Describe the mysteries of atomic spectra.Explain Bohr’s theory of the hydrogen atom.Explain Bohr’s planetary version of the atom.Illustrate power state using the energy-level diagram.Describe the triumphs and limits of Bohr’s theory.

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The good Danish physicist Niels Bohr (1885–1962) made instant use the Rutherford’s planetary version of the atom. (Figure 1). Bohr became convinced the its validity and also spent component of 1912 at Rutherford’s laboratory. In 1913, after returning to Copenhagen, he started publishing his theory of the easiest atom, hydrogen, based upon the planetary version of the atom. Because that decades, plenty of questions had actually been asked around atomic characteristics. From your sizes to your spectra, much was known around atoms, but small had been defined in terms of the laws of physics. Bohr’s theory explained the atomic spectrum of hydrogen and also established brand-new and generally applicable values in quantum mechanics.


Figure 1. Niels Bohr, Danish physicist, used the planetary version of the atom to describe the atomic spectrum and size of the hydrogen atom. His many contributions come the development of atomic physics and quantum mechanics, his an individual influence on numerous students and also colleagues, and his personal integrity, specifically in the confront of Nazi oppression, earned him a significant place in history. (credit: Unknown Author, via Wikimedia Commons)


Mysteries of atom Spectra

As provided in Quantization the Energy, the energies that some tiny systems are quantized. Atomic and also molecular emission and absorption spectra have been known for over a century to it is in discrete (or quantized). (See figure 2.) Maxwell and also others had actually realized that there should be a connection between the spectrum of one atom and its structure, something like the resonant frequencies of musical instruments. But, regardless of years of initiatives by many great minds, no one had a workable theory. (It to be a running joke that any theory that atomic and molecular spectra can be ruined by throwing a publication of data in ~ it, so facility were the spectra.) adhering to Einstein’s proposal that photons v quantized energies straight proportional to their wavelengths, it ended up being even more evident the electrons in atoms can exist only in discrete orbits.


Figure 2. Part (a) shows, indigenous left to right, a discharge tube, slit, and diffraction grating producing a heat spectrum. Component (b) reflects the emission heat spectrum because that iron. The discrete lines imply quantized energy states for the atom that develop them. The heat spectrum because that each element is unique, giving a an effective and lot used analytical tool, and many line spectra were famous for countless years prior to they can be described with physics. (credit for (b): Yttrium91, Wikimedia Commons)


In part cases, it had been feasible to devise formulas that described the emissions spectra. As you might expect, the most basic atom—hydrogen, v its single electron—has a reasonably simple spectrum. The hydrogen spectrum had actually been observed in the infrared (IR), visible, and also ultraviolet (UV), and also several series of spectral lines had actually been observed. (See number 3.) These collection are called after early on researchers that studied castle in certain depth.

The it was observed hydrogen-spectrum wavelengths deserve to be calculated utilizing the complying with formula:

\displaystyle\frac1\lambda=R\left(\frac1n_\textf^2-\frac1n_\texti^2\right)\\,

where λ is the wavelength the the emitted EM radiation and also R is the Rydberg constant, identified by the experiment to be R = 1.097 × 107 / m (or m−1).

The consistent nf is a hopeful integer connected with a specific series. Because that the Lyman series, nf = 1; because that the Balmer series, nf = 2; because that the Paschen series, nf = 3; and also so on. The Lyman collection is completely in the UV, while part of the Balmer series is visible through the remainder UV. The Paschen collection and all the remainder are entirely IR. Over there are apparently an unlimited number of series, although they lie significantly farther into the infrared and become challenging to observe together nf increases. The continuous ni is a confident integer, but it have to be better than nf. Thus, because that the Balmer series, nf = 2 and also ni = 3, 4, 5, 6, …. Note that ni can method infinity. While the formula in the wavelengths equation was simply a recipe designed come fit data and was not based on physical principles, it did imply a deeper meaning. Balmer very first devised the formula for his series alone, and also it to be later found to describe all the other collection by using different values that nf. Bohr to be the an initial to comprehend the deeper meaning. Again, we watch the interplay in between experiment and theory in physics. Experimentally, the spectra were well established, one equation was discovered to fit the experimental data, however the theoretical structure was missing.


Figure 3. A schematic of the hydrogen spectrum mirrors several collection named because that those who contributed most to their determination. Part of the Balmer collection is in the visible spectrum, when the Lyman collection is completely in the UV, and the Paschen series and others room in the IR. Worths of nf and also ni are displayed for few of the lines.


