Physics · Ch 8 — Atomic and Nuclear Physics

Bohr atom model

8.3.3

Bohr atom model

To overcome Rutherford's twin failures - instability and continuous emission - Niels Bohr introduced quantisation into the atomic model. His postulates for the hydrogen atom are:

Circular orbits under Coulomb attraction. The electron moves around the nucleus in a circular orbit, with the Coulomb electrostatic force of attraction supplying exactly the centripetal force needed for circular motion.
Stationary, non-radiating orbits with quantised angular momentum. Electrons can only occupy certain discrete orbits, called stationary orbits, in which - unlike in Rutherford's model - they do not radiate electromagnetic energy despite being in continuous circular motion. In these orbits the electron's angular momentum is quantised: it can only take integer multiples of $\hbar=h/2\pi$ (the reduced Planck constant), $l=n\hbar,\qquad n=1,2,3,\dots\qquad$ where $n$ is called the principal quantum number. This condition also has a wave-mechanical justification: treating the electron as a de Broglie wave of wavelength $\lambda=h/(m\upsilon)$ , a stable standing-wave orbit requires the orbit's circumference to be an exact integer number of wavelengths, $2\pi r=n\lambda$ (8.14), which upon substituting $\lambda=h/(m\upsilon)$ reproduces exactly the same angular-momentum quantisation condition $m\upsilon r=n\hbar$ .
Quantised energy and photon transitions. The orbit energies are discrete, not continuous. An electron jumps between two orbits by absorbing or emitting a single photon whose energy exactly equals the difference between the two orbital energies, $\Delta E=E_{final}-E_{initial}=h\nu=hc/\lambda$ , and the frequency of the emitted or absorbed radiation depends only on this energy difference, not on the electron's orbital motion frequency. Radius and velocity of the $n$ th orbit. Equating the Coulomb force $\tfrac{1}{4\pi\varepsilon_0}\tfrac{Ze^2}{r_n^2}$ to the centripetal requirement $\tfrac{m\upsilon_n^2}{r_n}$ , and combining with the quantisation condition $m\upsilon_n r_n=n\hbar$ , gives $r_n=\frac{4\pi\varepsilon_0 n^2\hbar^2}{Zme^2}=a_0\frac{n^2}{Z},\qquad a_0=\frac{4\pi\varepsilon_0\hbar^2}{me^2}=0.529\ \text{Å}\qquad (8.16)$ where $a_0$ , the radius of the first hydrogen orbit ( $n=1,Z=1$ ), is called the Bohr radius. So $r_1=a_0=0.529$ Å, $r_2=4a_0=2.116$ Å, $r_3=9a_0=4.761$ Å, and in general $r_n\propto n^2$ : successive orbits spread out rapidly. The corresponding orbital speed is $\upsilon_n=\frac{2\pi}{h}\cdot\frac{Ze^2}{4\pi\varepsilon_0}\cdot\frac{1}{n}\propto \frac{1}{n}$ , so the electron moves fastest in the ground state and progressively slower in higher orbits (a rectangular-hyperbola-shaped curve). Energy of the $n$ th orbit. The potential energy is $U_n=-\tfrac{1}{4\pi\varepsilon_0}\tfrac{Ze^2}{r_n}$ and the kinetic energy works out to $KE_n=-\tfrac{1}{2}U_n$ , so the total energy $E_n=KE_n+U_n=-KE_n$ is $E_n=-\frac{me^4Z^2}{8\varepsilon_0^2h^2n^2}=-\frac{13.6Z^2}{n^2}\ \text{eV}\qquad (8.17)$ For hydrogen ( $Z=1$ ): $E_1=-13.6$ eV (ground state), $E_2=-3.4$ eV (first excited state), $E_3=-1.51$ eV (second excited state), and so on - see Table 8.1. The negative sign reflects the convention that potential energy (and hence total energy) is defined to vanish only when the electron is removed to infinite separation from the nucleus. The lowest-energy orbit ( $n=1$ ) is the ground state, with the ground-state energy of hydrogen, $-13.6$ eV, used as a unit of energy called the Rydberg. …

Figure 8.13Spiral motion of an electron around the nucleus (instability of the classical atom)

