PX156 - B2 - bohr's model

electrons orbit around the nucleus

quantized angular momentum

orbiting electrons are standing waves, the orbits must be whole number multiples of the wavelength:

\begin{aligned} 2 π r & = n λ \\ = \frac{n h}{p} \\ = \frac{n h}{m_{e} v} \\ m_{e} v r = L & = \frac{n h}{2 π} = n ℏ \end{aligned}

bohr radius

v_{n} = \frac{n h}{2 π m_{e} r_{n}}

acceleration is caused due to electrostatic interaction, so from coulomb's law :

\begin{aligned} F_{E} & = F_{c} \\ \frac{e^{2}}{4 π ϵ_{0} r_{n}^{2}} & = \frac{m_{e} v_{n}^{2}}{r_{n}} \\ = \frac{n^{2} h^{2}}{4 π^{2} m_{e} r_{n}^{3}} \\ ∴ r_{n} & = \frac{ϵ_{0} n^{2} h^{2}}{π m_{e} e^{2}} \end{aligned}

bohr's radius $(a_{0}) :$

\begin{aligned} a_{0} & = \frac{ϵ_{0} h^{2}}{π m_{e} e^{2}} \\ r_{n} & = a_{0} n^{2} \\ v_{n} & = \frac{e^{2}}{2 n ϵ_{0} h} \end{aligned}

\begin{aligned} E_{n} & = K E + P E \\ = \frac{1}{2} m_{e} v_{n}^{2} - \frac{e^{2}}{4 π ϵ_{0} r_{n}} \\ = \frac{m_{e}}{2} \frac{e^{4}}{4 n^{2} ϵ_{0} h^{2}} - \frac{e^{2}}{4 π ϵ_{0}} \frac{π m_{e} e^{2}}{ϵ_{0} n^{2} h^{2}} \\ = - \frac{m_{e} e^{4}}{8 ϵ_{0}^{2} n^{2} h^{2}} \end{aligned}

comparing it with the rydberg formula:

E = \frac{h c}{λ} = h c R (\frac{1}{m^{2}} - \frac{1}{n^{2}})

R = 1.097 \times 10^{7} m^{- 1} = \frac{m_{e} e^{4}}{8 ϵ_{0}^{2} h^{3} c}

rydberg units of energy:
- $R h c \approx 13.6 e V$
- $E_{n} \approx - \frac{13.6}{n^{2}} e V$