Example 1. Calculating tide Interference the a Hydrogen Line

What is the distance in between the slits of a lattice that to produce a first-order maximum for the second Balmer heat at an edge of 15º?

Strategy and Concept

For an incorporated Concept problem, us must very first identify the physical ethics involved. In this example, we require to understand two things:

the wavelength of lightthe problems for one interference maximum for the sample from a twin slit

Part 1 deals through a topic of the present chapter, while component 2 considers the tide interference product of wave Optics.

Solution for Part 1

Hydrogen spectrum wavelength. The Balmer series requires the nf = 2. The first line in the series is taken to be because that ni = 3, and also so the second would have ni = 4.

The calculate is a straightforward applications of the wavelength equation. Entering the determined values for nf and also ni yields

\beginarraylll\frac1\lambda&=&R\left(\frac1n_\textf^2-\frac1n_\texti^2\right)\\\text &=&\left(1.097\times10^7\text m^-1\right)\left(\frac12^2-\frac14^2\right)\\\text &=&2.057\times10^6\text m^-1\endarray\\

Inverting to discover λ gives

\beginarraylll\lambda&=&\frac12.057\times10^6\text m^-1=486\times10^-9\text m\\\text &=&486\text nm\endarray\\

Discussion for Part 1

This is certainly the experimentally observed wavelength, matching to the second (blue-green) heat in the Balmer series. Much more impressive is the reality that the same basic recipe predicts all that the hydrogen spectrum lines, including new ones it was observed in subsequent experiments. What is nature informing us?

Solution for Part 2

Double-slit interference (Wave Optics). To attain constructive interference for a dual slit, the path length distinction from two slits must be one integral multiple of the wavelength. This problem was to express by the equation d sin θ =, where d is the distance in between slits and θ is the edge from the original direction of the beam. The number m is the stimulate of the interference; m=1 in this example. Solving for d and also entering known values yields

\displaystyled=\frac\left(1\right)\left(486\text nm\right)\sin15^\circ=1.88\times10^-6\text m\\

Discussion for component 2

This number is comparable to those offered in the interference instances of development to Quantum Physics (and is close come the spacing in between slits in typically used diffraction glasses).


Bohr’s systems for Hydrogen

Bohr was able to derive the formula because that the hydrogen spectrum using basic physics, the planetary version of the atom, and also some an extremely important brand-new proposals. His first proposal is the only specific orbits are allowed: we say that the orbits of electrons in atoms room quantized. Each orbit has actually a different energy, and electrons have the right to move come a higher orbit by taking in energy and also drop come a reduced orbit by emitting energy. If the orbits room quantized, the amount of energy took in or emitted is likewise quantized, creating discrete spectra. Photon absorption and also emission are amongst the major methods that transferring energy into and out of atoms. The energies the the photons space quantized, and also their energy is described as being equal to the adjust in power of the electron when it moves from one orbit come another. In equation form, this is Δ= hfEi − Ef.


Figure 4. The planetary model of the atom, as modified by Bohr, has the orbits the the electron quantized. Only specific orbits are allowed, explaining why atomic spectra space discrete (quantized). The energy brought away from an atom by a photon comes from the electron dropping from one allowed orbit to another and is thus quantized. This is likewise true for atomic absorb of photons.


Here, ΔE is the adjust in energy in between the initial and final orbits, and hf is the energy of the absorbed or emitted photon. The is quite logical (that is, meant from our day-to-day experience) that energy is connected in an altering orbits. A blast of energy is forced for the space shuttle, for example, to rise to a higher orbit. What is not intended is that atomic orbits have to be quantized. This is no observed for satellites or planets, which deserve to have any orbit provided the ideal energy. (See number 4.)

Figure 5 shows one energy-level diagram, a convenient method to display energy states. In the current discussion, us take these to it is in the allowed energy level of the electron. Power is plotted vertically through the shortest or ground state at the bottom and also with excited claims above. Offered the energies that the lines in an atomic spectrum, that is possible (although sometimes really difficult) to recognize the power levels of one atom. Energy-level diagrams are provided for numerous systems, consisting of molecules and also nuclei. A concept of the atom or any other mechanism must suspect its energies based on the physics that the system.


