# Radius with Bohr model [closed]

$$r_n = \left(\frac{h^2}{4\pi^2me^2}\right)\times\frac{n^2}{Z}$$

Would I be able to calculate the radius of sodium ion ($$\ce{Na+}$$) with the help of the above Bohr atomic model formula?

• This formula is for single electron systems, and $\ce{Na+}$ has 10 electrons. You could do $\ce{Na^10+}$, though. – Karsten Theis Feb 21 '19 at 17:55
• The atom is the chemical equivalent of the three body gravity problem in physics. There is no exact solution only numerical ones when you have more than one electron. – MaxW Feb 21 '19 at 18:15
• To follow-up on @MaxW's point, I believe the system is also chaotic, so even having a nice, closed form solution is kind of useless because very small changes to initial conditions can lead to wildly different behavior of the system. – Zhe Feb 21 '19 at 20:12

The hydrogen atom wavefunctions can be useful for multielectron atoms as a means of looking up their size (by means of a parameterization). The hydrogen atom wavefunctions form one possible basis for the definition of effective nuclear charges, see for instance here, where Hartree–Fock orbitals are used to compute effective charges:

$$Z_{eff} = \frac{_H}{_Z}$$

Here $$_H$$ and $$_Z$$ are the mean hydrogenic and Hartree-Fock radii (for nuclear charge Z).

The mean hydrogenic radius for ground state hydrogen is related to the Bohr radius $$a_0$$ as $$_H = \frac{3}{2}\frac{a_o}{Z}$$

Expressions for $$_H$$ are available for other orbitals, allowing $$_Z$$ to be computed from the tabulated values of $$Z_{eff}$$ as

$$_Z = \frac{_H}{Z_{eff}}$$

So using the hydrogen wavefunctions in this way is a little more complicated than using the equation for the Bohr model, but just the s-orbitals can already be useful to give you an idea about the size of atoms.