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L = m e vr n = n h 2 π n = 1, 2, 3, , size 12{ left (n=1,2,3, dotslow right )} {}

where L is the angular momentum, m e is the electron’s mass, r n is the radius of the n th orbit, and h is Planck’s constant. Note that angular momentum is L = . For a small object at a radius r , I = mr 2 and ω = v / r , so that L = mr 2 v / r = mvr . Quantization says that this value of mvr can only be equal to h / 2, 2 h / 2, 3 h / 2 size 12{h/2,` 2h/2, `3h/2} {} , etc. At the time, Bohr himself did not know why angular momentum should be quantized, but using this assumption he was able to calculate the energies in the hydrogen spectrum, something no one else had done at the time.

From Bohr’s assumptions, we will now derive a number of important properties of the hydrogen atom from the classical physics we have covered in the text. We start by noting the centripetal force causing the electron to follow a circular path is supplied by the Coulomb force. To be more general, we note that this analysis is valid for any single-electron atom. So, if a nucleus has Z size 12{Z} {} protons ( Z = 1 size 12{Z=1} {} for hydrogen, 2 for helium, etc.) and only one electron, that atom is called a hydrogen-like atom    . The spectra of hydrogen-like ions are similar to hydrogen, but shifted to higher energy by the greater attractive force between the electron and nucleus. The magnitude of the centripetal force is m e v 2 / r n size 12{m rSub { size 8{e} } v rSup { size 8{2} } /r rSub { size 8{n} } } {} , while the Coulomb force is k Zq e q e / r n 2 size 12{k left ( ital "Zq" rSub { size 8{e} } right ) left (q rSub { size 8{e} } right )/r rSub { size 8{n} } rSup { size 8{2} } } {} . The tacit assumption here is that the nucleus is more massive than the stationary electron, and the electron orbits about it. This is consistent with the planetary model of the atom. Equating these,

k Zq e 2 r n 2 = m e v 2 r n (Coulomb = centripetal). size 12{k { { ital "Zq" rSub { size 8{e} } rSup { size 8{2} } } over {r rSub { size 8{n} } rSup { size 8{2} } } } = { {m rSub { size 8{e} } v rSup { size 8{2} } } over {r rSub { size 8{n} } } } } {}

Angular momentum quantization is stated in an earlier equation. We solve that equation for v size 12{v} {} , substitute it into the above, and rearrange the expression to obtain the radius of the orbit. This yields:

r n = n 2 Z a B , for allowed orbits size 12{r rSub { size 8{n} } = { {n rSup { size 8{2} } } over {Z} } a rSub { size 8{B} } } {} n = 1, 2, 3, , size 12{ left (n=1, 2, 3, dotslow right )} {}

where a B size 12{a rSub { size 8{B} } } {} is defined to be the Bohr radius    , since for the lowest orbit n = 1 size 12{ left (n=1 right )} {} and for hydrogen Z = 1 size 12{ left (Z=1 right )} {} , r 1 = a B size 12{r rSub { size 8{1} } =a rSub { size 8{B} } } {} . It is left for this chapter’s Problems and Exercises to show that the Bohr radius is

a B = h 2 2 m e kq e 2 = 0.529 × 10 10 m . size 12{a rSub { size 8{B} } = { {h rSup { size 8{2} } } over {4π rSup { size 8{2} } m rSub { size 8{e} } ital "kq" rSub { size 8{e} } rSup { size 8{2} } } } =0 "." "529" times "10" rSup { size 8{ - "10"} } " m"} {}

These last two equations can be used to calculate the radii of the allowed (quantized) electron orbits in any hydrogen-like atom . It is impressive that the formula gives the correct size of hydrogen, which is measured experimentally to be very close to the Bohr radius. The earlier equation also tells us that the orbital radius is proportional to n 2 size 12{n rSup { size 8{2} } } {} , as illustrated in [link] .