Figure 5. An energy-level chart plots power vertically and also is helpful in visualizing the energy states the a system and the transitions between them. This diagram is for the hydrogen-atom electrons, showing a shift between two orbits having actually energies E4 and also E2.


Bohr was clever enough to find a method to calculate the electron orbital energies in hydrogen. This was an important first step that has been boosted upon, but it is fine worth repeating here, since it go correctly define many attributes of hydrogen. Suspect circular orbits, Bohr proposed the the angular inert L of an electron in that is orbit is quantized, the is, it has actually only specific, discrete values. The value for L is offered by the formula L=m_evr_n=n\frach2\pi\left(n=1,2,3,\dots\right)\\, where L is the angular momentum, me is the electron’s mass, rn is the radius the the n th orbit, and h is Planck’s constant. Keep in mind that angular momentum is . For a tiny object at a radius rImr2 and \omega=\fracvr\\, so the L=\left(mr^2\right)\fracvr=mvr\\. Quantization claims that this worth of mvr have the right to only be equal to \frach2,\frac2h2,\frac3h2\\, etc. In ~ the time, Bohr self did not recognize why angular momentum have to be quantized, yet using this assumption he had the ability to calculate the energies in the hydrogen spectrum, something no one else had actually done at the time.

From Bohr’s assumptions, we will now derive a variety of important nature of the hydrogen atom native the classic physics we have covered in the text. We begin by noting the centripetal force causing the electron to follow a circular route is offered by the Coulomb force. Come be more general, we keep in mind that this analysis is valid for any type of single-electron atom. So, if a nucleus has actually Z proton (= 1 because that hydrogen, 2 because that helium, etc.) and also only one electron, that atom is called a hydrogen-like atom. The spectra that hydrogen-like ions are similar to hydrogen, however shifted to higher energy by the higher attractive force in between the electron and nucleus. The magnitude of the centripetal force is \fracm_ev^2r_n\\, if the Coulomb force is k\frac\left(Zq_e\right)\left(q_e\right)r_n^2\\. The tacit assumption here is that the cell nucleus is much more massive 보다 the stationary electron, and also the electron orbits about it. This is continuous with the planetary design of the atom. Equating these,

k\fracZq_e^2r_n^2=\fracm_ev^2r_n\text (Coulomb = centripetal)\\.

Angular momentum quantization is proclaimed in an earlier equation. We fix that equation for v, substitute it into the above, and also rearrange the expression to achieve the radius that the orbit. This yields:

\displaystyler_n=\fracn^2Za_\textB,\text for allowed orbits \left(n=1,2,3\dots\right)\\,

where aB is identified to be the Bohr radius, because for the lowest orbit (= 1) and also for hydrogen (= 1), r1 = aB. It is left for this chapter’s Problems and also Exercises to present that the Bohr radius is

\displaystylea_\textB=\frach^24\pi^2m_ekq_e^2=0.529\times10^-10\text m\\.

These last 2 equations can be offered to calculation the radii that the permitted (quantized) electron orbits in any type of hydrogen-like atom. The is outstanding that the formula provides the correct size of hydrogen, i beg your pardon is measure experimentally come be really close come the Bohr radius. The previously equation additionally tells united state that the orbit radius is proportional come n2, as shown in number 6.


Figure 6. The enabled electron orbits in hydrogen have actually the radii shown. These radii were first calculated through Bohr and are given by the equation r_n=\fracn^2Za_\textB\\. The shortest orbit has actually the experimentally showed diameter that a hydrogen atom.


To acquire the electron orbit energies, we begin by noting the the electron energy is the amount of its kinetic and potential energy: En = KE + PE.

Kinetic power is the familiar KE=\frac12m_ev^2\\, presume the electron is not relocating at relativistic speeds. Potential energy for the electron is electrical, or PEqeV, whereby V is the potential because of the nucleus, which looks like a suggest charge. The nucleus has a hopeful charge Zqe ; thus, V=\frackZq_er_n\\, recalling an previously equation because that the potential due to a allude charge. Because the electron’s charge is negative, we watch that PE=-\frackZq_er_n\\. Start the expressions for KE and PE, we find

\displaystyleE_n=\frac12m_ev^2-k\fracZq_e^2r_n\\.

