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

You are watching: Radius of hydrogen atom in meters


The great Danish physicist Niels Bohr (1885–1962) made prompt use of Rutherford’s planetary design of the atom. (Figure 1). Bohr came to be convinced the its validity and also spent part of 1912 at Rutherford’s laboratory. In 1913, after return to Copenhagen, he began publishing his concept of the easiest atom, hydrogen, based on the planetary design of the atom. Because that decades, many questions had actually been asked around atomic characteristics. From your sizes to your spectra, much was known about atoms, but tiny had been explained in regards to the laws of physics. Bohr’s theory defined the atomic spectrum of hydrogen and also established new and broadly applicable values in quantum mechanics.


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


Mysteries of atomic Spectra

As listed in Quantization that Energy, the energies of some small systems are quantized. Atomic and molecular emission and absorption spectra have actually been recognized for end a century to be discrete (or quantized). (See number 2.) Maxwell and also others had actually realized that there need to be a connection in between the spectrum of an atom and also its structure, something prefer the resonant frequencies of music instruments. But, in spite of years of efforts by many an excellent minds, no one had a workable theory. (It to be a running hoax that any type of theory the atomic and also molecular spectra could be destroyed by throwing a book of data at it, so complex were the spectra.) adhering to Einstein’s proposal of photons with quantized energies directly proportional to your wavelengths, it ended up being even much more evident that electrons in atoms have the right to exist only in discrete orbits.


Figure 2. Component (a) shows, indigenous left come right, a discharge tube, slit, and also diffraction grating developing a line spectrum. Component (b) reflects the emission heat spectrum because that iron. The discrete lines suggest quantized power states because that the atoms that produce them. The line spectrum for each aspect is unique, giving a powerful and lot used analysis tool, and many heat spectra were famous for plenty of years before they could be defined with physics. (credit because that (b): Yttrium91, Wikimedia Commons)


In part cases, it had actually been feasible to devise formulas that defined the emissions spectra. Together you might expect, the simplest atom—hydrogen, with its solitary electron—has a reasonably simple spectrum. The hydrogen spectrum had been it was observed in the infrared (IR), visible, and ultraviolet (UV), and several collection of spectral lines had been observed. (See number 3.) These collection are named after at an early stage researchers that studied them in particular depth.

The observed hydrogen-spectrum wavelengths deserve to be calculated utilizing the following formula:

displaystylefrac1lambda=Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\,

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

The continuous nf is a optimistic integer connected with a certain series. For the Lyman series, nf = 1; because that the Balmer series, nf = 2; because that the Paschen series, nf = 3; and so on. The Lyman series is totally in the UV, while component of the Balmer collection is visible v the remainder UV. The Paschen series and every the remainder are completely IR. Over there are apparently an unlimited variety of series, although they lie progressively farther right into the infrared and also become daunting to observe as nf increases. The constant ni is a confident integer, yet it have to be better than nf. Thus, because that the Balmer series, nf = 2 and ni = 3, 4, 5, 6, …. Keep in mind that ni can method infinity. When the formula in the wavelengths equation was simply a recipe designed to fit data and was not based on physical principles, that did imply a depth meaning. Balmer an initial devised the formula because that his collection alone, and also it was later uncovered to describe all the other series by using different values of nf. Bohr to be the an initial to comprehend the deeper meaning. Again, we view the interplay between experiment and theory in physics. Experimentally, the spectra were well established, an equation was uncovered to right the experimental data, yet the theoretical foundation was missing.


Figure 3. A schematic of the hydrogen spectrum mirrors several collection named for those who contributed most to their determination. Component of the Balmer series is in the clearly shows spectrum, if the Lyman series is totally in the UV, and also the Paschen collection and others are in the IR. Values of nf and ni are displayed for some of the lines.


Example 1. Calculating tide Interference that a Hydrogen Line

What is the distance between the slits the a grating that produces a first-order maximum for the second Balmer line at an angle of 15º?

