What is an atomic orbital? How does an atomic orbital

What is electron configuration? Describe the roles that the Pauli exclusion principle and Hund's rule play in writing the electron configuration of elements.

Explain the meaning of the symbol \(4 d^{6}\).

Explain the meaning of diamagnetic and paramagnetic. Give an example of an element that is diamagnetic and one that is paramagnetic. What does it mean when we say that electrons are paired?

What is meant by the term "shielding of electrons" in an atom? Using the Li atom as an example, describe the effect of shielding on the energy of electrons in an atom.

Indicate which of the following sets of quantum numbers in an atom are unacceptable and explain why: (a) \(\left(1,\ 0,\ \frac{1}{2},\ \frac{1}{2}\right)\), (b) \(\left(3,\ 0,\ 0,\ +\frac{1}{2}\right)\), (c) \(\left(2,\ 2,\ 1,\ +\frac{1}{2}\right)\), (d) \(\left(4,\ 3,\ -2,\ +\frac{1}{2}\right)\), (e) \((3,\ 2,\ 1,\ 1)\).

The ground-state electron configurations listed here are incorrect. Explain what mistakes have been made in each and write the correct electron configurations. AI: \(1 s^{2} 2 s^{2} 2 p^{4} 3 s^{2} 3 p^{3}\) B: \(1 s^{2} 2 s^{2} 2 p^{5}\) F: \(1 s^{2} 2 s^{2} 2 p^{6}\)

The atomic number of an element is 73. Is this element diamagnetic or paramagnetic?

Indicate the number of unpaired electrons present in each of the following atoms: B, Ne, P, Sc, Mn, Se, Kr, Fe, Cd, I, Pb.

State the Aufbau principle and explain the role it plays in classifying the elements in the periodic table.

Describe the characteristics of the following groups of elements: transition metals, lanthanides, actinides.

What is the noble gas core? How does it simplify the writing of electron configurations?

What are the group and period of the element osmium?

Define the following terms and give an example of each: transition metals, lanthanides, actinides.

Explain why the ground-state electron configurations of Cr and Cu are different from what we might expect.

Problem 85P Explain why the ground-state electron configurations of Cr and Cu are different from what we might expect.

Explain what is meant by a noble gas core. Write the electron configuration of a xenon core.

Problem 87P Comment on the correctness of the following statement: The probability of finding two electrons with the same four quantum numbers in an atom is zero.

Problem 88P Use the Aufbau principle to obtain the ground-state electron configuration of selenium.

Problem 89P Use the Aufbau principle to obtain the ground-state electron configuration of technetium.

Problem 90P Write the ground-state electron configurations for the following elements: B, V, Ni, As, I, Au.

Problem 91P Write the ground-state electron configurations for the following elements: Ge, Fe, Zn, Ni, W, Tl.

Problem 92P The electron configuration of a neutral atom is 1s22s22p63s2. Write a complete set of quantum numbers for each of the electrons. Name the element.

A sample tube consisted of atomic hydrogens in their ground state. A student illuminated the atoms with monochromatic light, that is, light of a single wavelength. If only two spectral emission lines in the visible region are observed, what is the wavelength (or wavelengths) of the incident radiation?

Problem 94P A sample tube consisted of atomic hydrogens in their ground state. A student illuminated the atoms with monochromatic light, that is, light of a single wavelength. If only two spectral emission lines in the visible region are observed, what is the wavelength (or wavelengths) of the incident radiation?

Problem 95P A laser produces a beam of light with a wavelength of 532 nm. If the power output is 25.0 mW, how many photons does the laser emit per second? (1 W = 1 J/s.)

Problem 96P When a compound containing cesium ion is heated in a Bunsen burner flame, photons with an energy of 4.30 × 10?19 J are emitted. What color is the cesium flame?

Problem 97P Discuss the current view of the correctness of the following statements, (a) The electron in the hydrogen atom is in an orbit that never brings it closer than 100 pm to the nucleus, (b) Atomic absorption spectra result from transitions of electrons from lower to higher energy levels, (c) A many-electron atom behaves somewhat like a solar system that has a number of planets.

Identify the following individuals and their contributions to the development of quantum theory: Bohr, de Broglie, Einstein, Planck, Heisenberg, Schrödinger.

What is the maximum number of electrons in an atom that can have the following quantum numbers? Specify the orbitals in which the electrons would be found. (a) n=2, m=+5 ; (b) n=4, m=+1 ; (c) n=3, \(\ell=2\) ; (d) n=2, \(\ell=0, m_s=-\frac{1}{2}\) ; (e) n=4, \(\ell=3, m_{\ell}=-2\).

What properties of electrons are used in the operation of an electron microscope?

In a photoelectric experiment a student uses a light source whose frequency is greater than that needed to eject electrons from a certain metal. However, after continuously shining the light on the same area of the metal for a long period of time the student notices that the maximum kinetic energy of ejected electrons begins to decrease, even though the frequency of the light is held constant. How would you account for this behavior?