Now us substitute rn and v from previously equations into the above expression for energy. Algebraic manipulation yields

\displaystyleE_n=-\fracZ^2n^2E_0\left(n=1,2,3,\dots\right)\\

for the orbit energies the hydrogen-like atoms. Here, E0 is the ground-state energy (= 1) for hydrogen (= 1) and also is given by

\displaystyleE_0=\frac2\piq_e^4m_ek^2h^2=13.6\text eV\\

Thus, because that hydrogen,

\displaystyleE_n=-\frac13.6\text eVn^2\left(n=1,2,3\dots\right)\\


Figure 7. Energy-level diagram for hydrogen reflecting the Lyman, Balmer, and Paschen collection of transitions. The orbit energies space calculated using the above equation, very first derived through Bohr.


Figure 7 shows an energy-level diagram for hydrogen that additionally illustrates just how the various spectral collection for hydrogen are related to transitions between energy levels.

Electron full energies are negative, because the electron is bound come the nucleus, analogous to being in a hole without sufficient kinetic power to escape. As n approaches infinity, the complete energy i do not care zero. This synchronizes to a totally free electron v no kinetic energy, because rn it s okay very huge for large n, and also the electrical potential power thus i do not care zero. Thus, 13.6 eV is needed to ionize hydrogen (to go from –13.6 eV come 0, or unbound), one experimentally verified number. Given an ext energy, the electron becomes unbound through some kinetic energy. For example, giving 15.0 eV come an electron in the floor state of hydrogen strips it from the atom and also leaves it through 1.4 eV that kinetic energy.

Finally, permit us take into consideration the energy of a photon emitted in a downward transition, given by the equation to it is in ∆E = hfEi − Ef.

Substituting En = (–13.6 eV/n2), we view that

\displaystylehf=\left(13.6\text eV\right)\left(\frac1n_\textf^2-\frac1n_\texti^2\right)\\

Dividing both sides of this equation through hc gives one expression because that \frac1\lambda\\:

\displaystyle\frachfhc=\fracfc=\frac1\lambda=\frac\left(13.6\text eV\right)hc\left(\frac1n_\textf^2-\frac1n_\texti^2\right)\\

It have the right to be displayed that

\displaystyle\left(\frac13.6\text eVhc\right)=\frac\left(13.6\text eV\right)\left(1.602\times10^-19\text J/eV\right)\left(6.626\times10^-34\text J \cdot\text s\right)\left(2.998\times10^8\text m/s\right)=1.097\times10^7\text m^-1=R\\

is the Rydberg constant. Thus, we have used Bohr’s assumptions to derive the formula first proposed through Balmer years earlier as a recipe to fit experimental data.

\displaystyle\frac1\lambda=R\left(\frac1n_\textf^2-\frac1n_\texti^2\right)\\

We see that Bohr’s theory of the hydrogen atom answer the question regarding why this previously known formula describes the hydrogen spectrum. The is due to the fact that the power levels are proportional to \frac1n^2\\, where n is a non-negative integer. A downward change releases energy, and also so ni have to be higher than nf. The various collection are those where the transitions end on a certain level. Because that the Lyman series, nf = 1—that is, every the transitions end in the ground state (see additionally Figure 7). For the Balmer series, nf = 2, or every the transitions finish in the an initial excited state; and also so on. What was as soon as a recipe is now based in physics, and something brand-new is emerging—angular momentum is quantized.

Triumphs and Limits of the Bohr Theory

Bohr go what no one had actually been maybe to carry out before. Not only did he explain the spectrum of hydrogen, he correctly calculated the size of the atom from straightforward physics. Some of his concepts are broadly applicable. Electron orbit energies are quantized in all atoms and also molecules. Angular inert is quantized. The electrons carry out not spiral into the nucleus, as intended classically (accelerated charges radiate, so the the electron orbits classically would decay quickly, and the electrons would sit top top the nucleus—matter would collapse). This are significant triumphs.

But there are boundaries to Bohr’s theory. It cannot be applied to multielectron atoms, also one as an easy as a two-electron helium atom. Bohr’s model is what we call semiclassical. The orbits room quantized (nonclassical) yet are assumed come be an easy circular courses (classical). Together quantum mechanics to be developed, it came to be clear that there are no well-defined orbits; rather, there are clouds of probability. Bohr’s theory also did not describe that part spectral lines room doublets (split right into two) when examined closely. Us shall examine countless of these aspects of quantum mechanics in an ext detail, but it have to be retained in mind the Bohr did no fail. Rather, the made very important steps along the route to greater knowledge and also laid the foundation for every one of atomic physics that has because evolved.


