Where does the maximum electron density occur for 2s and 2p orbitals in hydrogen atom?

1 Answer
Dec 28, 2015

For hydrogen, we have to use spherical harmonics, so our dimensions are written as (r, theta, phi)(r,θ,ϕ). The wave function is defined as follows, via separation of variables:

color(green)(psi_(nlm_l)(r,theta,phi) = R_(nl)(r) Y_(l)^(m_l)(theta, phi))ψnlml(r,θ,ϕ)=Rnl(r)Ymll(θ,ϕ)

R_(nl)(r)Rnl(r) is the radial component of the wave function psi_(nlm_l)(r,theta,phi)ψnlml(r,θ,ϕ), Y_(l)^(m_l)(theta,phi)Ymll(θ,ϕ) is the angular component, nn is the principal quantum number, ll is the angular momentum quantum number, and m_lml is the projection of the angular momentum quantum number (i.e. 0, pm l0,±l). The wave function represents an orbital.

If you don't understand all of that, that's fine; it was just for context.

To get the maximum electron density, you have to look at probability density curves.

If we plot 4pir^2R_(nl)(r)^24πr2Rnl(r)2 against rr, we get the probability density curves for an atomic orbital.

The 2s2s orbital's plot looks like this:

http://chemwiki.ucdavis.edu/http://chemwiki.ucdavis.edu/

From this, you can tell that the maximum electron density occurs near 5a_05a0 (with a_0 ~~ 5.29177xx10^(-11) "m"a05.29177×1011m, the Bohr radius) from the center of the atom, and 4pir^2 R_(20)(r)^24πr2R20(r)2 is about 2.452.45 or so.

From this similar diagram, we can compare the 2s2s with the 2p2p orbital:

http://faculty.uml.edu/http://faculty.uml.edu/

Here, you should see that the 2p2p orbital has a maximum electron density near about 4a_04a0 from the center of the atom, and the value of 4pir^2 R_(21)(r)^24πr2R21(r)2 is perhaps around 2.52.5.

This should make more sense once you realize what the probability density plots of the 2s2s and 2p2p orbitals look like:

2s
http://cronodon.com/http://cronodon.com/

2p
http://cronodon.com/http://cronodon.com/

"The density of the [dark spots] is proportional to the probability of finding the electron in that region" (McQuarrie, Ch. 6-6).

Basically, start with a radius of 0, and expand your radius of vision outwards from the center of the orbital, and you should be constructing the probability density curves (radial distribution plots).