A certain pitcher's fastballs have been clocked at about 100 mph. (a) Calculate the wavelength of a 0.141-kg baseball (in nm) at this speed. (b) What is the wavelength of a hydrogen atom at the same speed? (1 mile = 1609 m.)

A student carried out a photoelectric experiment by shining visible light on a clean piece of cesium metal. The table here shows the kinetic energies (KE) of the ejected electrons as a function of wavelengths (\(\lambda\)). Determine graphically the work function and the Planck constant.

(a) What is the lowest possible value of the principal quantum number (n) when the angular momentum quantum number \((\ell)\) is 1? (b) What are the possible values of the angular momentum quantum number \((\ell)\) when the magnetic quantum number \(\left(m_{\ell}\right)\) is 0, given than \(n \leq 4\)?

Considering only the ground-state electron configuration, are there more diamagnetic or paramagnetic elements? Explain.

A ruby laser produces radiation of wavelength 633 nm in pulses whose duration is \(1.00 \times 10^{-9} \mathrm{~s}\). (a) If the laser produces 0.376 J of energy per pulse, how many photons are produced in each pulse? (b) Calculate the power (in watts) delivered by the laser per pulse. (1 W 1 J /s.)

A 368-g sample of water absorbs infrared radiation at \(1.06 \times 10^{4}\ \mathrm{nm}\) from a carbon dioxide laser. Suppose all the absorbed radiation is converted to heat. Calculate the number of photons at this wavelength required to raise the temperature of the water by \(5.00^{\circ} \mathrm{C}\).

Photodissociation of water \(\mathrm{H}_{2} \mathrm{O}(\mathrm{l})+h v \rightarrow \mathrm{H}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(g)\) has been suggested as a source of hydrogen. The \(\Delta H_{r x n}^{\circ}\) for the reaction, calculated from thermochemical data, is 285.8 kJ per mole of water decomposed. Calculate the maximum wavelength (in nm) that would provide the necessary energy. In principle, is it feasible to use sunlight as a source of energy for this process?

Spectral lines of the Lyman and Balmer series do not overlap. Verify this statement by calculating the longest wavelength associated with the Lyman series and the shortest wavelength associated with the Balmer series (in nm).

An atom moving at its root-mean-square speed at \(20^{\circ} \mathrm{C}\) has a wavelength of \(3.28 \times 10^{-11}\ \mathrm{m}\). Identify the atom.

Certain sunglasses have small crystals of silver chloride (AgCl) incorporated in the lenses. When the lenses are exposed to light of the appropriate wavelength, the following reaction occurs: \(A g C l \rightarrow A g+C l\) The Ag atoms formed produce a uniform gray color that reduces the glare. If \(\Delta H\) for the preceding reaction is 248 kJ/mol, calculate the maximum wavelength of light that can induce this process.

A helium atom and a xenon atom have the same kinetic energy. Calculate the ratio of the de Broglie wavelength of the helium atom to that of the xenon atom.

A laser is used in treating retina detachment. The wavelength of the laser beam is 514 nm and the power is 1.6 W. If the laser is turned on for 0.060 s during surgery, calculate the number of photons emitted by the laser. (1 W = 1 J/s.)

An electron in an excited state in a hydrogen atom can return to the ground state in two different ways: (a) via a direct transition in which a photon of wavelength \(\lambda_1\) is emitted and (b) via an intermediate excited state reached by the emission of a photon of wavelength \(\lambda_2\). This intermediate excited state then decays to the ground state by emitting another photon of wavelength \(\lambda_3\). Derive an equation that relates \(\lambda_1\) to \(\lambda_2\) and \(\lambda_3\).

A photoelectric experiment was performed by separately shining a laser at 450 nm (blue light) and a laser at 560 nm (yellow light) on a clean metal surface and measuring the number and kinetic energy of the ejected electrons. Which light would generate more electrons? Which light would eject electrons with greater kinetic energy? Assume that the same amount of energy is delivered to the metal surface by each laser and that the frequencies of the laser lights exceed the threshold frequency.

Draw the shapes (boundary surfaces) of the following orbitals: (a) \(2p_y\), (b) \(3d_{z^2}\), (c) \(3d_{x^2-y^2}\). (Show coordinate axes in your sketches.)

Problem 120P The electron configurations described in this chapter all refer to gaseous atoms in their ground states. An atom may absorb a quantum of energy and promote one of its electrons to a higher-energy orbital. When this happens, we say that the atom is in an excited state. The electron configurations of some excited atoms are given. Identify these atoms and write their ground-state configurations: (a) 1s12s1 (b) 1s22s22p23d1 (c) 1s22s22p64s1 (d) [Ar]4s13d104p4 (e) [Ne]3s23p43d1

Draw orbital diagrams for atoms with the following electron configurations: (a) \(1s^22s^22p^5\) (b) \(1s^22s^22p^63s^23p^3\) (c) \(1s^22s^22p^63s^23p^64s^23d^7\)

A particular form of electromagnetic radiation has a frequency of \(8.11\times 10^{14}\) Hz. (a) What is its wavelength in nanometers? In meters? (b) To what region of the electromagnetic spectrum would you assign it? (c) What is the energy (in joules) of one quantum of this radiation?