PhET Explorations: Models that the Hydrogen Atom

How go scientists figure out the structure of atoms without looking at them? shot out various models by shooting light at the atom. Check how the forecast of the version matches the speculative results.


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Click to download the simulation. Run utilizing Java.


Section Summary

The planetary design of the atom photos electrons orbiting the nucleus in the method that planets orbit the sun. Bohr used the planetary model to construct the an initial reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, v hydrogen spectrum wavelengths offered by the formula\frac1\lambda =R\left(\frac1n_\textf^2-\frac1n_\texti^2\right)\\,where λ is the wavelength the the emitted EM radiation and R is the Rydberg constant, which has actually the value = 1.097 × 107 m−1.The constants ni and also nf are confident integers, and ni have to be greater than nf.Bohr effectively proposed the the energy and also radii of the orbits of electron in atoms space quantized, with energy for transitions in between orbits given by ∆= hf = Ei − Ef, where ∆E is the readjust in energy between the initial and final orbits and also hf is the energy of an absorbed or emitted photon. That is helpful to plot orbit energies top top a vertical graph referred to as an energy-level diagram.Bohr proposed that the enabled orbits room circular and also must have quantized orbit angular momentum provided by L=m_e\textvr_n=n\frach2\pi \left(n=1, 2, 3 \dots \right)\\, where L is the angular momentum, rn is the radius the the nth orbit, and h is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is offered by r_n=\fracn^2Za_\textB\left(\textallowed orbits n=1, 2, 3, …\right)\\Z is the atomic variety of an facet (the variety of electrons is has actually when neutral) and also aB is identified to be the Bohr radius, which is a_\textB=\frach^24\pi ^2m_e\textkq_e^2=\text0.529\times \text10^-\text10\text m\\.Furthermore, the energies of hydrogen-like atom are given by E_n=-\fracZ^2n^2E_0\left(n=1, 2, 3 …\right)\\, where E0 is the ground-state energy and is given by E_0=\frac2\pi ^2q_e^4m_ek^2h^2=\text13.6 eV\\.Thus, because that hydrogen, E_n=-\frac\text13.6 eVn^2\left(n,=,1, 2, 3 …\right)\\.The Bohr Theory provides accurate values for the power levels in hydrogen-like atoms, however it has actually been enhanced upon in number of respects.

Conceptual Questions

How perform the permitted orbits for electrons in atoms differ native the permitted orbits for planets around the sun? explain how the correspondence principle uses here.Explain just how Bohr’s ascendancy for the quantization that electron orbit angular momentum different from the yes, really rule.What is a hydrogen-like atom, and how room the energies and also radii the its electron orbits pertained to those in hydrogen?

Problems & Exercises

By calculating the wavelength, present that the an initial line in the Lyman series is UV radiation.Find the wavelength that the 3rd line in the Lyman series, and identify the kind of EM radiation.Look up the values of the amounts in a_\textB=\frach^24\pi ^2m_e\textkq_e^2\\ , and also verify the the Bohr radius aB is 0.529 × 10−10 m.Verify that the soil state power E0 is 13.6 eV by making use of E_0=\frac2\pi ^2q_e^4m_ek^2h^2\\.If a hydrogen atom has its electron in the = 4 state, just how much energy in eV is needed to ionize it?A hydrogen atom in an excited state deserve to be ionized with less power than once it is in its soil state. What is n for a hydrogen atom if 0.850 eV of energy can ionize it?Find the radius of a hydrogen atom in the = 2 state follow to Bohr’s theory.Show the \frac\left(13.6 \texteV\right)hc=1.097\times10^7\text m=R\\ (Rydberg’s constant), as discussed in the text.What is the smallest-wavelength line in the Balmer series? Is that in the visible component of the spectrum?Show that the entire Paschen series is in the infrared component of the spectrum. To carry out this, you only need to calculate the shortest wavelength in the series.Do the Balmer and also Lyman series overlap? come answer this, calculate the shortest-wavelength Balmer line and also the longest-wavelength Lyman line.(a) Which line in the Balmer collection is the first one in the UV component of the spectrum? (b) How numerous Balmer series lines room in the visible part of the spectrum? (c) How countless are in the UV?A wavelength that 4.653 µm is it was observed in a hydrogen spectrum because that a change that ends in the nf = 5 level. What to be ni because that the early level the the electron?A singly ionized helium ion has actually only one electron and also is denoted He+. What is the ion’s radius in the ground state compared to the Bohr radius that hydrogen atom?A beryllium ion v a single electron (denoted Be3+) is in one excited state through radius the very same as the of the soil state of hydrogen. (a) What is n for the Be3+ ion? (b) how much power in eV is essential to ionize the ion native this excited state?Atoms have the right to be ionized by thermal collisions, such together at the high temperatures found in the solar corona. One such ion is C+5, a carbon atom with just a solitary electron. (a) through what element are the energies that its hydrogen-like levels higher than those that hydrogen? (b) What is the wavelength that the an initial line in this ion’s Paschen series? (c) What type of EM radiation is this?Verify Equations r_n=\fracn^2Za_\textB\\ and a_B=\frach^24\pi ^2m_ekq_e^2=0.529\times10^-10\text m\\ using the technique stated in the text. The is, equate the Coulomb and also centripetal forces and then insert an expression for velocity indigenous the problem for angular momentum quantization.The wavelength the the 4 Balmer collection lines because that hydrogen are uncovered to it is in 410.3, 434.2, 486.3, and 656.5 nm. What median percentage distinction is found between these wavelength numbers and also those guess by \frac1\lambda=R\left(\frac1n_\textf^2-\frac1n_\texti^2\right)\\? the is amazing exactly how well a an easy formula (disconnected originally from theory) can duplicate this phenomenon.

