Help I have to derive an expression for the allowed energy levels using Bohr's model

Question

I have a question that asks, following Bohr's model derive an expression for the allowed energy levels of an electron bound to a nucleus of charge Ze. At each stage, state explicity any laws, postulates or assunptions you employ. Express your answer in terms of E1. I just dont know where to start with it can anybody help!!!!

Answer 1

The Bohr Theory is as follows: -

1. The electron revolves in a circular orbit with the centripetal force supplied by the coulomb interaction between the electron and the nucleus, which may be expressed as: -

Ze²..  = mv² = mω²r

___ .... __

4πε0r².. r

where r = radius of allowed orbit

v = electron velocity

ω = corresponding angular velocity

2. The angular momentum A of the electron takes on only values which are integer multiples of  Ã„§, such that: -

A = mvr = mωr² = nħ

3. When an electron makes a transition from one allowed stationary orbit state to another, the Einstein frequency condition hν =E(i) - E(f) is satisfied (i -> initial and f -> final). Thus, the allowed radii are: -

r(n) = 4πε0ħ²n²

........________

....... me²Z

The energy of the electron is partially kinetic and partially potential. If we call the energy zero when the electron is at rest at infinity, its potential energy in the presence of the nucleus is: -

P = - 1 . Ze²

..... __  . ____

.....4πε0 . r  

It kinetic energy is: -

K = ½mv² = ½mr²ω²  

Since, from Bohr's postulates (above): -

v = nħ/mr  

The velocity of the electron in the n'th orbit is: -

v(n) = e²Z

....... ____

....... 2ε0hn

Hence, the kinetic energy is: -

K = .. 1..  Ze²

....... __ ..  ___

..... 4πε0 2r

The total energy is E = K + P, and so for the n'th state (after summing the two equations and reducing the expression): -

E(n) = - me⁴Z²

........ ________

........ 8ε0²h²nZ²

This calculation needs to be corrected for reduced mass of an orbital system. Thus, for a nucleus of mass ‘M’ and electron of mass ‘m’, the reduced mass is given by: -

m(r)  = m/(1 + m/M)

Hence: -

E(n) = - m(r)e⁴Z²

........ ________

........ 8ε0²h²nZ²

With approximate values, inserted, the equation becomes : -

E(n)  = - 13.58Z²/n²  eV

I hope this is of some help!

Answer 2

No you've lost me. Ask me something about cars.

Answer 3

See

http://en.wikipedia.org/wiki/Bohr_model#...