Ideal gasses, bonds and partition functions

Hi
I am struggling immensely with understand some aspects of chemical thermodynamics:
1) Lets say I have a solid with N atoms and am examining the ionization of individual atoms, and I am supposed to think of the electrons as ideal gasses.
Or,
2) a solid or liquid is in thermal equilibrium with its gas form (again the gas is an ideal gas). So, I am examining the vapor pressure.
In such exercises, I am sometimes expected to use the grand canonical ensemble and other times the canonical ensemble. However, I do not get:

Why do I use the grand canonical ensemble in 1 but not in 2? How can I tell if the chemical potential is significant enough to use the grand canonical ensemble?
Why is the chemical potential for electrons bonded to the solid the same as the ideal gas?
What is the energy of electrons bonded to atoms versus the ones that are not bonded?

What chapter in what book should I read in order to understand this?

TeethWhitener
Gold Member
I'll try to answer the easy part first.
Why is the chemical potential for electrons bonded to the solid the same as the ideal gas?
In most cases, the ionization of an atom requires much more energy than is thermally available at room temperature. So in the case of situation #2 in your original post, you can essentially ignore the electron contribution to the energetics of the system, and simply assume that all atoms are neutral and in their ground electronic energy states. Since the electronic states of the gaseous atoms are the same as the electronic states of the atoms in the solid, their chemical potential is equal.
What is the energy of electrons bonded to atoms versus the ones that are not bonded?
Depends on the atom. The system $A^+ + e^-$ is higher in energy than the neutral atom $A$ because the electron falls into the potential well of the positively charged $A^+$, moving to a lower energy state by binding to the positive center. The amount of energy needed to bounce an electron back out of this potential well:
$$A\longrightarrow A^+ + e^-$$
is known as the ionization energy, and varies from atom to atom.
Why do I use the grand canonical ensemble in 1 but not in 2? How can I tell if the chemical potential is significant enough to use the grand canonical ensemble?
People tend to struggle with these concepts quite a bit, so you're not alone. The basic idea behind these different ensembles is this: for a box full of 1023 particles (like something you'd find in a chemistry lab), it's waaaaay too complicated to calculate exactly what's going on. So what we do instead is try to figure out what's likely to occur statistically. In order to do that, we have to examine a whole bunch of copies of that original box of 1023 particles to map out a probability distribution of what will happen with the system.
Now, in order to make sure that each of those copies behave more or less the same, we have to make sure they're in equilibrium. It turns out that there's a few different ways to do this. For your purposes, you can make sure that all the boxes are at the same temperature, same volume, and have the same number of particles (NVT). This arrangement gives you a canonical ensemble.
Another way to make sure the boxes are in equilibrium is to let particles transfer from box to box, as long as the chemical potential remains the same (μVT). This is known as the grand canonical ensemble.
The canonical ensemble allows energy to be transferred from box to box, but requires that temperature and particle number remain constant. This corresponds to cases like #2 above because, even though you have a solid and a gas in the same box, that box always stays at the same temperature and never loses or gains any particles. Contrast this with the case of an atom that's ionized. If you consider that the atom is your entire box, then ionizing it causes it to lose a particle (an electron). This means that you can no longer use the canonical ensemble. Instead you have to use the grand canonical ensemble, which allows particles to be lost and gained.

Avatrin