The work function of potassium is \(3.68 \times 10^{-19} \mathrm{~J}\). (a) What is the minimum frequency of light needed to eject electrons from the metal? (b) Calculate the kinetic energy of the ejected electrons when light of frequency equal to \(8.62 \times 10^{14} \mathrm{~s}^{-1}\) is used for irradiation.

When light of frequency equal to \(2.11 \times 10^{15}\ s^{-1}\) shines on the surface of gold metal, the kinetic energy of ejected electrons is found to be \(5.83 \times 10^{-19}\ \mathrm{J}\). What is the work function of gold?

(a) Briefly describe Bohr's theory of the hydrogen atom and how it explains the appearance of an emission spectrum. How does Bohr's theory differ from concepts of classical physics? (b) Explain the meaning of the negative sign in Equation (7.5).

(a) What is an energy level? Explain the difference between ground state and excited state. (b) What are emission spectra? How do line spectra differ from continuous spectra?

Explain why elements produce their own characteristic colors when they emit photons?

Some copper compounds emit green light when they are heated in a flame. How would you determine whether the light is of one wavelength or a mixture of two or more wavelengths?

Is it possible for a fluorescent material to emit radiation in the ultraviolet region after absorbing visible light? Explain your answer.

Explain how astronomers are able to tell which elements are present in distant stars by analyzing the electromagnetic radiation emitted by the stars.

Consider the following energy levels of a hypothetical atom: \(E_{4}\underline{\hspace{3cm}}-1.0 \times 10^{-19}\ \mathrm{J}\) \(E_{3}\underline{\hspace{3cm}}-5.0 \times 10^{-19}\ \mathrm{J}\) \(E_{2}\underline{\hspace{3cm}}-10 \times 10^{-19}\ \mathrm{J}\) \(E_{1}\underline{\hspace{3cm}}-15 \times 10^{-19}\ \mathrm{J}\) (a) What is the wavelength of the photon needed to excite an electron from \(E_{1}\) to \(E_{4}\) (b) What is the energy (in joules) a photon must have in order to excite an electron from \(E_{2}\) to \(E_{3}\)? (c) When an electron drops from the \(E_{3}\) level to the \(E_{1}\) level, the atom is said to undergo emission. Calculate the wavelength of the photon emitted in this process.

The first line of the Balmer series occurs at a wavelength of 656.3 nm. What is the energy difference between the two energy levels involved in the emission that results in this spectral line?

Calculate the wavelength (in nanometers) of a photon emitted by a hydrogen atom when its electron drops from the n = 5 state to the n = 3 state.

Calculate the frequency (Hz) and wavelength (nm) of the emitted photon when an electron drops from the n = 4 to the n = 2 level in a hydrogen atom.

Careful spectral analysis shows that the familiar yellow light of sodium lamps (such as street lamps) is made up of photons of two wavelengths, 589.0 nm and 589.6 nm. What is the difference in energy (in joules) between photons with these wavelengths?

An electron in the hydrogen atom makes a transition from an energy state of principal quantum numbers n; to the n = 2 state. If the photon emitted has a wavelength of 434 nm, what is the value of \(n_{i}\)?

Explain the statement, Matter and radiation have a "dual nature."

How does de Broglie's hypothesis account for the fact that the energies of the electron in a hydrogen atom are quantized?

Why is Equation (7.8) meaningful only for submicroscopic particles, such as electrons and atoms, and not for macroscopic objects?

(a) If a H atom and a He atom are traveling at the same speed, what will be the relative wavelengths of the two atoms? (b) If a H atom and a He atom have the same kinetic energy, what will be the relative wavelengths of the two atoms?

Thermal neutrons are neutrons that move at speeds comparable to those of air molecules at room temperature. These neutrons are most effective in initiating a nuclear chain reaction among \({ }^{235} U\) isotopes. Calculate the wavelength (in nm) associated with a beam of neutrons moving at \(7.00 \times 10^{2}\ \mathrm{m} / \mathrm{s}\). (Mass of a neutron = \(1.675 \times 10^{-27}\ \mathrm{kg}\).)

Protons can be accelerated to speeds near that of light in particle accelerators. Estimate the wavelength (in nm) of such a proton moving at \(2.90 \times 10^{8}\ \mathrm{m} / \mathrm{s}\). (Mass of a proton = \(1.675 \times 10^{-27}\ \mathrm{kg}\).)

What is the de Broglie wavelength, in cm, of a 12.4-g hummingbird flying at \(1.20 \times 10^{2}\ \mathrm{mph}\)? (1 mile = 1.61 km.)

What is the de Broglie wavelength (in nm) associated with a 2.5-g Ping-Pong ball traveling 35 mph?

What are the inadequacies of Bohr’s theory?

What is the Heisenberg uncertainty principle? What is the Schrodinger equation?

