Bohr's Atom Model

Bohr's postulates

Bohr's model of the Hydrogen atom is based on 3 postulates (https://socratic.org/questions/what-are-the-postulates-of-bohr-s-atomic-model)

Electrons revolve around the nucleus in stationary orbits, without radiating carrying, energy $E_{n}$ where $n = 1, 2, . . .$
In a transition from a higher to a lower level $E_{n} \to E_{n - 1}$ the electron emits a photon of frequency $ν$ $$\nu =\frac{E_n - E_{n-1} }{h } \tag{1}$$
The only possible orbits are such that the angular momentum of the electron is a multiple of $h / 2 π$ $$L = \frac {nh}{2\pi} = n\hbar ;; n= 1,2,3, . . . \tag{2}$$
In a stable orbit the centripetal force and the Coulomb force must balance $$\frac{m_ev^2}{r}= \frac{1}{4\pi\epsilon_0} \frac{z e^2}{r^2} \tag{3}$$
where $e$ is the elementary charge, $z$ is the atomic number $ϵ_{0}$ is the permittivity of free space .

Characteristics of the Bohr's atom

The third postulate gives $L = m_{e} v_{n} r_{n} = n ℏ$ and therefore $$v_n = \frac{n\hbar}{m_er_n} \tag{4}$$
From (3) using Coulomb's constant $k_{c} = \frac{1}{4 π ϵ_{0}}$ and for the Hydrogen Atom $$ \frac{\cancel{m_e}}{r} \frac{n^2\hbar^2}{m_e^\cancel{2}\cancel{r^2}} = k_C \frac{ e^2}{\cancel{r^2}} \tag{5} $$

\begin{matrix} (6) & r_{n} = \frac{n ² ℏ^{2}}{k_{c} m_{e} z e^{2}} \end{matrix}

The energy $E_{n}$ has a kinetic and a electrostatic component Electromagnetism#Electric Potential Energy

\begin{matrix} (7) & E_{n} = 1 / 2 m_{e} v_{n}^{2} - k_{C} e^{2} / r_{n} = \frac{2 π^{2} m_{e} e^{4}}{n^{2} h ²} \end{matrix}

Replacing (6) in (4) $$v_n = \frac{\cancel{n\hbar} k_c \cancel{m_e} ze^2}{\cancel{m_e} n^\cancel2 \hbar^\cancel2} $$

\begin{matrix} (8) & v_{n} = \frac{k_{C} z e^{2}}{n ℏ} \end{matrix}

Replacing (8) for $v_{n}$ and (6) for $r_{n}$ in (7)

\begin{matrix} (9) & E_{n} = \frac{m_{e} k_{c}^{2} z^{2} e^{4}}{2 n^{2} ℏ^{2}} - \frac{(k_{C} z e^{2}) (k_{c} m_{e} z e^{2})}{n^{2} ℏ^{2}} \end{matrix}

E_n = \frac{m_e k_c^2 z^2e^4}{2 n^2 \hbar^2} -\frac{m_e k_c^2 z^2e^4}{ n^2 \hbar^2} \tag{10}$$$$\boxed{E_n = -\frac{m_e k_c^2 z^2e^4}{2 n^2 \hbar^2}} \tag{10}

(“Schaum Quantum Mechanics,” n.d., chap. 1 for a brief introduction)
Quantum Physics exercices#Calculate the radius of the Hydrogen atom according to Bohr's model