Answer to Question #169412 in Atomic and Nuclear Physics for AFOLABI SODIQ

Question #169412

Calculate the radius and energy of the electron in the nth orbital in the hydrogen atom


1
Expert's answer
2021-03-08T08:21:57-0500

The radius and energy of the electron in the nth orbital in the hydrogen atom can be calculated in a semiclassical approximation using Bohr's postulates.

Start from the classical case. The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force:


mev2r=Zkee2r2{m_\mathrm{e} v^2\over r} = {Zk_\mathrm{e} e^2 \over r^2}

where mem_e is the mass of the electron, vv is its speed, rr is the radius of the orbit, ZZ is the atom's atomic number (Z=1Z = 1 for hydrogen), kek_e is the Coulomb constant, and ee is the charge of the electron.

Thus, the speed is:


v=kee2merv = \sqrt{ k_\mathrm{e} e^2 \over m_\mathrm{e} r}

and the energy is:


E=12mev2E= -{1\over 2} m_\mathrm{e} v^2

Now, let's use the Bohr's assumption - angular momentum is an integer multiple of \hbar :


mevr=nm_evr = n\hbar

Substituting the expression for the velocity into the last expression gives an equation for r in terms of n:


mekee2merr=nrn=n22kee2mem_{\text{e}}\sqrt{\dfrac{k_{\text{e}}e^2}{m_{\text{e}}r}}r=n\hbar\\ r_n = {n^2\hbar^2\over k_\mathrm{e} e^2 m_\mathrm{e}}

The same procedure for the energy gives:


En=kee22rn=ke2e4me22n2E_n = -{k_\mathrm{e} e^2 \over 2r_n } = - { k_\mathrm{e}^2 e^4 m_\mathrm{e} \over 2\hbar^2 n^2}

Answer. En=ke2e4me22n2,rn=n22kee2meE_n= - { k_\mathrm{e}^2 e^4 m_\mathrm{e} \over 2\hbar^2 n^2}, r_n = {n^2\hbar^2\over k_\mathrm{e} e^2 m_\mathrm{e}}.


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