What is the physical significance of the wave function?

How is the concept of electron density used to describe the position of an electron in the quantum mechanical treatment of an atom?

What is an atomic orbital? How does an atomic orbital differ from an orbit?

Describe the shapes of s, p, and d orbitals. How are these orbitals related to the quantum numbers \(n,\ \ell\), and \(m_{\ell}\)?

List the hydrogen orbitals in increasing order of energy.

Describe the characteristics of an s orbital, a p orbital, and a d orbital. Which of the following orbitals do not exist: 1p, 2s, 2d, 3p, 3d, 3f, 4g?

Why is a boundary surface diagram useful in representing an atomic orbital?

Describe the four quantum numbers used to characterize an electron in an atom.

Which quantum number defines a shell? Which quantum numbers define a subshell?

Which of the four quantum numbers \(\left(n, l, m_{\ell}, m_{s}\right)\) determine (a) the energy of an electron in a hydrogen atom and in a many-electron atom, (b) the size of an orbital, (c) the shape of an orbital, (d) the orientation of an orbital in space?

An electron in a certain atom is in the n = 2 quantum level. List the possible values of \(\ell\) and \(m_{\ell}\) that it can have.

Give the values of the quantum numbers associated with the following orbitals: (a) 2p, (b) 3s, (c) 5d.

An electron in an atom is in the n = 3 quantum level. List the possible values of ? and \(m_{?}\) that it can have.

Give the values of the four quantum numbers of an electron in the following orbitals: (a) 3s, (b) 4p, (c) 3d.

Discuss the similarities and differences between a 1s and a 2s orbital.

What is the difference between a \(2 p_{x}\) and a \(2 p_{y}\) orbital?

List all the possible subshells and orbitals associated with the principal quantum number n, if n = 5.

List all the possible subshells and orbitals associated with the principal quantum number n, if n = 6.

Calculate the total number of electrons that can occupy (a) one s orbital, (b) three p orbitals, (c) five d orbitals, (d) seven f orbitals.

What is the total number of electrons that can be held in all orbitals having the same principal quantum number n?

Determine the maximum number of electrons that can be found in each of the following subshells: 3s, 3d, 4p, 4f, 5f.

Indicate the total number of (a) p electrons in N (Z = 7); (b) s electrons in Si (Z = 14); and (c) 3d electrons in S (Z = 16).

A clean metal surface is irradiated with light of three different wavelengths \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\). The kinetic energies of the ejected electrons are as follows: \(\lambda_{1}:\ 2.9 \times 10^{-20}\ \mathrm{J} ;\ \lambda_{2}\): approximately zero; \(\lambda_{3}:\ 4.2 \times 10^{-19}\ \mathrm{J}\). Which light has the shortest wavelength and which has the longest wavelength?

Give the high and low wavelength values that define the visible region of the electromagnetic spectrum.

What is the wavelength (in nanometers) of a photon emitted during a transition from \(n_{i}=6\) to \(n_{f}=4\) state in the H atom?

Which transition in the hydrogen atom would emit light of a shorter wavelength? (a) \(n_{i}=5 \rightarrow n_{f}=3\) or (b) \(n_{i}=4 \rightarrow n_{f}=2\).

Briefly explain Planck’s quantum theory and explain what a quantum is. What are the units for Planck’s constant?

Calculate the wavelength (in nanometers) of a H atom \(\left(\text { mass }=1.674 \times 10^{-27}\ \mathrm{kg}\right)\) moving at \(7.00 \times 10^{2}\ \mathrm{cm} / \mathrm{s}\).

Which quantity in Equation (7.8) is responsible for the fact that macroscopic objects do not show observable wave properties?

Give two everyday examples that illustrate the concept of quantization.

Estimate the uncertainty in the speed of an oxygen molecule if its position is known to be \(\pm 3\ \mathrm{nm}\). The mass of an oxygen molecule is \(5.31 \times 10^{-26}\ \mathrm{kg}\).

What is the difference between \(\psi\) and \(\psi^{2}\) for the electron in a hydrogen atom?

(a) What is the wavelength (in nanometers) of light having a frequency of \(8.6 \times 10^{13}\ \mathrm{Hz}\)? (b) What is the frequency (in Hz) of light having a wavelength of 566 nm?

Give the values of the quantum numbers associated with the orbitals in the 3p subshell.

Give the four quantum numbers for each of the two electrons in a 6s orbital.

(a) What is the frequency of light having a wavelength of 456 nm? (b) What is the wavelength (in nanometers) of radiation having a frequency of \(2.45 \times 10^{9}\ \mathrm{Hz}\)? (This is the type of radiation used in microwave ovens.)

What is the total number of orbitals associated with the principal quantum number n = 4?

Why is it not possible to have a 2d orbital but a 3d orbital is allowed?

The average distance between Mars and Earth is about \(1.3 \times 10^{8}\ \text { miles }\). How long would it take TV pictures transmitted from the Viking space vehicle on Mars' surface to reach Earth? (1 mile = 1.61 km.)

