Why Use Nuclear Charge In Finding Energy Value of Singular Electron?

[member 659869]

My Textbook says this is the formula to find energy values for electron shells:

$$E_{mol of electrons} = \frac{-1312kJ}{n^2}$$

where $n$ is in electron shell number

But when we divide by 1 mol to get the energy value for each electron we get

$$E_{electron} = \frac{-2.178 \cdot 10^{-18}}{n^2} J$$

but the actual equation is (as given by textbook) rather

$$E_{electron} = \frac{-2.178 \cdot 10^{-18} \cdot Z^2}{n^2} J$$

Where $Z$ is nuclear charge. Why must we introduce $Z$ and why is it not used in the first equation?

 

[DrClaude]

Usually, if an equation for the energy an electron doesn't take into account the nuclear charge, it is because it is an equation for a hydrogen ($Z=1$) or for a multi-electron atom where only the outermost electron is taken into account, such that one can take the nucleus plus the other electrons as a single entity of net charge +1 (which is always an approximation).

 

[member 659869]

Thanks! Second question: if i wanted to find the net change in an electron's energy for an arbitrary atom when going from say, $n=4$ to $n=2$, would I use Rydberg's Equation? I have asked this question on another site where people have said no closed form exists, and you would have to use numerical methods, but that was for finding the electron energy values of $n=4$ and $n=2$, not for finding the net difference.

 

[DrClaude]

It doesn't matter whether you want to difference or the absolute value. There is no simple formula for multi-electron atoms. In addition, the energy does not depend only on $n$, but also on $l$.

There are some approximate formulas that can be used, especially for atoms with a single valence electron (alkali atoms), using quantum defect theory.