# What atomic number of an element "X" would have to become so that the 4th orbit around X would fit inside the 1st Bohr orbit of H-atom?

##### 1 Answer

Well, the radial distance from the nucleus of a hydrogen-like atom is given by

#r = n^2 (a_0/Z)# .

Interestingly enough, one often sees the substitution

#sigma = (Zr)/(a_0)#

in hydrogen atom wave functions. I derive this result below. Upon solving this you should get

We are asking, what would the atomic number

For a given Bohr orbit, the **wavelength** is given by

#2pir = nlambda_e# #" "bb((1))# where

#r# is the radial distance from the nucleus,#n# is the principal quantum number, and#lambda_e# is the electron wavelength.

Since electrons have rest mass **de Broglie relation**:

#lambda_e = h/(m_ev)# #" "bb((2))# where

#v# is its velocity and#h = 6.626 xx 10^(-34) "J"cdot"s"# is Planck's constant.

We can thus rewrite

#2pir = n(h/(m_ev))#

#=> vr = (nℏ)/m_e# ,#" "bb((3))# where

#ℏ = h//2pi# is the reduced Planck's constant.

Next, the **coulomb force** for a hydrogen-like atom is given by the derivative of the potential

#-(delV)/(delr) = -del/(delr)[-(Ze^2)/(4pi epsilon_0 r)]#

#= -(Ze^2)/(4pi epsilon_0 r^2)# #" "bb((4))# ,where

#epsilon_0 = 8.854 xx 10^(-12) "F/m"# is the vacuum permittivity and#e = 1.602 xx 10^(-19) "C"# is the elementary charge.

From uniform circular motion in physics, the **centripetal force** (in the Bohr picture) brought about by the coulomb force (with inwards, or attraction, being negative) is given by

#F_c = -(m_ev^2)/r# .#" "bb((5))#

Equating

#(m_ev^2)/r = (Ze^2)/(4pi epsilon_0r^2)#

#=> (Ze^2)/(4pi epsilon_0m_e) = v^2r# #" "bb((6))#

Comparing

#v^2r^2 = ((nℏ)/m_e)^2 = (Ze^2)/(4pi epsilon_0m_e) cdot r#

Now if we solve this for

#r = ((nℏ)/m_e)^2 cdot (4pi epsilon_0m_e)/(Ze^2)#

#= n^2 ((4pi epsilon_0 ℏ^2)/(m_e e^2)) cdot 1/Z #

By comparison with Wikipedia, we use the definition of the Bohr radius,

#color(blue)(barul(|stackrel(" ")(" "r = n^2 (a_0/Z)" ")|))#

So for a ** hydrogen-like** atom, if we want

#Z = (n^2 cdot a_0)/(r)#

#= n^2 cdot cancel(a_0/a_0)^(1)#

For the fourth Bohr orbit, we have **sulfur cation**,