Glossary

hydrogen spectrum wavelengths: the wavelengths of visible light from hydrogen; can be calculate by

\displaystyle\frac1\lambda =R\left(\frac1n_\textf^2-\frac1n_\texti^2\right)\\

Rydberg constant: a physical constant related come the atom spectra v an created value of 1.097 × 107 m−1

double-slit interference: an experiment in which tide or particles from a single source impinge upon 2 slits so the the result interference pattern might be observed

energy-level diagram: a diagram provided to analysis the power level of electron in the orbits of an atom

Bohr radius: the median radius of the orbit of one electron roughly the cell nucleus of a hydrogen atom in its soil state

hydrogen-like atom: any atom with only a solitary electron

energies that hydrogen-like atoms: Bohr formula because that energies the electron states in hydrogen-like atoms: E_n=-\fracZ^2n^2E_0\left(n=\text1, 2, 3,\dots \right)\\


Selected services to Problems & Exercises

1. \displaystyle\frac1\lambda=R\left(\frac1n_\textf^2-\frac1n_\texti^2\right)\Rightarrow \lambda =\frac1R\left<\frac\left(n_\texti\cdotn_\textf\right)^2n_\texti^2-n_\textf^2\right>;n_\texti=2,n_\textf=1\\, so that

\displaystyle\lambda =\left(\fracm1.097\times \text10^7\right)\left<\frac\left(2\times1\right)^22^2-1^2\right>=1\text.\text22\times \text10^-7\textm=\text122 nm\\ , i beg your pardon is UV radiation.

3. \beginarrayllla_\textB&=&\frach^24\pi ^2m_e\textkZq_e^2\\\text &=&\frac\left(\text6.626\times \text10^-\text34\textJ \cdot\text s\right)^24\pi ^2\left(9.109\times \text10^-\text31\textkg\right)\left(8.988\times \text10^9\textN\cdot\textm^2/C^2\right)\left(1\right)\left(1.602\times \text10^-\text19\textC\right)^2\\\text &=&\text0.529\times \text10^-\text10\textm\endarray\\

5. 0.850 eV

7. 2.12 × 10−10 m

9. 365 nm; it is in the ultraviolet.

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11. No overlap; 365 nm; 122 nm

13. 7

15. (a) 2; (b) 54.4 eV

17. \displaystyle\frac\textkZq_e^2r_n^2=\fracm_eV^2r_n\\, so that \displaystyler_n=\frac\textkZq_e^2m_eV^2=\frac\textkZq_e^2m_e\frac1V^2\\. Native the equation \displaystylem_evr_n=n\frach2\pi\\, we can substitute for the velocity, giving:

\displaystyler_n=\frac\textkZq_e^2m_e\cdot \frac4\pi ^2m_e^2r_n^2n^2h^2\\

so that

\displaystyler_n=\fracn^2Z\frach^24\pi ^2m_e\textkq_e^2=\fracn^2Za_\textB\\,

where

\displaystylea_\textB=\frach^24\pi ^2m_e\textkq_e^2\\.