Strategy and Concept

For an combined Concept problem, we must an initial identify the physical principles involved. In this example, we need to know two things:

the wavelength the lightthe conditions for one interference maximum for the pattern from a dual slit

Part 1 deals v a object of the existing chapter, while part 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 for ni = 3, and so the 2nd would have ni = 4.

The calculation is a straightforward applications of the wavelength equation. Entering the established values because that nf and ni yields

eginarraylllfrac1lambda&=&Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\ ext &=&left(1.097 imes10^7 ext m^-1 ight)left(frac12^2-frac14^2 ight)\ ext &=&2.057 imes10^6 ext m^-1endarray\

Inverting to discover λ gives

eginarrayllllambda&=½.057 imes10^6 ext m^-1=486 imes10^-9 ext m\ ext &=&486 ext nmendarray\

Discussion for Part 1

This is indeed the experimentally it was observed wavelength, equivalent to the second (blue-green) heat in the Balmer series. An ext impressive is the fact that the same an easy recipe predicts all the the hydrogen spectrum lines, including new ones observed in succeeding experiments. What is nature telling us?

Solution for Part 2

Double-slit interference (Wave Optics). To obtain constructive interference because that a twin slit, the path length difference from two slits need to be one integral lot of of the wavelength. This condition was express by the equation d sin θ =, where d is the distance between slits and θ is the angle from the initial direction of the beam. The number m is the order of the interference; m=1 in this example. Solving for d and entering recognized values yields

displaystyled=fracleft(1 ight)left(486 ext nm ight)sin15^circ=1.88 imes10^-6 ext m\

Discussion for part 2

This number is comparable to those supplied in the interference instances of advent to Quantum Physics (and is close to the spacing between slits in generally used diffraction glasses).


Bohr’s equipment for Hydrogen

Bohr had the ability to derive the formula because that the hydrogen spectrum using straightforward physics, the planetary version of the atom, and some an extremely important new proposals. His first proposal is the only particular orbits space allowed: we say the the orbits of electrons in atoms space quantized. Every orbit has a different energy, and electrons have the right to move come a greater orbit by taking in energy and drop come a lower orbit by emitting energy. If the orbits are quantized, the lot of energy absorbed or emitted is likewise quantized, creating discrete spectra. Photon absorption and also emission are among the primary methods the transferring energy into and also out that atoms. The energies the the photons space quantized, and also their energy is defined as being same to the change in energy of the electron once it moves from one orbit to another. In equation form, this is Δ= hfEi − Ef.


Figure 4. The planetary version of the atom, as modified by Bohr, has the orbits that the electrons quantized. Only specific orbits are allowed, explaining why atom spectra room discrete (quantized). The energy lugged away indigenous an atom by a photon originates from the electron dropping from one allowed orbit come another and is therefore quantized. This is an in similar way true for atomic absorption of photons.


Here, ΔE is the change in energy in between the initial and final orbits, and also hf is the energy of the soaked up or emitted photon. That is quite logical (that is, supposed from our everyday experience) that energy is affiliated in transforming orbits. A blast of energy is forced for the an are shuttle, for example, to climb to a greater orbit. What is not meant is the atomic orbits must be quantized. This is no observed because that satellites or planets, which deserve to have any orbit provided the appropriate energy. (See number 4.)

Figure 5 shows one energy-level diagram, a convenient way to display energy states. In the current discussion, us take this to be the allowed energy levels of the electron. Energy is plotted vertically with the lowest or ground state in ~ the bottom and also with excited claims above. Offered the energies that the currently in an atomic spectrum, that is possible (although sometimes an extremely difficult) to determine the power levels of an atom. Energy-level diagrams are used for countless systems, including molecules and also nuclei. A theory of the atom or any kind of other system must predict its energies based on the physics that the system.


Figure 5. One energy-level chart plots energy vertically and also is valuable in visualizing the power states the a system and the transitions in between them. This diagram is for the hydrogen-atom electrons, reflecting a transition between two orbits having energies E4 and also E2.


