Physics: Principles with Applications

Douglas C. Giancoli

Chapter 29

MOLECULES AND SOLIDS - all with Video Answers

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Chapter Questions

Problem 1

(I) Estimate the binding energy of a KCl molecule by calculating the electrostatic potential energy when the K$^+$ and CI$^-$ ions are at their stable separation of 0.28 nm. Assume each has a charge of magnitude 1.0$e$.

Deepak Kohli

Deepak Kohli

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Problem 2

(II) The measured binding energy of KCl is 4.43 eV. From the result of Problem 1, estimate the contribution to the binding energy of the repelling electron clouds at the equilibrium distance $r_0 =$ 0.28 nm.

Ben Nicholson

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Problem 3

(II) The equilibrium distance $r_0$ between two atoms in a molecule is called the $\textbf{bond length}$. Using the bond lengths of homogeneous molecules (like H$_2$, O$_2$, and N$_2$), one can estimate the bond length of heterogeneous molecules (like CO, CN, and NO). This is done by summing half of each bond length of the homogenous molecules to estimate that of the heterogeneous molecule. Given the following bond lengths: H$_2$ (= 74 pm), N$_2$ (= 145 pm), O$_2$ (= 121 pm), C$_2$ (= 154 pm), estimate the bond lengths for: HN, CN, and NO.

Bryce Samwel

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Problem 4

(II) Binding energies are often measured experimentally in kcal per mole, and then the binding energy in eV per molecule is calculated from that result. What is the conversion factor in going from kcal per mole to eV per molecule?What is the binding energy of KCl (= 4.43 eV) in kcal per mole?

Ben Nicholson

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Problem 5

(III) Estimate the binding energy of the H$_2$ molecule, assuming the two H nuclei are 0.074 nm apart and the two electrons spend 33% of their time midway between them.

Qasim Sadiq

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Problem 6

III) ($a$) Apply reasoning similar to that in the text for the $S =$ 0 and $S =$ 1 states in the formation of the molecule to show why the molecule He$_2$ is $not$ formed. ($b$) Explain why the He$_2^+$ molecular ion $could$ form. (Experiment shows it has a binding energy of 3.1 eV at $r_0 =$ 0.11 nm.)

Ben Nicholson

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Problem 7

(I) Show that the quantity $\hslash^2/I$ has units of energy.

Qasim Sadiq

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Problem 8

(II) (a) Calculate the "characteristic rotational energy," $\hslash/2I$ for the O$_2$ molecule whose bond length is 0.121 nm. (b) What are the energy and wavelength of photons emitted in an $\ell =$ 3 to $\ell =$ 2 transition?

Ben Nicholson

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Problem 9

(II) The "characteristic rotational energy," $\hslash/2I$, for N$_2$ is 2.48 $\times$ 10$^{-4}$eV. Calculate the bond length.

Qasim Sadiq

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Problem 10

(II) The equilibrium separation of H atoms in the molecule is 0.074 nm (Fig. 29-8). Calculate the energies and wavelengths of photons for the rotational transitions ($a$) $\ell =$ 1 to $\ell =$ 0, ($b$) $\ell =$ 2 to $\ell =$ 1, and ($c$) $\ell =$ 3 to $\ell =$ 2.

Ben Nicholson

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Problem 11

(II) Determine the wavelength of the photon emitted when the CO molecule makes the rotational transition $\ell = 5$ to $\ell = 4$ . [$Hint:$ See Example 29-2.]

Qasim Sadiq

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Problem 12

(II) Calculate the bond length for the NaCl molecule given that three successive wavelengths for rotational transitions are 23.1 mm, 11.6 mm, and 7.71 mm.

Ben Nicholson

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Problem 13

(II) ($a$) Use the curve of Fig. 29-17 to estimate the stiffness constant $k$ for the H$_2$ molecule. (Recall that $_{\small\mathrm{PE}} = \frac{1}{2}kx^2$) (b) Then estimate the fundamental wavelength for vibrational transitions using the classical formula (Chapter 11), but use only $\frac{1}{2}$ the mass of an H atom (because both H atoms move).

Qasim Sadiq

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Problem 14

(II) Common salt, NaCl, has a density of 2.165 g/cm$^3$. The molecular weight of NaCl is 58.44. Estimate the distance between nearest neighbor Na and Cl ions. [$Hint: Each$ ion can be considered to be at the corner of a cube.]

Ben Nicholson

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Problem 15

II) Repeat Problem 14 for KCl whose density is 1.99 g/cm$^3$.

Qasim Sadiq

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Problem 16

(II) The spacing between "nearest neighbor" Na and Cl ions in a NaCl crystal is 0.24 nm.What is the spacing between two nearest neighbor Na ions?

Ben Nicholson

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Problem 17

(I) A semiconductor is struck by light of slowly increasing frequency and begins to conduct when the wavelength of the light is 620 nm. Estimate the energy gap $E$g .