What this figure shows. This figure illustrates why a purely classical picture of the atom fails: since an accelerated (circulating) charge continuously radiates electromagnetic energy according to classical electrodynamics, an orbiting electron would steadily lose energy, causing its orbital radius to shrink continuously in a spiral path shown winding inward toward the nucleus. The diagram traces this shrinking spiral all the way to the point where the electron would collide with and fall into the nucleus, which - since atoms are observed to be stable and do not collapse - is exactly the paradox that Bohr's postulate of stationary, non-radiating orbits wa …

Figure 8.14The line spectrum of hydrogen

What this figure shows. This figure shows a photograph or rendering of the hydrogen emission spectrum as a series of sharp, discrete bright lines at specific wavelengths against a dark background, in clear contrast to a smooth rainbow-like continuous spectrum. It is placed at the start of the Bohr model discussion precisely because explaining why hydrogen emits light only at these particular discrete wavelengths - rather than at all wavelengths - was the central puzzle that motivated Bo …

Figure 8.15Standing wave pattern for an electron in a stable orbit

What this figure shows. This figure shows a circular orbit of radius r on which a de Broglie wave of wavelength lambda is drawn fitting exactly n complete wavelengths around the circumference, illustrated for n = 5 and also for a non-integer case such as n = 3.3 labelled 'forbidden' to show that only whole-number fits produce a self-reinforcing standing wave. This wave-mechanical picture gives a physical justification for Bohr's angular-momentum quantisation condition: an orbit is stable only when the electron's de Broglie wave closes on itself smoothly after one full revolution, which happens precisely when the circumferenc …

Figure 8.16Absorption and emission of radiation

What this figure shows. This figure shows an electron making a transition between two energy levels E_initial and E_final, drawn with a photon of energy hv arriving and being absorbed as the electron jumps to a higher level, and separately with a photon of energy hv being emitted as the electron falls to a lower level. Both processes are drawn side by side to emphasise that the photon's energy always equals the difference in the atom's energy between the two orbital levels involved, exactly as required by Bohr's third postulate, regardless of whether the tra …

Figure 8.17Electron revolving around the nucleus

What this figure shows. This figure shows the basic geometry used throughout the Bohr-model derivation: a nucleus of charge +Ze is drawn at the centre, assumed to remain stationary because it is far more massive than the electron, while an electron of charge -e and mass m revolves around it in a circular orbit of radius r_n with speed v_n. An arrow labelled F on the electron represents the inward electrostatic (Coulomb) force of attraction from the nucleus, which is exactly the centripetal force needed to keep the electron on its circular path, and this single picture is the starting point from which the radius, …

Figure 8.18Variation of the radius of the orbit with principal quantum number

What this figure shows. This figure draws four concentric circular orbits of increasing radius, labelled r1, r2 = 4r1, r3 = 9r1 and r4 = 16r1, to make visually concrete the result that orbit radius grows as the square of the principal quantum number (r_n proportional to n squared). Seeing the orbits drawn to scale like this makes clear how rapidly the electron's distance from the nucleus increases for higher excited states compared to the tightly bound ground-state …

Figure 8.19Variation of the velocity of the electron in the orbit with principal quantum number

What this figure shows. This figure plots the electron's orbital speed v_n on the vertical axis against the principal quantum number n on the horizontal axis, producing a curve that falls off steeply at first and then flattens - the shape of a rectangular hyperbola - because v_n is proportional to 1/n. The graph makes clear that the electron moves fastest in the tightly bound ground state (n = 1) and progressively slower in each successive excited state as n increases. …

Figure 8.20Energy levels of a hydrogen atom

What this figure shows. This figure is the standard hydrogen energy-level diagram: horizontal lines are drawn at the allowed energies for n = 1 through n = 5 and n = infinity, labelled both in joules and in electron-volts (for example -13.6 eV for the ground state n = 1, rising through -3.40 eV, -1.51 eV, -0.85 eV and -0.54 eV up to 0 eV for the free, ionised electron at n = infinity). The diagram visually distinguishes the tightly spaced 'ground state' line at the bottom from the 'excited states' bunching closer and closer together as n grows, up to the continuous band of 'free electron' energies above zero, giving a complete visual summary …

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