Problems & Exercises

Using the given charge-to-mass ratios for electrons and protons, and knowing the magnitudes of their charges are equal, what is the ratio of the proton’s mass to the electron’s? (Note that since the charge-to-mass ratios are given to only three-digit accuracy, your answer may differ from the accepted ratio in the fourth digit.)

If someone wanted to build a scale model of the atom with a nucleus 1.00 m in diameter, how far away would the nearest electron need to be?

Rutherford found the size of the nucleus to be about ${\text{10}}^{-\text{15}}\text{m}$. This implied a huge density. What would this density be for gold?

(a) An aspiring physicist wants to build a scale model of a hydrogen atom for her science fair project. If the atom is 1.00 m in diameter, how big should she try to make the nucleus?

(b) How easy will this be to do?

By calculating its wavelength, show that the first line in the Lyman series is UV radiation.

Look up the values of the quantities in $a_{\text{B}}=\frac{h^2}{{4\pi }^2{m}_e{\text{kq}}_e^2}$, and verify that the Bohr radius $a_{\text{B}}$ is $\text{0.529}\times {\text{10}}^{-\text{10}}\text{m}$.

If a hydrogen atom has its electron in the $n=4$ state, how much energy in eV is needed to ionize it?

Find the radius of a hydrogen atom in the $n=2$ state according to Bohr’s theory.

What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?

Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.

A wavelength of $4\text{.}\text{653 μm}$ is observed in a hydrogen spectrum for a transition that ends in the $n_{\text{f}}=5$ level. What was $n_{\text{i}}$ for the initial level of the electron?

A beryllium ion with a single electron (denoted ${\text{Be}}^{3+}$) is in an excited state with radius the same as that of the ground state of hydrogen.

(a) What is $n$ for the ${\text{Be}}^{3+}$ ion?

(b) How much energy in eV is needed to ionize the ion from this excited state?

Verify Equations $r_n=\frac{n^2}{Z}{a}_{\text{B}}$ and $a_{\mathrm{B}}=\frac{h^2}{{4\pi }^2{m}_e{\text{kq}}_e^2}=\text{0.529}\times {\text{10}}^{-\text{10}}\text{m}$ using the approach stated in the text. That is, equate the Coulomb and centripetal forces and then insert an expression for velocity from the condition for angular momentum quantization.

(a) What is the shortest-wavelength x-ray radiation that can be generated in an x-ray tube with an applied voltage of 50.0 kV? (b) Calculate the photon energy in eV. (c) Explain the relationship of the photon energy to the applied voltage.

An x ray tube has an applied voltage of 100 kV. (a) What is the most energetic x-ray photon it can produce? Express your answer in electron volts and joules. (b) Find the wavelength of such an X–ray.

What are the approximate energies of the $K_{\alpha }$ and $K_{\beta }$ x rays for copper?

Figure 30.37 shows the energy-level diagram for neon. (a) Verify that the energy of the photon emitted when neon goes from its metastable state to the one immediately below is equal to 1.96 eV. (b) Show that the wavelength of this radiation is 633 nm. (c) What wavelength is emitted when the neon makes a direct transition to its ground state?

Ruby lasers have chromium atoms doped in an aluminum oxide crystal. The energy level diagram for chromium in a ruby is shown in Figure 30.59. What wavelength is emitted by a ruby laser?

Some of the most powerful lasers are based on the energy levels of neodymium in solids, such as glass, as shown in Figure 30.60. (a) What average wavelength light can pump the neodymium into the levels above its metastable state? (b) Verify that the 1.17 eV transition produces $1\text{.}\text{06 μm}$ radiation.

If an atom has an electron in the $n=5$ state with $m_l=3$, what are the possible values of $l$?

What are the possible values of $m_l$ for an electron in the $n=4$ state?

(a) Calculate the magnitude of the angular momentum for an $l=1$ electron. (b) Compare your answer to the value Bohr proposed for the $n=1$ state.

Repeat Exercise 30.40 for $l=3$.

What angles can the spin $s$ of an electron make with the $z$-axis?

(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) $5s^1$ (b) $1d^1$ (c) $4s^3$ (d) $3p^7$ (e) $5g^{15}$. 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 $2n^2$.

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

Integrated Concepts

The electric and magnetic forces on an electron in the CRT in Figure 30.7 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).

Integrated Concepts

A galaxy moving away from the earth has a speed of $\text{0.0100}c$. What wavelength do we observe for an $n_{\text{i}}=7$ to $n_{\text{f}}=2$ transition for hydrogen in that galaxy?

Integrated Concepts

In a Millikan oil-drop experiment using a setup like that in Figure 30.9, 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?

Integrated Concepts

In a laboratory experiment designed to duplicate Thomson’s determination of $q_e/m_e$, a beam of electrons having a velocity of $6\text{.}\text{00}\times {\text{10}}^7\text{m/s}$ enters a $\text{5.00}\times {\text{10}}^{-3}\text{T}$ 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.

Integrated Concepts

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 $a_{\text{B}}=\frac{h^2}{{\mathrm{4\pi }}^2{m}_e{\text{kq}}_e^2}=\text{0.529}\times {\text{10}}^{-\text{10}}\text{m}$ is 207 $m_e$.

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

(b) Compare this with the 7.5-fm radius of a uranium nucleus. Note that since the muon orbits inside the electron, it falls into a hydrogen-like orbit. Since your answer is less than the radius of the nucleus, you can see that the photons emitted as the muon falls into its lowest orbit can give information about the nucleus.

Integrated Concepts

A carbon dioxide laser used in surgery emits infrared radiation with a wavelength of $\text{10.6 μm}$. In 1.00 ms, this laser raised the temperature of ${\text{1.00 cm}}^3$ of flesh to $\text{100ºC}$ 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?

Integrated Concepts

(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 kilogram.

(c) Convert this to eV and compare to the photon energy. Discuss the implications.

Integrated Concepts

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 $\pm \text{50}\mathrm{.0\; km/s}$. What range of wavelengths will we observe for the 91.20-nm line in the Lyman series of hydrogen? (Such line broadening is observed and actually provides part of the evidence for rapid rotation.)

Unreasonable Results

(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?

Critical Thinking (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?