Calculating Effective Nuclear Charge using Ionization Energy

[doublavay]

Find the effective nuclear charge of the outermost electron in sodium using its ionization energy.

Ionization energy of sodium = 5.14 eV = 8.23e-19 J
Zeff = Z - S

I tried using the equation Zeff = Z - S, but when I followed all the rules, I found

S = (8*0.85)+(2*1.00) = 8.8
Zeff = 11 - 8.8 = 2.2

but the answer in my book says that it should be 1.8, and that method didn't use the ionization energy of sodium. I need to find an equation that explains quantitatively how ionization energy relates to Zeff.

 

[Redbelly98]

Welcome to Physics Forums! And sorry for the delay in responding.

Up until the last 10 minutes, I was not familiar with this type of problem. After a google search, I am now familiar with two ways to solve this problem.

One way is the way you did it, but they don't seem to want that. The other way is shown here, look at the equation for Z_eff:

https://www.physicsforums.com/showthread.php?t=266326

That way does make use of the ionization potential. You'll have to use the appropriate value of n as well, of course.

 

[doublavay]

I'm still having trouble with this problem, because E = R_H (Z_eff / n)² doesn't seem to have units that cancel out. Energy is measured in Joules or eV, but R_H is the only thing with units on the other side of the equation, and it has units of m^-1. Or maybe I'm using the wrong value of R_H... I thought it was just the Rydberg constant multiplied by 1(because it's Hydrogen). Is that incorrect?

 

[Redbelly98]

It's good to know that some people still pay attention to units

R_H, with the units of m^-1, is 1/λ for ionization of the hydrogen atom. Instead, use the Rydberg constant in eV, which may be found in the Atomic constants section here:

http://www.pnc.aps.anl.gov/info/Physical_constants_1986.html

(This is simply the ionization energy of Hydrogen.)