Bohr was clever enough to discover a means to calculation the electron orbital energies in hydrogen. This was crucial first step that has actually been enhanced upon, but it is fine worth repeating here, due to the fact that it go correctly explain many features of hydrogen. Suspect circular orbits, Bohr proposed that the angular momentum L of one electron in that is orbit is quantized, that is, it has actually only specific, discrete values. The value for L is given by the formula L=m_evr_n=nfrach2pileft(n=1,2,3,dots ight)\, where L is the angular momentum, me is the electron’s mass, rn is the radius of the n th orbit, and also h is Planck’s constant. Keep in mind that angular momentum is . For a tiny object in ~ a radius rImr2 and omega=fracvr\, so the L=left(mr^2 ight)fracvr=mvr\. Quantization states that this worth of mvr can only be same to frach2,frac2h2,frac3h2\, etc. In ~ the time, Bohr self did not know why angular momentum should be quantized, however using this presumption he was able to calculate the energies in the hydrogen spectrum, something no one else had actually done at the time.

From Bohr’s assumptions, us will now derive a number of important properties of the hydrogen atom from the classic physics we have covered in the text. We start by note the centripetal force causing the electron to monitor a circular course is offered by the Coulomb force. To be an ext general, we note that this evaluation is precious for any kind 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 dubbed a hydrogen-like atom. The spectra that hydrogen-like ions are comparable to hydrogen, however shifted to higher energy by the better attractive force between the electron and also nucleus. The size of the centripetal force is fracm_ev^2r_n\, if the Coulomb pressure is kfracleft(Zq_e ight)left(q_e ight)r_n^2\. The tacit assumption here is that the nucleus is much more massive than the stationary electron, and also the electron orbits about it. This is continual with the planetary design of the atom. Equating these,

kfracZq_e^2r_n^2=fracm_ev^2r_n ext (Coulomb = centripetal)\.

Angular momentum quantization is stated in an previously equation. We solve that equation because that v, substitute it into the above, and also rearrange the expression to attain the radius the the orbit. This yields:

displaystyler_n=fracn^2Za_ extB, ext for allowed orbits left(n=1,2,3dots ight)\,

where aB is defined to be the Bohr radius, since for the lowest orbit (= 1) and also for hydrogen (= 1), r1 = aB. The is left for this chapter’s Problems and Exercises to display that the Bohr radius is

displaystylea_ extB=frach^24pi^2m_ekq_e^2=0.529 imes10^-10 ext m\.

These last 2 equations deserve to be supplied to calculation the radii that the enabled (quantized) electron orbits in any kind of hydrogen-like atom. The is exceptional that the formula offers the correct dimension of hydrogen, i m sorry is measured experimentally to be really close come the Bohr radius. The earlier equation likewise tells united state that the orbital radius is proportional come n2, as illustrated in figure 6.


Figure 6. The allowed electron orbits in hydrogen have the radii shown. This radii were very first calculated through Bohr and are provided by the equation r_n=fracn^2Za_ extB\. The lowest orbit has actually the experimentally confirmed diameter of a hydrogen atom.


To get the electron orbital energies, we begin by noting the the electron energy is the sum of its kinetic and potential energy: En = KE + PE.

Kinetic power is the acquainted KE=frac12m_ev^2\, assuming the electron is not moving at relativistic speeds. Potential power for the electron is electrical, or PEqeV, where V is the potential due to the nucleus, i m sorry looks prefer a point charge. The nucleus has a positive charge Zqe ; thus, V=frackZq_er_n\, recalling an previously equation for the potential due to a allude charge. Because the electron’s charge is negative, we view that PE=-frackZq_er_n\. Beginning the expressions because that KE and PE, us find

displaystyleE_n=frac12m_ev^2-kfracZq_e^2r_n\.