Write the four quantum numbers for an electron in a 4d orbital.

The ground-state electron configuration of an atom is \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{3}\). Which of the four quantum numbers would be the same for the three 3p electrons?

How many minutes would it take a radio wave to travel from the planet Venus to Earth? (Average distance from Venus to Earth = 28 million miles.)

Calculate the total number of electrons that can be present in the principal level for which n = 4.

Identify the atom that has the following ground-state electron configuration: \([A r] 4 s^{2} 3 d^{6}\)

The SI unit of time is the second, which is defined as 9,192,631,770 cycles of radiation associated with a certain emission process in the cesium atom. Calculate the wavelength of this radiation (to three significant figures). In which region of the electromagnetic spectrum is this wavelength found?

Write a complete set of quantum numbers for each of the electrons in boron (B).

The SI unit of length is the meter, which is defined as the length equal to 1,650,763.73 wavelengths of the light emitted by a particular energy transition in krypton atoms. Calculate the frequency of the light to three significant figures.

Write the ground-state electron configuration for phosphorus (P).

What are photons? What role did Einstein's explanation of the photoelectric effect play in the development of the particle-wave interpretation of the nature of electromagnetic radiation?

Consider the plots shown here for the photoelectric effect of two different metals A (green line) and B (red line). (a) Which metal has a greater work function? (b) What does the slope of the lines tell us?

A photon has a wavelength of 624 nm. Calculate the energy of the photon in joules.

The blue color of the sky results from the scattering of sunlight by air molecules. The blue light has a frequency of about \(7.5 \times 10^{14} \mathrm{~Hz}\). (a) Calculate the wavelength, in nm, associated with this radiation, and (b) calculate the energy, in joules, of a single photon associated with this frequency.

A photon has a frequency of \(6.0 \times 10^{4}\ \mathrm{Hz}\). (a) Convert this frequency into wavelength (nm). Does this frequency fall in the visible region? (b) Calculate the energy (in joules) of this photon. (c) Calculate the energy (in joules) of 1 mole of photons all with this frequency.

What is the wavelength, in nm, of radiation that has an energy content of \(1.0 \times 10^{3}\ \mathrm{kJ} / \mathrm{mol}\)? In which region of the electromagnetic spectrum is this radiation found?

When copper is bombarded with high-energy electrons, X rays are emitted. Calculate the energy (in joules) associated with the photons if the wavelength of the X rays is 0.154 nm.

If Rutherford and his coworkers had used electrons instead of alpha particles to probe the structure of the nucleus as described in Section 2.2, what might they have discovered?

Scientists have found interstellar hydrogen atoms with quantum number n in the hundreds. Calculate the wavelength of light emitted when a hydrogen atom undergoes a transition from n = 236 to n = 235. In what region of the electromagnetic spectrum does this wavelength fall?

Calculate the wavelength of a helium atom whose speed is equal to the root-mean-square speed at \(20^\circ \mathrm C\).

Ionization energy is the minimum energy required to remove an electron from an atom. It is usually expressed in units of kJ/mol, that is, the energy in kilojoules required to remove one mole of electrons from one mole of atoms, (a) Calculate the ionization energy for the hydrogen atom, (b) Repeat the calculation, assuming in this second case that the electrons are removed from the n = 2 state.

An electron in a hydrogen atom is excited from the ground state to the n = 4 state. Comment on the correctness of the following statements (true or false). (a) n = 4 is the first excited state. (b) It takes more energy to ionize (remove) the electron from n = 4 than from the ground state. (c) The electron is farther from the nucleus (on average) in n = 4 than in the ground state. (d) The wavelength of light emitted when the electron drops from n = 4 to n = 1 is longer than that from n = 4 to n = 2. (e) The wavelength the atom absorbs in going from n = 1 to n = 4 is the same as that emitted as it goes from n = 4 to n = 1.

The ionization energy of a certain element is 412 kJ/mol (see Problem 7.125). However, when the atoms of this element are in the first excited state, the ionization energy is only 126 kJ/mol. Based on this information, calculate the wavelength of light emitted in a transition from the first excited state to the ground state.

Alveoli are the tiny sacs of air in the lungs (see Problem 5.136) whose average diameter is \(5.3 \times 10^{-5}~ \mathrm m\) . Consider an oxygen molecule \((5.3 \times 10^{?26} ~\mathrm{kg})\) trapped within a sac. Calculate the uncertainty in the velocity of the oxygen molecule. (Hint: The maximum uncertainty in the position of the molecule is given by the diameter of the sac.)

How many photons at 660 nm must be absorbed to melt \(5.0 \times 10^2~ \mathrm g\) of ice? On average, how many \(\mathrm{H_2O}\) molecules does one photon convert from ice to water? (Hint: It takes 334 J to melt 1 g of ice at \(0 ^\circ \mathrm C\).)

Shown are portions of orbital diagrams representing the ground-state electron configurations of certain elements. Which of them violate the Pauli exclusion principle? Hund's rule?

