01-26-2008, 06:31 PM
You were on track with the wavelength of the electron getting smaller with increasing atomic number.
If you focus on just the innermost spherical s-shell of the electron cloud around an atomic nucleus, as you move up the periodic table, the shell is kept closer in toward the center. As you increase the atomic number and as you add electrons into a partially full s-shell, the new electrons occupy locations at the same distance as their peers in the s-shell, but the entire s-shell is now a smaller radius. It is a smaller radius because it is in a deeper potential energy well.
In the Bohr model, the energy required to remove an electron in a particular shell from an atom goes as the square of the atomic number, Z, over the square of the index, n, of the shell.
E_n = Z^2 *R_e / n^2
This is the depth of the energy well.
For the innermost s-shell, the index n=1. So for n=1,
E is proportional to Z^2.
An atom with a more protons will have its innermost s-electrons in a lower energy state (a deeper well) than an atom with less protons.
This is relevant because the probability wave function gets "tighter" with a deeper energy well. The DeBroglie wavelength for the electron is
lambda = h/p (where h = constant and p = momentum)
A higher (magnitude) energy electron has a higher momentum and thus a smaller wavelength. A smaller wavelength means the average radius for the electron around the atom is smaller. The shell shrinks as you add protons to the nucleus.
Now examine how these shells are populated. The innermost s-shell is mostly empty. Add an electron (and a nuclear proton). That electron is at the same energy state as its peers, but the s-shell itself decreases in radius a little. Add another electron and the same thing happens. The atom decreases in size as you go right along a row of the periodic table until the shell fills up.
There are other effects such as electron screening where the electrostatic repulsion from an s-state will counter the pull of the nucleus on a p-state. But that is beyond the scope of the initial question.
Edit:
Just in case anybody was wondering, I wasn't a "Moron Major" in college.
If you focus on just the innermost spherical s-shell of the electron cloud around an atomic nucleus, as you move up the periodic table, the shell is kept closer in toward the center. As you increase the atomic number and as you add electrons into a partially full s-shell, the new electrons occupy locations at the same distance as their peers in the s-shell, but the entire s-shell is now a smaller radius. It is a smaller radius because it is in a deeper potential energy well.
In the Bohr model, the energy required to remove an electron in a particular shell from an atom goes as the square of the atomic number, Z, over the square of the index, n, of the shell.
E_n = Z^2 *R_e / n^2
This is the depth of the energy well.
For the innermost s-shell, the index n=1. So for n=1,
E is proportional to Z^2.
An atom with a more protons will have its innermost s-electrons in a lower energy state (a deeper well) than an atom with less protons.
This is relevant because the probability wave function gets "tighter" with a deeper energy well. The DeBroglie wavelength for the electron is
lambda = h/p (where h = constant and p = momentum)
A higher (magnitude) energy electron has a higher momentum and thus a smaller wavelength. A smaller wavelength means the average radius for the electron around the atom is smaller. The shell shrinks as you add protons to the nucleus.
Now examine how these shells are populated. The innermost s-shell is mostly empty. Add an electron (and a nuclear proton). That electron is at the same energy state as its peers, but the s-shell itself decreases in radius a little. Add another electron and the same thing happens. The atom decreases in size as you go right along a row of the periodic table until the shell fills up.
There are other effects such as electron screening where the electrostatic repulsion from an s-state will counter the pull of the nucleus on a p-state. But that is beyond the scope of the initial question.
Edit:
Just in case anybody was wondering, I wasn't a "Moron Major" in college.