Chapter 7

Draw orbital diagrams for atoms with the following electron configurations: (a) \(1 s^{2} 2 s^{2} 2 p^{5}\) (b) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{3}\) (c) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{7}\)

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?

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 \(n=1\)

Alveoli are the tiny sacs of air in the lungs (see Problem 5.136 ) whose average diameter is \(5.0 \times\) \(10^{-5} \mathrm{~m} .\) Consider an oxygen molecule \(\left(5.3 \times 10^{-26} \mathrm{~kg}\right)\) 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 \(660 \mathrm{nm}\) must be absorbed to melt \(5.0 \times 10^{2} \mathrm{~g}\) of ice \(?\) On average, how many \(\mathrm{H}_{2} \mathrm{O}\) molecules does one photon convert from ice to water? (Hint: It takes \(334 \mathrm{~J}\) to melt \(1 \mathrm{~g}\) of ice at \(0^{\circ} \mathrm{C}\).)

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 \(2.0 \mathrm{~h}\), 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 \(400 \mathrm{nm}\) ) 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 \(360 \mathrm{nm}\) 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 \(3.5 \times 10^{4} \mathrm{~kJ} / \mathrm{mol}\) 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 \(\left.\frac{3}{2} R T .\right)\)

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?

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 \(1 \mathrm{~V}\) imparts a kinetic energy of \(1.602 \times 10^{-19} \mathrm{C} \times \mathrm{V}\) or \(1.602 \times\) \(10^{-19} \mathrm{~J}\). Calculate the wavelength associated with electrons accelerated by \(5.00 \times 10^{3} \mathrm{~V}\).

The radioactive \(\mathrm{Co}-60\) 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 \(1.29 \times 10^{11} \mathrm{~J} / \mathrm{mol} .\)

(a) An electron in the ground state of the hydrogen atom moves at an average speed of \(5 \times 10^{6} \mathrm{~m} / \mathrm{s} .\) 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 \(0.073 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s} .\) 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.

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

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 \(298 \mathrm{~K}\). (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 \(\left.\left(\frac{3}{2}\right) R T .\right]\)

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 \(\mathrm{nm}\) ) of the emitted photon in a transition from \(n=5\) to \(n=1 ?\) The Rydberg constant for hydrogen like ions is \((2.18 \times\) \(\left.10^{-18} \mathrm{~J}\right) Z^{2},\) where \(Z\) is the atomic number.

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

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.)

All molecules undergo vibrational motions. Quantum mechanical treatment shows that the vibrational energy, \(E_{\mathrm{vib}},\) of a diatomic molecule like \(\mathrm{HCl}\) is given by $$ E_{\mathrm{vib}}=\left(n+\frac{1}{2}\right) h \nu $$ where \(n\) is a quantum number given by \(n=0,1,2,\) \(3, \ldots\) and \(\nu\) is the fundamental frequency of vibration. (a) Sketch the first three vibrational energy levels for \(\mathrm{HCl}\). (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 \(\mathrm{HCl}\) 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 \(\frac{1}{2} h v\) 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 \(2 s\) 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 \(0.529 \mathrm{nm}, \rho\) is \(Z\left(r / a_{0}\right),\) and \(r\) is the distance from the nucleus in meters. Calculate the location of the node of the \(2 s\) wave function from the nucleus.

A student placed a large unwrapped chocolate bar in a microwave oven without a rotating glass plate. After turning the oven on for less than a minute, she noticed there were evenly spaced dents (due to melting) about \(6 \mathrm{~cm}\) apart. Based on her observations, calculate the speed of light given that the microwave frequency is \(2.45 \mathrm{GHz}\). (Hint: The energy of a wave is proportional to the square of its amplitude.)

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.

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 \(200 \mathrm{~kJ} / \mathrm{mol}\) or greater. \()\)

Referring to the Chemistry in Action essay "Quantum Dots" in Section 7.9 , estimate the wavelength of light that would be emitted by a cadmium selenide (CdSe) quantum dot with a diameter of \(10 \mathrm{nm} .\) Would the emitted light be visible to the human eye? The diameter and emission wavelength for a series of quantum dots are given here.