Dading Chen

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Problem 18

(I) Calculate the longest-wavelength photon that can cause an electron in silicon ($Eg =$ 1.12 eV) to jump from the valence band to the conduction band.

Ben Nicholson

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Problem 19

(II) The energy gap between valence and conduction bands in germanium is 0.72 eV. What range of wavelengths can a photon have to excite an electron from the top of the valence band into the conduction band?

Vishal Gupta

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Problem 20

(II) The band gap of silicon is 1.12 eV. (a) For what range of wavelengths will silicon be transparent? (See Example 29-5.) In what region of the electromagnetic spectrum does this transparent range begin? (b) If window glass is transparent for all visible wavelengths, what is the minimum possible band gap value for glass (assume $\lambda =$ 400nm to 700nm)? [$Hint:$ If the photon has less energy than the band gap, the photon will pass through the solid without being absorbed.]

Ben Nicholson

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Problem 21

(II) The energy gap $E$g in germanium is 0.72 eV. When used as a photon detector, roughly how many electrons can be made to jump from the valence to the conduction band by the passage of an 830-keV photon that loses all its energy in this fashion?

Qasim Sadiq

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Problem 22

(III) We saw that there are $2N$ possible electron states in the 3$s$ band of Na, where $N$ is the total number of atoms. How many possible electron states are there in the ($a$) 2$s$ band, ($b$) 2$p$ band, and ($c$) 3p band? ($d$) State a general formula for the total number of possible states in any given electron band.

Ben Nicholson

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Problem 23

(III) Suppose that a silicon semiconductor is doped with phosphorus so that one silicon atom in 1.5 $\times$ 10$^6$ is replaced by a phosphorus atom. Assuming that the "extra" electron in every phosphorus atom is donated to the conduction band, by what factor is the density of conduction electrons increased? The density of silicon is 2330 kg/m$^3$, and the density of conduction electrons in pure silicon is about 10$^{16}$ m$^{-3}$ at room temperature.

Qasim Sadiq

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Problem 24

(I) At what wavelength will an LED radiate if made from a material with an energy gap $E$g = 1.3 eV?

Ben Nicholson

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Problem 25

(I) If an LED emits light of wavelength $\lambda =$ 730 nm, what is the energy gap (in eV) between valence and conduction bands?

Qasim Sadiq

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Problem 26

(I) A semiconductor diode laser emits 1.3-$\mu$m light. Assuming that the light comes from electrons and holes recombining, what is the band gap in this laser material?

Ben Nicholson

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Problem 27

(II) A silicon diode, whose current-voltage characteristics are given in Fig. 29-30, is connected in series with a battery and a 960-$\Omega$ resistor. What battery voltage is needed to produce a 14-mA current?

Qasim Sadiq

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Problem 28

(II) An ac voltage of 120-V rms is to be rectified. Estimate very roughly the average current in the output resistor $R$ (= 31 k$\Omega$) for ($a$) a half-wave rectifier (Fig. 29-31), and ($b$) a full-wave rectifier (Fig. 29-32) without capacitor.

Ben Nicholson

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Problem 29

(III) Suppose that the diode of Fig. 29-30 is connected in series to a 180-$\Omega$ resistor and a 2.0-V battery. What current flows in the circuit? [$Hint$: Draw a line on Fig. 29-30 representing the current in the resistor as a function of the voltage across the diode; the intersection of this line with the characteristic curve will give the answer.]

Qasim Sadiq

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Problem 30

(III) Sketch the resistance as a function of current, for $V > 0$, for the diode shown in Fig. 29-30.

Ben Nicholson

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Problem 31

(III) A 120-V rms 60-Hz voltage is to be rectified with a full-wave rectifier as in Fig. 29-32, where $R = 33 \mathrm{k}\Omega$, and $C = 28 \mu$F. ($a$) Make a rough estimate of the average current. ($b$) What happens if $C = 0.1 \mu$F [$Hint:$ See Section 19-6.]

Qasim Sadiq

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Problem 32

(I) From Fig. 29-41, write an equation for the relationship between the base current ($I_B$), the collector current ($I_C$), and the emitter current ($I_E$ not labeled in Fig. 29-41). Assume $i_B = i_C = 0$.

Ben Nicholson

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Problem 33

(I) Draw a circuit diagram showing how a $pnp$ transistor can operate as an amplifier, similar to Fig. 29-41 showing polarities, etc.

Qasim Sadiq

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Problem 34

(II) If the current gain of the transistor amplifier in Fig. 29-41 is $\beta = i_C/i_B = 95$, what value must $R_C$ have if a 1.0-$\mu$A ac base current is to produce an ac output voltage of 0.42 V?

Ben Nicholson

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Problem 35

(II) Suppose that the current gain of the transistor in Fig. 29-41 is $\beta = i_C/i_B = 85$. If $R_C = 3.8 \mathrm{k}\Omega$, calculate the ac output voltage for an ac input current of 2.0 $\mu$A.

