30.9 The Pauli Exclusion Principle

(a) How many electrons can be in the $n = 4$ shell?

(b) What are its subshells, and how many electrons can be in each?

(a) If one subshell of an atom has 9 electrons in it, what is the minimum value of $l$? (b) What is the spectroscopic notation for this atom, if this subshell is part of the $n = 3$ shell?

Which of the following spectroscopic notations are not allowed? (a) $5 s^{1}$ (b) $1 d^{1}$ (c) $4 s^{3}$ (d) $3 p^{7}$ (e) $5 g^{15}$. State which rule is violated for each that is not allowed.

(a) Using the Pauli exclusion principle and the rules relating the allowed values of the quantum numbers $\left(n , l , m_{l} , m_{s}\right)$, prove that the maximum number of electrons in a subshell is $2 n^{2}$.

(b) In a similar manner, prove that the maximum number of electrons in a shell is 2n².

The electric and magnetic forces on an electron in the CRT in the reference are supposed to be in opposite directions. Verify this by determining the direction of each force for the situation shown. Explain how you obtain the directions (that is, identify the rules used).

A galaxy moving away from the earth has a speed of $\text{0}.\text{0100} c$. What wavelength do we observe for an $n_{\text{i}} = 7$ to $n_{\text{f}} = 2$ transition for hydrogen in that galaxy?

In a Millikan oil-drop experiment using a setup like that in the reference, a 500-V potential difference is applied to plates separated by 2.50 cm. (a) What is the mass of an oil drop having two extra electrons that is suspended motionless by the field between the plates? (b) What is the diameter of the drop, assuming it is a sphere with the density of olive oil?

In a laboratory experiment designed to duplicate Thomson’s determination of $q_{e} / m_{e}$, a beam of electrons having a velocity of $6 . \text{00} \times \text{10}^{7} \text{m}/\text{s}$ enters a $\text{5}.\text{00} \times \text{10}^{- 3} \text{T}$ magnetic field. The beam moves perpendicular to the field in a path having a 6.80-cm radius of curvature. Determine $q_{e} / m_{e}$ from these observations, and compare the result with the known value.

Particles called muons exist in cosmic rays and can be created in particle accelerators. Muons are very similar to electrons, having the same charge and spin, but they have a mass 207 times greater. When muons are captured by an atom, they orbit just like an electron but with a smaller radius, since the mass in $a_{\text{B}} = \frac{h^{2}}{4π^{2} m_{e} \text{kq}_{e}^{2}} = \text{0}.\text{529} \times \text{10}^{- \text{10}} \text{ m}$ is 207 $m_{e}$.

(a) Calculate the radius of the $n = 1$ orbit for a muon in a uranium ion ($Z = \text{92}$).

(b…

A carbon dioxide laser used in surgery emits infrared radiation with a wavelength of $\text{10}.\text{6 }\mu\text{m}$. In 1.00 ms, this laser raised the temperature of $\text{1}.\text{00 cm}^{3}$ of flesh to $\text{100}º\text{C}$ and evaporated it.

(a) How many photons were required? You may assume flesh has the same heat of vaporization as water. (b) What was the minimum power output during the flash?

(a) An excimer laser used for vision correction emits 193-nm UV. Calculate the photon energy in eV.

(b) These photons are used to evaporate corneal tissue, which is very similar to water in its properties. Calculate the amount of energy needed per molecule of water to make the phase change from liquid to gas. That is, divide the heat of vaporization in kJ/kg by the number of water molecules in a…

A pulsar is a rapidly spinning remnant of a supernova. It rotates on its axis, sweeping hydrogen along with it so that hydrogen on one side moves toward us as fast as 50.0 km/s, while that on the other side moves away as fast as 50.0 km/s. This means that the EM radiation we receive will be Doppler shifted over a range of $\pm \text{50} .0 km/s$. What range of wavelengths will we observe for the 91.20-nm line in the L…

(a) What voltage must be applied to an X-ray tube to obtain 0.0100-fm-wavelength X-rays for use in exploring the details of nuclei? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) For carbon, calculate the energy when an electron falls from $n = 3$ to $n = 2$, then from $n = 2$ to $n = 1$. Then add these for the total energy released in the process. (b) Calculate the energy released when the electron falls directly from $n = 3$ to $n = 1$. (c) Are the energies in part a and in part b the same?