You learned in Chapter 38 that, except for hydrogen, the...

You learned in Chapter 38 that, except for hydrogen, the mass of a nucleus with atomic number $Z$ is larger than the mass of the Z protons. The additional mass was ultimately discovered to be due to neutrons, but prior to the discovery of the neutron it was suggested that a nucleus with mass number $A$ might contain $A$ protons and $(A-Z)$ electrons. Such a nucleus would have the mass of $A$ protons, but its net charge would be only Ze. a. We know that the diameter of a nucleus is approximately 10 fm. Model the nucleus as a one-dimensional box and find the minimum range of speeds that an electron would have in such a box. b. What does your answer imply about the possibility that the nucleus contains electrons? Explain.

You learned in Chapter 38 that, except for hydrogen, the mass of a nucleus with atomic number $Z$ is larger than the mass of the Z protons. The additional mass was ultimately discovered to be due to neutrons, but prior to the discovery of the neutron it was suggested that a nucleus with mass number $A$ might contain $A$ protons and $(A-Z)$ electrons. Such a nucleus would have the mass of $A$ protons, but its net charge would be only Ze.
a. We know that the diameter of a nucleus is approximately
10 fm. Model the nucleus as a one-dimensional box and find the minimum range of speeds that an electron would have in such a box.
b. What does your answer imply about the possibility that the nucleus contains electrons? Explain.

Key Concepts

Particle in a Box Model

This concept is a foundational quantum mechanical model in which a particle is confined to a finite region (a 'box') with impenetrable boundaries. The model leads to quantized energy levels, and the confinement of the electron to a very small region (like the nucleus) forces the electron to adopt very high momenta and correspondingly high kinetic energies. This helps illustrate why confining an electron to a nucleus-sized region results in non-physical speeds when using non-relativistic estimates.

Heisenberg Uncertainty Principle

This principle states that there is an inherent limit to how precisely we can know both the position and momentum of a particle simultaneously. For an electron confined to a very small space (on the order of 10 femtometers), the uncertainty in its momentum must be very large, which implies it would have a very high minimum speed. This reasoning is critical in understanding why electrons cannot exist in the nucleus, as the allowed speeds would be extremely high, often even approaching or exceeding the speed of light in non-relativistic calculations.

Relativistic Effects

When calculations for phenomena at very small scales predict speeds that exceed or approach the speed of light, it indicates that relativistic mechanics must be taken into account. In the case of an electron confined within a nucleus, the enormous momentum implied by quantum confinement would require a relativistic treatment, revealing that such electrons would possess prohibitively high kinetic energies, further reinforcing that electrons are not constituents of the nucleus.

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