The UV light that is responsible for tanning the skin falls in the 320- to 400-nm region. Calculate the total energy (in joules) absorbed by a person exposed to this radiation for 2.0 h, given that there are \(2.0 \times 10^{16}\) photons hitting Earth's surface per square centimeter per second over a 80-nm (320 nm to 400 nm) range and that the exposed body area is \(0.45~\mathrm m^2\). Assume that only half of the radiation is absorbed and the other half is reflected by the body. (Hint: Use an average wavelength of 360 nm in calculating the energy of a photon.)

The sun is surrounded by a white circle of gaseous material called the corona, which becomes visible during a total eclipse of the sun. The temperature of the corona is in the millions of degrees Celsius, which is high enough to break up molecules and remove some or all of the electrons from atoms. One way astronomers have been able to estimate the temperature of the corona is by studying the emission lines of ions of certain elements. For example, the emission spectrum of \(\mathrm {Fe^{14+}}\) ions has been recorded and analyzed. Knowing that it takes \(3.5 \times 10^4~ \mathrm{kJ/mol}\) to convert \(\mathrm {Fe^{13+}}\) to \(\mathrm {Fe^{14+}}\), estimate the temperature of the sun's corona. (Hint: The average kinetic energy of one mole of a gas is \(\frac{3}{2}RT\).

In 1996 physicists created an anti-atom of hydrogen. In such an atom, which is the antimatter equivalent of an ordinary atom, the electrical charges of all the component particles are reversed. Thus, the nucleus of an anti-atom is made of an anti-proton, which has the same mass as a proton but bears a negative charge, while the electron is replaced by an anti- electron (also called positron) with the same mass as an electron, but bearing a positive charge. Would you expect the energy levels, emission spectra, and atomic orbitals of an antihydrogen atom to be different from those of a hydrogen atom? What would happen if an anti-atom of hydrogen collided with a hydrogen atom?

Use Equation (5.16) to calculate the de Broglie wavelength of a \(\mathrm N_2\) molecule at 300 K.

When an electron makes a transition between energy levels of a hydrogen atom, there are no restrictions on the initial and final values of the principal quantum number n. However, there is a quantum mechanical rule that restricts the initial and final values of the orbital angular momentum \(\ell\). This is the selection rule, which states that \(\Delta \ell=\pm 1\); that is, in a transition, the value of \(\ell\) can only increase or decrease by one. According to this rule, which of the following transitions are allowed: (a) \(2s \longrightarrow 1s\), (b) \(3p \longrightarrow 1s\), (c) \(3d \longrightarrow 4f\), (d) \(4d \longrightarrow 3s\)? In view of this selection rule, explain why it is possible to observe the various emission series shown in Figure 7.11.

In an electron microscope, electrons are accelerated by passing them through a voltage difference. The kinetic energy thus acquired by the electrons is equal to the voltage times the charge on the electron. Thus, a voltage difference of 1 V imparts a kinetic energy of \(\mathrm {1.602 \times 10^{?19}~ C \times V ~or~ 1.602 \times 10^{?19}~ J}\). Calculate the wavelength associated with electrons accelerated by \(5.00 \times 10^3~ \mathrm V\).

A microwave oven operating at \(1.22 \times 10^8~\mathrm {nm}\) is used to heat 150 mL of water (roughly the volume of a tea cup) from \(20^\circ \mathrm C\) to \(100^\circ \mathrm C\). Calculate the number of photons needed if 92.0 percent of microwave energy is converted to the thermal energy of water.

The radioactive Co-60 isotope is used in nuclear medicine to treat certain types of cancer. Calculate the wavelength and frequency of an emitted gamma photon having the energy of \(1.29 \times 10^{11}~ \mathrm{ J/mol}\).

(a) An electron in the ground state of the hydrogen atom moves at an average speed of \(5 \times 10^6~\mathrm{m/s}\). If the speed is known to an uncertainty of 1 percent, what is the uncertainty in knowing its position? Given that the radius of the hydrogen atom in the ground state is \(5.29 \times 10^-{11}~\mathrm m\), comment on your result. The mass of an electron is \(9.1094 \times 10^{-31}~\mathrm {kg}\). (b) A 3.2-g Ping-Pong ball moving at 50 mph has a momentum of \(\mathrm {0.073~ kg \cdot ~ m/s}\). If the uncertainty in measuring the momentum is \(1.0 \times 10^{?7}\) of the momentum, calculate the uncertainty in the Ping-Pong ball's position.

One wavelength in the hydrogen emission spectrum is 1280 nm. What are the initial and final states of the transition responsible for this emission?

Owls have good night vision because their eyes can detect a light intensity as low as \(5.0 \times 10^{?13}~\mathrm {W/m^2}\). Calculate the number of photons per second that an owl's eye can detect if its pupil has a diameter of 9.0 mm and the light has a wavelength of 500 nm. (1 W = 1 J/s.)