Now we substitute rn and also v from earlier equations right into the above expression for energy. Algebraic manipulation yields

displaystyleE_n=-fracZ^2n^2E_0left(n=1,2,3,dots ight)\

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

displaystyleE_0=frac2piq_e^4m_ek^2h^2=13.6 ext eV\

Thus, because that hydrogen,

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


Figure 7. Energy-level diagram because that hydrogen reflecting the Lyman, Balmer, and also Paschen series of transitions. The orbit energies room calculated using the over equation, first derived through Bohr.


Figure 7 shows an energy-level diagram because that hydrogen that also illustrates how the miscellaneous spectral series for hydrogen are regarded transitions in between energy levels.

Electron total energies are negative, since the electron is bound come the nucleus, analogous to being in a feet without enough kinetic energy to escape. Together n ideologies infinity, the complete energy i do not care zero. This corresponds to a totally free electron v no kinetic energy, because rn it s okay very large for large n, and also the electric potential energy thus i do not care zero. Thus, 13.6 eV is necessary to ionize hydrogen (to walk from –13.6 eV to 0, or unbound), one experimentally proved number. Given an ext energy, the electron becomes unbound with some kinetic energy. Because that example, giving 15.0 eV come an electron in the floor state of hydrogen strips it from the atom and also leaves it v 1.4 eV that kinetic energy.

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

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

displaystylehf=left(13.6 ext eV ight)left(frac1n_ extf^2-frac1n_ exti^2 ight)\

Dividing both sides of this equation by hc gives an expression because that frac1lambda\:

displaystylefrachfhc=fracfc=frac1lambda=fracleft(13.6 ext eV ight)hcleft(frac1n_ extf^2-frac1n_ exti^2 ight)\

It deserve to be shown that

displaystyleleft(frac13.6 ext eVhc ight)=fracleft(13.6 ext eV ight)left(1.602 imes10^-19 ext J/eV ight)left(6.626 imes10^-34 ext J cdot ext s ight)left(2.998 imes10^8 ext m/s ight)=1.097 imes10^7 ext m^-1=R\

is the Rydberg constant. Thus, we have used Bohr’s presumptions to have the formula first proposed by Balmer years earlier as a recipe to fit speculative data.

displaystylefrac1lambda=Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\

We see that Bohr’s concept of the hydrogen atom answers the question regarding why this previously known formula defines the hydrogen spectrum. The is because the energy levels space proportional to frac1n^2\, whereby n is a non-negative integer. A downward change releases energy, and also so ni should be higher than nf. The various collection are those whereby the transitions end on a specific level. Because that the Lyman series, nf = 1—that is, all the transitions end in the soil state (see likewise Figure 7). For the Balmer series, nf = 2, or every the transitions end in the an initial excited state; and also so on. What was as soon as a cooking recipes is now based in physics, and something new is emerging—angular inert is quantized.

Triumphs and Limits that the Bohr Theory

Bohr did what no one had actually been may be to do before. Not just did he define the spectrum that hydrogen, he properly calculated the dimension of the atom from an easy physics. Several of his ideas are broadly applicable. Electron orbital energies are quantized in all atoms and also molecules. Angular inert is quantized. The electrons perform not spiral into the nucleus, as supposed classically (accelerated dues radiate, so the the electron orbits classically would decay quickly, and the electrons would sit top top the nucleus—matter would collapse). This are major triumphs.

But over there are borders to Bohr’s theory. It can not be used to multielectron atoms, even one as an easy as a two-electron helium atom. Bohr’s model is what we call semiclassical. The orbits are quantized (nonclassical) however are assumed to be straightforward circular jajalger2018.org (classical). As quantum mechanics to be developed, it came to be clear that there room no well-defined orbits; rather, there space clouds the probability. Bohr’s theory likewise did not describe that some spectral lines space doublets (split right into two) as soon as examined closely. Us shall examine countless of these elements of quantum mechanics in more detail, but it have to be preserved in mind that Bohr did not fail. Rather, that made very important measures along the path to higher knowledge and also laid the structure for every one of atomic physics the has because evolved.


