Q.State the postulates of Bohr's atomic model and derive the expression for the energy of an electron in the atom.

[!TLDR] Bohr's 3 postulates (circular orbits under Coulomb attraction; quantized angular momentum $nh/2\pi$ defining non-radiating stable orbits; photon emission on transitions) combine with the centripetal-force condition to give $E_n=-\frac{m_eZ^2e^4}{8\epsilon_0^2h^2n^2}=-\frac{13.6Z^2}{n^2}$ eV. [!ANSWER] $E_n=-\dfrac{13.6\,Z^2}{n^2}$ eV.

Bohr's three postulates are: (1) electrons revolve in circular orbits, with the centripetal force provided by electrostatic attraction to the nucleus; (2) only orbits with angular momentum $m_ev_nr_n=\frac{nh}{2\pi}$ are allowed, and electrons in these stable orbits do not radiate; (3) a transition between orbits emits or absorbs a photon whose energy equals the orbit-energy difference. To derive the energy formula, first combine postulates (1) and (2): the centripetal-force condition $\frac{m_ev_n^2}{r_n}=\frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r_n^2}$ and the angular-momentum condition $m_ev_nr_n=\frac{nh}{2\pi}$ are two equations in $v_n$ and $r_n$; solving them together gives $r_n=\frac{n^2h^2\epsilon_0}{\pi m_eZe^2}$ and $v_n=\frac{Ze^2}{2\epsilon_0hn}$. The electron's total energy is the sum of kinetic and (negative) potential energy, $E_n=\frac{1}{2}m_ev_n^2-\frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r_n}$; substituting the expressions for $v_n$ and $r_n$ and simplifying gives $E_n=-\frac{m_eZ^2e^4}{8\epsilon_0^2h^2n^2}$. Substituting the numerical values of $m_e$, e, $\epsilon_0$ and h converts this into the practically useful form $E_n=-\frac{13.6Z^2}{n^2}$ eV, valid for hydrogen (Z=1) and any hydrogen-like (single-electron) ion. [!ANSWER] $E_n=-\dfrac{13.6\,Z^2}{n^2}$ eV.

State the three postulates, combine the angular-momentum and centripetal-force conditions to eliminate $v_n$ and $r_n$ in favour of n, and substitute back into the total-energy expression $\frac{1}{2}m_ev_n^2-\frac{Ze^2}{4\pi\epsilon_0r_n}$.

Forgetting the negative sign on the potential energy term (Coulomb attraction, not repulsion), which would otherwise flip the sign of the whole final energy expression and destroy its physical meaning as a bound-state energy.