Energy Formula: $E_n = -\frac{13.6Z^2}{n^2} \text{ eV}$

• Always negative; represents bound state
• Z = atomic number (1 for H, 2 for $He^{+}$, 3 for $Li^{2+}$)
• n = principal quantum number
• Ground state (n=1) is most stable (most negative)

Key Energy Values:

Note: $He^{+}$ (n=2) = H (n=1) = −13.6 eV; $Li^{2+}$ (n=3) = H (n=1) = −13.6 eV.

Radius Formula: $r_n = \frac{0.529n^2}{Z} \text{ Å}$

• H ground state: 0.529 Å (Bohr radius a_{0})
• r scales as $n^{2}$ (quadruples from n=1 to n=2)
• r decreases with higher Z ($He^{+}$ is more compact than H)

Velocity Formula: $v_n = \frac{2.18 \times 10^6 Z}{n} \text{ m/s}$

• Decreases with n; increases with Z
• H ground state: v_{1} ≈ 0.73% of c

Energy of Transition (Emission): $\Delta E = 13.6Z^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \text{ eV}, \quad n_1 < n_2$

Virial Theorem Relations:

• KE = −E_n = +|E_n|
• PE = 2E_n (negative, twice total)
• Total = KE + PE = −KE

Ionization Energy from nth orbit: IE = |E_n| = 13.6$Z^{2}$/$n^{2}$ eV

Time Period Ratio: $T_n \propto \frac{n^3}{Z^2} \Rightarrow T_1:T_2:T_3 = 1:8:27$