PhET Explorations: Models the the Hydrogen Atom

How walk scientists figure out the framework of atom without looking in ~ them? try out different models through shooting irradiate at the atom. Inspect how the prediction of the design matches the speculative results.


*

Click to download the simulation. Run utilizing Java.


Section Summary

The planetary model of the atom pictures electrons orbiting the cell core in the means that planets orbit the sun. Bohr supplied the planetary model to develop the very first reasonable theory of hydrogen, the easiest atom. Atomic and molecular spectra room quantized, through hydrogen spectrum wavelengths provided by the formulafrac1lambda =Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\,where λ is the wavelength of 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 hopeful integers, and also ni need to be higher than nf.Bohr effectively proposed that the energy and radii that the orbits of electrons in atoms space quantized, with energy for transitions in between orbits provided by ∆= hf = Ei − Ef, where ∆E is the change in energy between the initial and also final orbits and hf is the energy of an took in or emitted photon. It is useful to plot orbital energies top top a upright graph called an energy-level diagram.Bohr proposed the the allowed orbits space circular and must have actually quantized orbital angular momentum provided by L=m_e extvr_n=nfrach2pi left(n=1, 2, 3 dots ight)\, where L is the angular momentum, rn is the radius of the nth orbit, and also h is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of one orbit is provided by r_n=fracn^2Za_ extBleft( extallowed orbits n=1, 2, 3, … ight)\Z is the atomic variety of an aspect (the number of electrons is has actually when neutral) and aB is characterized to it is in the Bohr radius, i m sorry is a_ extB=frach^24pi ^2m_e extkq_e^2= ext0.529 imes ext10^- ext10 ext m\.Furthermore, the energies that hydrogen-like atom are provided by E_n=-fracZ^2n^2E_0left(n=1, 2, 3 … ight)\, where E0 is the ground-state energy and is offered by E_0=frac2pi ^2q_e^4m_ek^2h^2= ext13.6 eV\.Thus, for hydrogen, E_n=-frac ext13.6 eVn^2left(n,=,1, 2, 3 … ight)\.The Bohr Theory provides accurate values for the power levels in hydrogen-like atoms, but it has been improved upon in number of respects.

Conceptual Questions

How do the allowed orbits for electrons in atom differ native the permitted orbits for planets approximately the sun? define how the correspondence principle uses here.Explain exactly how Bohr’s ascendancy for the quantization of electron orbital angular momentum differs from the yes, really rule.What is a hydrogen-like atom, and how space the energies and radii of its electron orbits concerned those in hydrogen?

