Rutherford’s nuclear model of the atom matched the experimental evidence about the structure of atoms, but it had one serious shortcoming. According to Maxwell’s theory of electricity and magnetism, the orbiting electrons in a Rutherford atom should act as small antennas and radiate electromagnetic waves. That sounds encouraging, because we know that atoms can emit light, but the radiation of electromagnetic waves means the atoms would continuously lose energy. As FIGURE 37.24 shows, this would cause the electrons to spiral into the nucleus! Calculations showed that a Rutherford atom can last no more than about a microsecond. This clearly does not happen.

FIGURE 37.24 The fate of a Rutherford atom.

The experimental efforts of the late 19th and early 20th centuries had been impressive, and there could be no doubt about the existence of electrons, about the small positive nucleus, and about the unique discrete spectrum emitted by each atom. But the theoretical framework for understanding such observations had lagged behind. As the new century dawned, physicists could not explain the structure of atoms, could not explain the stability of matter, could not explain discrete spectra or blackbody radiation, and could not explain the origin of x rays or radioactivity.

Classical physics had reached its limit, and a whole new generation of brilliant young physicists, with new ideas, was about to take the stage. Among the first was an unassuming young man in Bern, Switzerland. His scholastic record had been mediocre, and the best job he could find upon graduation was as a clerk in the patent office, examining patent applications. His name was Albert Einstein.

Chapter 37 Challenge Example Radioactive decay

The cesium isotope ${}^{137}\text{Cs,}$ with $Z=55$, is radioactive and decays by beta decay. A beta particle is observed in the laboratory with a kinetic energy of 300 keV. With what kinetic energy was the beta particle ejected from the 12.4-fm-diameter nucleus?

MODEL A beta particle is an electron that is ejected from the nucleus of an atom during a radioactive decay. Because the negative electron is attracted to the positive nucleus, it must be ejected at a speed greater than the escape speed in order to reach the detector rather than fall back to the nucleus. Energy is conserved in the decay.

VISUALIZE FIGURE 37.25 shows a before-and-after pictorial representation. The electron starts by being ejected from the nucleus with kinetic energy $K_{\text{i}}$. It has electric potential energy $U_{\text{i}}$ due to its interaction with the nucleus. The potential energy due to the atom’s orbiting electrons is negligible because they are so far away in comparison to a nuclear radius. The detected electron is very far from the nucleus, so $U_{\text{f}}=0$.

FIGURE 37.25 A before-and-after pictorial representation of the beta decay.

FIGURE 37.25 Full Alternative Text

SOLVE The conservation of energy statement is $K_{\text{f}}+U_{\text{f}}=K_{\text{i}}+U_{\text{i}}$. The electron starts outside the nucleus, even though at the surface, so the spherical nucleus can be treated as a point charge with $q_1=55e$. The electron has $q_2=-e$, so the initial electron-nucleus potential energy is

$$\begin{array}{l}{U}_{\text{i}}\begin{array}{cc}=& {\displaystyle \frac{K{q}_1{q}_2}{r_{\text{i}}}}\end{array}\\ \begin{array}{ccc}& =& {\displaystyle \frac{(9.0\times {10}^9{\text{ N m}}^2{\text{/C}}^2)\left(55\times 1.60\times {10}^{-19}\text{ C}\right)\left(-1.60\times {10}^{-19}\text{ C}\right)}{6.20\times {10}^{-15}\text{ m}}}\end{array}\\ \begin{array}{ccc}& =& (-2.04\times {10}^{-12}\text{ J})\times {\displaystyle \frac{1\text{ eV}}{1.60\times {10}^{-19}\text{ J}}}=-12.8\text{ MeV}\end{array}\end{array}$$

To be detected in the laboratory with $K_{\text{f}}=300\text{keV}=0.3\text{MeV}$, the electron had to be ejected from the nucleus with

$$\begin{array}{ccc}\hfill K_{\text{i}}& \hfill =\hfill & K_{\text{f}}+U_{\text{f}}-U_{\text{i}}=0.3\text{MeV}+0\text{MeV}+12.8\text{MeV}\hfill \\ \hfill & \hfill =\hfill & 13.1\text{MeV}\hfill \end{array}$$

REVIEW A negative electron is very strongly attracted to the nucleus. It’s not surprising that it has to be ejected from the nucleus with an enormous amount of kinetic energy to be able to escape at all.