Most Probable Radius of a 2s Hydrogenic Atom?

[pcheminsanity]

Homework Statement

Determine the most probable radius of an electron for a 2s hydrogenic atom.

Homework Equations

r*=a0/Z where a0 is Bohr radius

The Attempt at a Solution

Well, I know that the answer is ~5.2a0/Z. The book tells me that the most probable radius for hydrogen (EDIT - a 2s hydrogen shell I mean) is 5.2a0 - thus I know that the answer is simply 5.2a0/Z (275pm/Z) for the series of hydrogenic atoms. My only problem is this: I don't know how to get the coefficient of 5.2, and presumably simply getting it out of the book is not the right way to do it. I think it may have to do with taking the derivative of the wavefunction but I'm unsure...Any help would be appreciated, thanks!

 

[G01]

The probability for an electron to be found in the radial range r+dr can be determined using the wavefunction for the 2S state:

$$P(r+dr)=|\Psi_{200}|^24\pi r^2dr$$

Now if you want to find the most probable value, your looking for the value of r, such that P(r+dr) is a maximum. Think back to calculus. What is the condition for a function, P(r) to be maximum?

HINT: The condition involves $$\frac{dP}{dr}$$