Problems & Exercises

By calculating that is wavelength, present that the first line in the Lyman collection is UV radiation.Find the wavelength the the third line in the Lyman series, and also identify the form of EM radiation.Look up the worths of the amounts in a_ extB=frach^24pi ^2m_e extkq_e^2\ , and verify the the Bohr radius aB is 0.529 × 10−10 m.Verify that the floor state power E0 is 13.6 eV by making use of E_0=frac2pi ^2q_e^4m_ek^2h^2\.If a hydrogen atom has actually its electron in the = 4 state, exactly how much energy in eV is needed to ionize it?A hydrogen atom in one excited state can be ionized with less energy than when 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 the a hydrogen atom in the = 2 state follow to Bohr’s theory.Show the fracleft(13.6 exteV ight)hc=1.097 imes10^7 ext m=R\ (Rydberg’s constant), as questioned in the text.What is the smallest-wavelength heat in the Balmer series? Is it in the visible part of the spectrum?Show that the whole Paschen series is in the infrared component of the spectrum. To do this, you only have to calculate the shortest wavelength in the series.Do the Balmer and also Lyman collection overlap? come answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.(a) Which line in the Balmer series is the very first one in the UV component of the spectrum? (b) How plenty of Balmer collection lines are in the visible component of the spectrum? (c) How numerous are in the UV?A wavelength that 4.653 µm is it was observed in a hydrogen spectrum for a change that end in the nf = 5 level. What was ni because that the early stage level of the electron?A singly ionized helium ion has only one electron and also is denoted He+. What is the ion’s radius in the soil state contrasted to the Bohr radius that hydrogen atom?A beryllium ion through a single electron (denoted Be3+) is in an excited state v radius the very same as that of the soil state of hydrogen. (a) What is n for the Be3+ ion? (b) exactly how much energy in eV is needed to ionize the ion indigenous this excited state?Atoms deserve to be ionized by thermal collisions, such as at the high temperatures found in the solar corona. One together ion is C+5, a carbon atom with just a single electron. (a) through what element are the energies of its hydrogen-like levels higher than those the hydrogen? (b) What is the wavelength the the an initial line in this ion’s Paschen series? (c) What type of EM radiation is this?Verify Equations r_n=fracn^2Za_ extB\ and a_B=frach^24pi ^2m_ekq_e^2=0.529 imes10^-10 ext m\ using the approach stated in the text. That is, equate the Coulomb and also centripetal forces and then insert one expression for velocity native the condition for angular inert quantization.The wavelength that the 4 Balmer collection lines because that hydrogen are found to it is in 410.3, 434.2, 486.3, and 656.5 nm. What median percentage distinction is found in between these wavelength numbers and those suspect by frac1lambda=Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\? it is amazing just how well a straightforward formula (disconnected initially from theory) might duplicate this phenomenon.

Glossary

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

displaystylefrac1lambda =Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\

Rydberg constant: a physical consistent related to the atomic spectra v an established value of 1.097 × 107 m−1

double-slit interference: an experiment in which tide or corpuscle from a solitary source impinge upon 2 slits so that the resulting interference pattern may be observed

energy-level diagram: a diagram used to analysis the power level of electrons in the orbits of one atom

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

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

energies of hydrogen-like atoms: Bohr formula for energies the electron states in hydrogen-like atoms: E_n=-fracZ^2n^2E_0left(n= ext1, 2, 3,dots ight)\


Selected services to Problems & Exercises

1. displaystylefrac1lambda=Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)Rightarrow lambda =frac1Rleft;n_ exti=2,n_ extf=1\, therefore that

displaystylelambda =left(fracm1.097 imes ext10^7 ight)left=1 ext. ext22 imes ext10^-7 extm= ext122 nm\ , which is UV radiation.

3. eginarrayllla_ extB&=&frach^24pi ^2m_e extkZq_e^2\ ext &=&fracleft( ext6.626 imes ext10^- ext34 extJ cdot ext s ight)^24pi ^2left(9.109 imes ext10^- ext31 extkg ight)left(8.988 imes ext10^9 extNcdot extm^2/C^2 ight)left(1 ight)left(1.602 imes ext10^- ext19 extC ight)^2\ ext &=& ext0.529 imes ext10^- ext10 extmendarray\

5. 0.850 eV

7. 2.12 × 10−10 m

9. 365 nm; it is in the ultraviolet.

See more: Which Event Helped Increase Chinese Immigration To The United States, 1851

11. No overlap; 365 nm; 122 nm

13. 7

15. (a) 2; (b) 54.4 eV

17. displaystylefrac extkZq_e^2r_n^2=fracm_eV^2r_n\, so the displaystyler_n=frac extkZq_e^2m_eV^2=frac extkZq_e^2m_efrac1V^2\. Native the equation displaystylem_evr_n=nfrach2pi\, we can substitute for the velocity, giving:

displaystyler_n=frac extkZq_e^2m_ecdot frac4pi ^2m_e^2r_n^2n^2h^2\

so that

displaystyler_n=fracn^2Zfrach^24pi ^2m_e extkq_e^2=fracn^2Za_ extB\,

where

displaystylea_ extB=frach^24pi ^2m_e extkq_e^2\.