decided one way or the other, and made your own argument. What do you think?

 

[Niles]

Well, first I think my question need to be re-stated: Isn't it better to ask if this amount of energy is equal to the energy that the particle needs in order for it to enter the system?

Personally, I would say yes. Because when looking at a gas of fermions at T=0, then the chemical potential is the Fermi-energy. Thus in order for the outside particle to occupy the highest un-occupied state (which has energy e_F), it needs to acquire an energy e_F. This is also the amount of energy which the system needs to get rid of in order for the entropy not to increase - but since T=0, the entropy is zero.

So yes, I believe that the energy that a system needs to get rid of in order for the entropy and volume to stay constant when a particle is added is equal to the amount of energy that the particle has to acquire in order to get into the system.

Am I correct ?

 

[Mapes]

I don't think it's as good as the original definition, because the amount of energy a particle has is meaningless without a reference. The original definition refers only to a difference in two energy levels, and is equivalent to

$$dU=T\,dS-p\,dV+\sum \mu_i\,dN_i$$

The chemical potential controls the way that mass diffuses, just as electrical potential controls the way that charge diffuses. Matter tends to go where the chemical potential is lowest. And the chemical potential can be negative, which makes it difficult to visualize what your definition means (a required negative energy?).

 

[Niles]

Ahh, yes. I see your point regarding the negative required energy.. not good.

So the answer is no? I should just stick to the formal definition: "The energy that a system needs to get rid of in order for the entropy and volume to stay constant when a particle is added"?

 

[Mapes]

I've found it helpful (for any relevant scientific concept, not just chemical potential) to try to assemble and keep in mind:

1) A good definition, like the one you gave

2) Applicable equations (e.g., the one above, and knowing that

$$\mu_i=\left(\frac{\partial U}{\partial N_i}\right)_{S,V}=\left(\frac{\partial G}{\partial N_i}\right)$$

which means that chemical potential of anything is its partial molar Gibbs free energy)

3) What it means when a number is positive, negative, or zero (e.g., photons, phonons, and any other nonconserved particles have a chemical potential of zero)

4) How to explain it to the layman (e.g., matter tends to go where the chemical potential is low)

If you can do this for a particular concept, you've gone a long way towards understanding it.

 

[Niles]

Unfortunately, I am not sure I can account for #5: Explaining it to one self

I'm a little confused here.

1) When a particle enters the system (let's say it occupies the state with energy E_i), does one have to give the particle an energy equal to E_i or equal to E_i-E_j, where "j" is the state below "i"?

2) In my notes it says that the change in the chemical potential is E_i, where a particle previously in state "i" with energy E_i is taken out of the system. Furthermore the change in chemical potential is E_j, when a particle is added to state "j" with energy E_j.

 

[Mapes]

I'm not sure I follow you here. If we add a particle to a system with energy $E_i$, then do work on the system to restore its original volume and add/remove heat to restore its original entropy, leaving the system with energy $E_j$, then the chemical potential of the particle was $E_j-E_i$. But I'm not sure if that's what you mean.

 

[Niles]

We have the system with an energy $E_j$, and we wish to add one particle to this system: In particular, we want to add the particle to the state with energy $E_i$.

1) Does the particle have to be given an energy $E_i$ to be added to this state in the system?

2) When the particle is added to the state $E_i$, does the systems energy also increase with this amount? I.e. the systems energy is now $E_i$?

3) What amount of work must be done on the system to keep the entropy unchanged?

By the way, now you talk of the chemical potential of the particle?

 

[Mapes]

Strictly, I should have phrased it as the increase in chemical potential when adding the particle to the system. The value depends both on that particular particle and that particular system.

1) This seems right, but see (2).

2) It doesn't seem like the system's energy would necessarily go to $E_i$. I'm thinking about the quantum well ("particle in a box") idealization. You might have two electrons at $E_1$ and one at $E_2$, for example. If you add an electron with energy $E_{10}$, it would relax inside the system rather than bringing the system to energy $E_{10}$.

3) Work doesn't involve the transfer of entropy, so the purpose of doing work is to restore the volume only. Heat transfer would be used to restore the system to its original entropy.

I haven't thought about the chemical potential in exactly this way before, so I could be wrong. Hope this is helpful nonetheless.

 

[Niles]

Thanks for taking the time to help. It is helpful!