Qasim Sadiq

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Problem 36

(II) An amplifier has a voltage gain of 75 and a 25-k$\Omega$ load (output) resistance. What is the peak output current through the load resistor if the input voltage is an ac signal with a peak of 0.080 V?

Ben Nicholson

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Problem 37

(II) A transistor, whose current gain $\beta = i_C/i_B = 65$, is connected as in Fig. 29-41 with $R_B = 3.8 \mathrm{k}\Omega$ and $R_C = 7.8 \mathrm{k}\Omega$ Calculate ($a$) the voltage gain, and ($b$) the power amplification.

Qasim Sadiq

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Problem 38

Use the uncertainty principle to estimate the binding energy of the molecule by calculating the difference in kinetic energy of the electrons between (i) when they are in separate atoms and (ii) when they are in the molecule. Take $\Delta x$ for the electrons in the separated atoms to be the radius of the first Bohr orbit, 0.053 nm, and for the molecule take $\Delta x$ to be the separation of the nuclei, 0.074 nm. [$Hint$:Let $\Delta p \approx \Delta p_x$.]

Ben Nicholson

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Problem 39

The average translational kinetic energy of an atom or molecule is about $_{\mathrm{KE}} = \frac{3}{2} kT$ (see Section 13-9), where $k =$ 1.38 $\times$ 10$^{-23}$ J/K is Boltzmann's constant. At what temperature $T$ will be $_{\mathrm{KE}}$ on the order of the bond energy (and hence the bond easily broken by thermal motion) for ($a$) a covalent bond (say H$_2$ ) of binding energy 4.0 eV, and ($b$) a "weak" hydrogen bond of binding energy 0.12 eV?

Qasim Sadiq

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Problem 40

A diatomic molecule is found to have an activation energy of 1.3 eV. When the molecule is disassociated, 1.6 eV of energy is released. Draw a potential energy curve for this molecule.

Ben Nicholson

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Problem 41

In the ionic salt KF, the separation distance between ions is about 0.27 nm. ($a$) Estimate the electrostatic potential energy between the ions assuming them to be point charges (magnitude 1$e$). ($b$) When F "grabs" an electron, it releases 3.41 eV of energy, whereas 4.34 eV is required to ionize K. \ Find the binding energy of KF relative to free K and F atoms, neglecting the energy of repulsion.

Qasim Sadiq

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Problem 42

The rotational absorption spectrum of a molecule displays peaks about 8.9 $\times$ 10$^{11}$ Hz apart. Determine the moment of inertia of this molecule.

Ben Nicholson

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Problem 43

For O$_2$ with a bond length of 0.121 nm, what is the moment of inertia about the center of mass?

Qasim Sadiq

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Problem 44

Must we consider quantum effects for everyday rotating objects? Estimate the differences between rotational energy levels for a spinning baton compared to the energy of the baton. Assume the baton consists of a uniform 32-cm-long bar with a mass of 230 g and two small end masses, each of mass 380 g, and it rotates at 1.8 rev/s about the bar's center.

Ben Nicholson

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Problem 45

For a certain semiconductor, the longest wavelength radiation that can be absorbed is 2.06 mm. What is the energy gap in this semiconductor?

Qasim Sadiq

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Problem 46

When EM radiation is incident on diamond, it is found that light with wavelengths shorter than 226 nm will cause the diamond to conduct. What is the energy gap between the valence band and the conduction band for diamond?

Ben Nicholson

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Problem 47

The energy gap between valence and conduction bands in zinc sulfide is 3.6 eV. What range of wavelengths can a photon have to excite an electron from the top of the valence band into the conduction band?

Brian Francisco

Brian Francisco

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Problem 48

Most of the Sun's radiation has wavelengths shorter than 1100 nm. For a solar cell to absorb all this, what energy gap ought the material have?

Ben Nicholson

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Problem 49

A TV remote control emits IR light. If the detector on the TV set is not to react to visible light, could it make use of silicon as a "window" with its energy gap $Eg =$ 1.12 eV? What is the shortest-wavelength light that can strike silicon without causing electrons to jump from the valence band to the conduction band?

Qasim Sadiq

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Problem 50

Green and blue LEDs became available many years after red LEDs were first developed. Approximately what energy gaps would you expect to find in green (525 nm) and in blue (465 nm) LEDs?

Ben Nicholson

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Problem 51

Consider a monatomic solid with a weakly bound cubic lattice, with each atom connected to six neighbors, each bond having a binding energy of 3.4 $\times$ 10$^{-3}$ eV. When this solid melts, its latent heat of fusion goes directly into breaking the bonds between the atoms. Estimate the latent heat of fusion for this solid, in J/mol. [$Hint:$ Show that in a simple cubic lattice (Fig. 29-44), there are $three$ times as many bonds as there are atoms, when the number of atoms is large.]

Qasim Sadiq

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