The electron orbits are shown in the form of four concentric circles. The radius of each circle is marked as r sub one, r sub two, up to r sub four.
The allowed electron orbits in hydrogen have the radii shown. These radii were first calculated by Bohr and are given by the equation r n = n 2 Z a B size 12{r rSub { size 8{n} } = { {n rSup { size 8{2} } } over {Z} } a rSub { size 8{B} } } {} . The lowest orbit has the experimentally verified diameter of a hydrogen atom.

To get the electron orbital energies, we start by noting that the electron energy is the sum of its kinetic and potential energy:

E n = KE + PE . size 12{E rSub { size 8{n} } =" KE "+" PE"} {}

Kinetic energy is the familiar KE = 1 / 2 m e v 2 size 12{ ital "KE"= left (1/2 right )m rSub { size 8{e} } v rSup { size 8{2} } } {} , assuming the electron is not moving at relativistic speeds. Potential energy for the electron is electrical, or PE = q e V size 12{ ital "PE"=q rSub { size 8{e} } V} {} , where V size 12{V} {} is the potential due to the nucleus, which looks like a point charge. The nucleus has a positive charge Zq e size 12{ ital "Zq" rSub { size 8{e} } } {} ; thus, V = kZq e / r n , recalling an earlier equation for the potential due to a point charge. Since the electron’s charge is negative, we see that PE = kZq e / r n size 12{ ital "PE"= - ital "kZq" rSub { size 8{e/r rSub { size 6{n} } } } } {} . Entering the expressions for KE size 12{ ital "KE"} {} and PE size 12{ ital "PE"} {} , we find

Questions & Answers

A charge of 1.6*10^-6C is placed in a uniform electric field in a density 2*5^10Nc^-1, what is the magnitude of the electric force exerted on the charge?
Omotosho Reply
what's phenomena
Enoch Reply
Phenomena is an observable fact or event.
Love
Prove that 1/d+1/v=1/f
James Reply
What interference
Moyinoluwa Reply
What is a polarized light called?
Moyinoluwa
what is a half life
Mama Reply
the time taken for a radioactive element to decay by half of its original mass
ken
what is radioactive element
mohammed
Half of the total time required by a radioactive nuclear atom to totally disintegrate
Justice
radioactive elements are those with unstable nuclei(ie have protons more than neutrons, or neutrons more than protons
Justice
in other words, the radioactive atom or elements have unequal number of protons to neutrons.
Justice
state the laws of refraction
Fabian
state laws of reflection
Fabian
Why does a bicycle rider bends towards the corner when is turning?
Mac
When do we say that the stone thrown vertically up wards accelerate negatively?
Mac
Give two importance of insulator placed between plates of a capacitor.
Mac
Macho had a shoe with a big sole moving in mudy Road, shanitah had a shoe with a small sole. Give reasons for those two cases.
Mac
when was the name taken from
Biola Reply
retardation of a car
Biola
when was the name retardation taken
Biola
did you mean a motion with velocity decreases uniformly by the time? then, the vector acceleration is opposite direction with vector velocity
Sphere
Atomic transmutation
Basirat Reply
An atom is the smallest indivisible particular of an element
mosco Reply
what is an atomic
Awene Reply
reference on periodic table
Titus Reply
what Is resonance?
Mozam Reply
phenomena of increasing amplitude from normal position of a substance due to some external source.
akif
What is a black body
Amey Reply
Black body is the ideal body can absorb and emit all radiation
Ahmed
the emissivity of black body is 1. it is a perfect absorber and emitter of heat.
Busayo
Why is null measurement accurate than standard voltmeter
Neemat Reply
that is photoelectric effect ?
Sabir Reply
It is the emission of electrons when light hits a material
Anita
Yeah
yusuf
is not just a material
Neemat
it is the surface of a metal
Neemat
what is the formula for time of flight ,maxjmum height and range
agangan Reply
what is an atom
Awene
an atom is the smallest particle of a element which can take part in chemical reaction.
Israel
Practice Key Terms 7

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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