What is Physics ?
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Classical physics failed to explain the stability of atoms and the discrete nature of atomic spectra.
Rutherford’s nuclear model predicted that orbiting electrons should continuously radiate energy and spiral into the nucleus. To resolve these contradictions, Niels Bohr (1913) proposed a quantum model for the hydrogen atom.
Electrons revolve around the nucleus only in certain allowed circular orbits without radiating energy.
The angular momentum of the electron is quantized:
\[mvr = n\hbar, \quad n = 1,2,3,\dots\]Radiation is emitted or absorbed only during transitions between allowed orbits:
\[h\nu = E_i - E_f\]For an electron in a circular orbit, the electrostatic force provides the centripetal force:
\[\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2} = \frac{mv^2}{r}\]Using force balance and angular momentum quantization, the radius of the $n^{\text{th}}$ orbit is:
\[\boxed{ r_n = \frac{4\pi\varepsilon_0 \hbar^2}{m e^2} \, n^2 }\]The smallest allowed radius (Bohr radius):
\[\boxed{ a_0 = 0.529 \, \text{Å} }\]Hence,
\[\boxed{ r_n = n^2 a_0 }\]The velocity of the electron in the $n^{\text{th}}$ orbit is:
\[\boxed{ v_n = \frac{e^2}{2\varepsilon_0 h} \frac{1}{n} }\]For the ground state ($n=1$):
\[v_1 \approx 2.18 \times 10^6 \, \text{m s}^{-1}\]\(K = \frac{1}{2}mv^2\)
\(U = -\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r}\)
\(E = K + U = -\frac{1}{8\pi\varepsilon_0}\frac{e^2}{r}\)
Substituting $r_n$:
\[\boxed{ E_n = -\frac{m e^4}{8 \varepsilon_0^2 h^2} \frac{1}{n^2} }\]Numerically,
\[\boxed{ E_n = -\frac{13.6}{n^2} \, \text{eV} }\]For a transition from $n_i$ to $n_f$:
\[h\nu = E_{n_i} - E_{n_f}\]Thus,
\[\boxed{ \nu = \frac{13.6\,\text{eV}}{h} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) }\]where
\[R = 1.097 \times 10^7 \, \text{m}^{-1}\]Ionization corresponds to $n=1 \rightarrow n=\infty$:
\[\boxed{ E_{\text{ion}} = 13.6 \, \text{eV} }\]