For hydrogenlike ions, that is, ions containing only one electron, Equation (7.5) is modified as follows: \(E_n = ?R_\mathrm H Z^2(1/n^2)\), where Z is the atomic number of the parent atom. The figure here represents the emission spectrum of such a hydrogenlike ion in the gas phase. All the lines result from the electronic transitions from the excited states to the n = 2 state. (a) What electronic transitions correspond to lines B and C? (b) If the wavelength of line C is 27.1 nm, calculate the wavelengths of lines A and B. (c) Calculate the energy needed to remove the electron from the ion in the n = 4 state, (d) What is the physical significance of the continuum?

When two atoms collide, some of their kinetic energy may be converted into electronic energy in one or both atoms. If the average kinetic energy is about equal to the energy for some allowed electronic transition, an appreciable number of atoms can absorb enough energy through an inelastic collision to be raised to an excited electronic state. (a) Calculate the average kinetic energy per atom in a gas sample at 298 K. (b) Calculate the energy difference between the n = 1 and n = 2 levels in hydrogen. (c) At what temperature is it possible to excite a hydrogen atom from the n = 1 level to n = 2 level by collision? [The average kinetic energy of 1 mole of an ideal gas is \((\frac{3}{2})RT\).]

Calculate the energies needed to remove an electron from the n = 1 state and the n = 5 state in the \(\mathrm {Li}^{2+}\) ion. What is the wavelength (in nm) of the emitted photon in a transition from n = 5 to n = 1? The Rydberg constant for hydrogenlike ions is \((\mathrm {2.18 \times 10^{?18} J)Z^2}\), where Z is the atomic number.

The de Broglie wavelength of an accelerating proton in the Large Hadron Collider is \(2.5 \times 10^{-14}~\mathrm m\). What is the kinetic energy (in joules) of the proton?

The minimum uncertainty in the position of a certain moving particle is equal to its de Broglie wavelength. If the speed of the particle is \(1.2 \times 10^5~\mathrm{m/s}\), what is the minimum uncertainty in its speed?

According to Einstein's special theory of relativity, the mass of a moving particle, \(m_\mathrm{moving}\), is related to its mass at rest, \(m_\mathrm{rest}\), by the following equation \(m_\mathrm{moving}=\frac{m_\mathrm{rest}}{\sqrt {1-(\frac{u}{c})^2}}\) where u and c are the speeds of the particle and light, respectively. (a) In particle accelerators, protons, electrons, and other charged particles are often accelerated to speeds close to the speed of light. Calculate the wavelength (in nm) of a proton moving at 50.0 percent the speed of light. The mass of a proton is \(1.673 \times 10^{-27}~\mathrm{kg}\). (b) Calculate the mass of a \(6.0 \times 10^{?2}~\mathrm{kg}\) tennis ball moving at 63 m/s. Comment on your results.

The mathematical equation for studying the photoelectric effect is \(hv=W+\frac{1}{2}m_\mathrm e u^2\) where v is the frequency of light shining on the metal, W is the work function, and me and u are the mass and speed of the ejected electron. In an experiment, a student found that a maximum wavelength of 351 nm is needed to just dislodge electrons from a zinc metal surface. Calculate the speed (in m/s) of an ejected electron when she employed light with a wavelength of 313 nm.

In the beginning of the twentieth century, some scientists thought that a nucleus may contain both electrons and protons. Use the Heisenberg uncertainty principle to show that an electron cannot be confined within a nucleus. Repeat the calculation for a proton. Comment on your results. Assume the radius of a nucleus to be \(1.0 \times 10^{15}~\mathrm m\). The masses of an electron and a proton are \(9.109 \times 10^{-31}~\mathrm {kg}\) and \(1.673 \times 10^{-27}~\mathrm {kg}\), respectively. (Hint: Treat the diameter of the nucleus as the uncertainty in position.)

Blackbody radiation is the term used to describe the dependence of the radiation energy emitted by an object on wavelength at a certain temperature. Planck proposed the quantum theory to account for this dependence. Shown in the figure is a plot of the radiation energy emitted by our sun versus wavelength. This curve is characteristic of the temperature at the surface of the sun. At a higher temperature, the curve has a similar shape but the maximum will shift to a shorter wavelength. What does this curve reveal about two consequences of great biological significance on Earth?

All molecules undergo vibrational motions. Quantum mechanical treatment shows that the vibrational energy, \(E_\mathrm{vib}\), of a diatomic molecule like HCl is given by \(E_\mathrm{vib}=(n+\frac{1}{2})hv\) where n is a quantum number given by n = 0, 1, 2, 3, . . . and v is the fundamental frequency of vibration. (a) Sketch the first three vibrational energy levels for HCl. (b) Calculate the energy required to excite a HCl molecule from the ground level to the first excited level. The fundamental frequency of vibration for HCl is \(8.66 \times 10^{13}~ \mathrm s^{-1}\). (c) The fact that the lowest vibrational energy in the ground level is not zero but equal to \(\frac{1}{2}hv\) means that molecules will vibrate at all temperatures, including the absolute zero. Use the Heisenberg uncertainty principle to justify this prediction. (Hint: Consider a nonvibrating molecule and predict the uncertainty in the momentum and hence the uncertainty in the position.)

The wave function for the 2s orbital in the hydrogen atom is \(\psi_{2 s}=\frac{1}{\sqrt{2 a_{0}^{3}}}\left(1-\frac{\rho}{2}\right) e^{-\rho / 2}\) where \(a_0\) is the value of the radius of the first Bohr orbit, equal to 0.529 nm, \(\rho\) is \(Z(r/a_0)\), and r is the distance from the nucleus in meters. Calculate the location of the node of the 2s wave function from the nucleus.

Atoms of an element have only two accessible excited states. In an emission experiment, however, three spectral lines were observed. Explain. Write an equation relating the shortest wavelength to the other two wavelengths.

According to Wien's law, the wavelength of maximum intensity in blackbody radiation, \(\lambda_\mathrm{max}\), is given by \(\lambda_\mathrm{max}=\frac{b}{T}\) where b is a constant \((2.898 \times 10^6~\mathrm{ nm \cdot K})\) and T is the temperature of the radiating body in kelvins. (a) Estimate the temperature at the surface of the sun. (b) How are astronomers able to determine the temperature of stars in general? (See Problem 7.150 for a definition of blackbody radiation.)

Only a fraction of the electrical energy supplied to an incandescent-tungsten lightbulb is converted to visible light. The rest of the energy shows up as infrared radiation (that is, heat). A 60-W lightbulb converts about 15.0 percent of the energy supplied to it into visible light. Roughly how many photons are emitted by the lightbulb per second? (1 W = 1 J/s.)

Photosynthesis makes use of photons of visible light to bring about chemical changes. Explain why heat energy in the form of infrared photons is ineffective for photosynthesis. (Hint: Typical chemical bond energies are 200 kJ/mol or greater.)

A typical red laser pointer has a power of 5 mW. How long would it take a red laser pointer to emit the same number of photons emitted by a 1-W blue laser in 1 s? (1 W = 1 J/s.)

Referring to the Chemistry in Action on p. 314, estimate the wavelength of light that would be emitted by a cadmium selenide (CdSe) quantum dot with a diameter of 10 nm. Would the emitted light be visible to the human eye? The diameter and emission wavelength for a series of quantum dots are given here.

What is a wave? Explain the following terms associated with waves: wavelength, frequency, amplitude.

What is the wavelength (in meters) of an electromagnetic wave whose frequency is \(3.64 \times 10^{7}\ \mathrm{Hz}\)?

Which of the waves shown here has (a) the highest frequency, (b) the longest wavelength, (c) the greatest amplitude?

What are the units for wavelength and frequency of electromagnetic waves? What is the speed of light in meters per second and miles per hour?

The energy of a photon is \(5.87 \times 10^{-20}\ J\). What is its wavelength (in nanometers)?

Why is radiation only in the UV but not the visible or infrared region responsible for sun tanning?

List the types of electromagnetic radiation, starting with the radiation having the longest wavelength and ending with the radiation having the shortest wavelength.

The work function of titanium metal is \(6.93 \times 10^{-19}\ \mathrm{J}\). Calculate the kinetic energy of the ejected electrons if light of frequency \(2.50 \times 10^{15}\ \mathrm{s}^{-1}\) is used to irradiate the metal.

The \(\mathrm{He}^{+}\) ion contains only one electron and is therefore a hydrogenlike ion. Calculate the wavelengths, in increasing order, of the first four transitions in the Balmer series of the \(\mathrm{He}^{+}\) ion. Compare these wavelengths with the same transitions in a H atom. Comment on the differences. (The Rydberg constant for He is \(8.72 \times 10^{-18}\ \mathrm{J}\).)

Ozone \(\left(O_{3}\right)\) in the stratosphere absorbs the harmful radiation from the sun by undergoing decomposition: \(O_{3} \rightarrow O+O_{2}\). (a) Referring to Table 6.4, calculate the \(\Delta H^{\circ}\) for this process. (b) Calculate the maximum wavelength of photons (in nm) that possess this energy to cause the decomposition of ozone photochemically.

The retina of a human eye can detect light when radiant energy incident on it is at least \(4.0 \times 10^{-17}~\mathrm J\). For light of 600-nm wavelength, how many photons does this correspond to?

Problem 67P Make a chart of all allowable orbitals in the first four principal energy levels of the hydrogen atom. Designate each by type (for example, s, p) and indicate how many orbitals of each type there are.

Why do the 3s, 3p, and 3d orbitals have the same energy in a hydrogen atom but different energies in a many-electron atom?

For each of the following pairs of hydrogen orbitals, indicate which is higher in energy: (a) 1s, 2s; (b) 2p, 3p; (c) \(3 d_{x y},\ 3 d_{y z}\); (d) 3s, 3d; (e) 4f, 5s.

Which orbital in each of the following pairs is lower in energy in a many-electron atom? (a) 2s, 2p; (b) 3p, 3d; (c) 3s, 4s; (d) 4d, 5f.