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But there are limits to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s model is what we call semiclassical . The orbits are quantized (nonclassical) but are assumed to be simple circular paths (classical). As quantum mechanics was developed, it became clear that there are no well-defined orbits; rather, there are clouds of probability. Bohr’s theory also did not explain that some spectral lines are doublets (split into two) when examined closely. We shall examine many of these aspects of quantum mechanics in more detail, but it should be kept in mind that Bohr did not fail. Rather, he made very important steps along the path to greater knowledge and laid the foundation for all of atomic physics that has since evolved.

Phet explorations: models of the hydrogen atom

How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results.

Models of the Hydrogen Atom

Section summary

  • The planetary model of the atom pictures electrons orbiting the nucleus in the way that planets orbit the sun. Bohr used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula
    1 λ = R 1 n f 2 1 n i 2 , size 12{ { {1} over {λ} } =R left ( { {1} over {n rSub { size 8{f} } rSup { size 8{2} } } } - { {1} over {n rSub { size 8{i} } rSup { size 8{2} } } } right )} {}
    where λ size 12{λ} {} is the wavelength of the emitted EM radiation and R size 12{R} {} is the Rydberg constant, which has the value
    R = 1.097 × 10 7 m −1 .
  • The constants n i size 12{n rSub { size 8{i} } } {} and n f size 12{n rSub { size 8{f} } } {} are positive integers, and n i must be greater than n f size 12{n rSub { size 8{f} } } {} .
  • Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by
    Δ E = hf = E i E f , size 12{ΔE= ital "hf"=E rSub { size 8{i} } - E rSub { size 8{f} } } {}
    where Δ E size 12{ΔE} {} is the change in energy between the initial and final orbits and hf size 12{ ital "hf"} {} is the energy of an absorbed or emitted photon. It is useful to plot orbital energies on a vertical graph called an energy-level diagram.
  • Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum given by
    L = m e vr n = n h 2 π n = 1, 2, 3 … ,
    where L size 12{L} {} is the angular momentum, r n size 12{r rSub { size 8{n} } } {} is the radius of the n th size 12{n"th"} {} orbit, and h size 12{h} {} is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is given by
    r n = n 2 Z a B (allowed orbits n = 1, 2, 3, ...),
    Z size 12{Z} {} is the atomic number of an element (the number of electrons is has when neutral) and a B size 12{a rSub { size 8{B} } } {} is defined to be the Bohr radius, which is
    a B = h 2 4 π 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" "." } {}
  • Furthermore, the energies of hydrogen-like atoms are given by
    E n = Z 2 n 2 E 0 n = 1, 2, 3 ... , size 12{ left (n=1, 2, 3 "." "." "." right )} {}
    where E 0 size 12{E rSub { size 8{0} } } {} is the ground-state energy and is given by
    E 0 = 2 q e 4 m e k 2 h 2 = 13.6 eV. size 12{E rSub { size 8{0} } = { {2π rSup { size 8{2} } q rSub { size 8{e} } rSup { size 8{4} } m rSub { size 8{e} } k rSup { size 8{2} } } over {h rSup { size 8{2} } } } ="13" "." 6" eV"} {}
    Thus, for hydrogen,
    E n = 13.6 eV n 2 size 12{E rSub { size 8{n} } = - { {"13" "." 6" eV"} over {n rSup { size 8{2} } } } } {} n = 1, 2, 3 ... . size 12{ left (n=1, 2, 3 "." "." "." right ) "." } {}
  • The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.

Conceptual questions

How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.

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Questions & Answers

what is a wave
Williams Reply
wave means. A field of study
aondohemba
what are Atoms
aondohemba
is the movement back and front or up and down
sani
how ?
aondohemba
wave is a disturbance that transfers energy through matter or space with little or no associated mass.
lots
A wave is a motion of particles in disturbed medium that carry energy from one midium to another
conist
an atom is the smallest unit( particle) of an element that bares it's chemical properties
conist
what is electromagnetic induction?
conist
How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?
Larissa Reply
How do you convert 0.0045kgcm³ to the si unit?
EDYKING Reply
how many state of matter do we really have like I mean... is there any newly discovered state of matter?
Falana Reply
I only know 5: •Solids •Liquids •Gases •Plasma •Bose-Einstein condensate
Thapelo
Alright Thank you
Falana
Which one is the Bose-Einstein
James
can you explain what plasma and the I her one you mentioned
Olatunde
u can say sun or stars are just the state of plasma
Mohit
but the are more than seven
Issa
list it out I wanna know
Cristal
what the meaning of continuum
Akhigbe Reply
What state of matter is fire
Thapelo Reply
fire is not in any state of matter...fire is rather a form of energy produced from an oxidising reaction.
Xenda
Isn`t fire the plasma state of matter?
Walter
all this while I taught it was plasma
Victor
How can you define time?
Thapelo Reply
Time can be defined as a continuous , dynamic , irreversible , unpredictable quantity .
Tanaya
unpredictable? but I can say after one o'clock its going to be two o'clock predictably!
Victor
how can we define vector
mahmud
I would define it as having a magnitude (size)with a direction. An example I can think of is a car traveling at 50m/s (magnitude) going North (direction)
Hanzo
as for me guys u would say time is quantity that measures how long it takes for a specific condition to happen e.g how long it takes for the day to end or how it takes for the travelling car to cover a km.
conist
what is the relativity of physics
Paul Reply
How do you convert 0.0045kgcm³ to the si unit?
flint
What is the formula for motion
Anthony Reply
V=u+at V²=u²-2as
flint
S=ut+½at
flint
they are eqns of linear motion
King
S=Vt
Thapelo
v=u+at s=ut+at^\2 v^=u^+2as where ^=2
King
hi
Mehadi
hello
King
Explain dopplers effect
Jennifer Reply
Not yet learnt
Bob
Explain motion with types
Bob
Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
Alabi Reply
Scalar quantity Because acceleration has only magnitude
Bob
acleration is vectr quatity it is found in a spefied direction and it is product of displcemnt
bhat
its a scalar quantity
Paul
velocity is speed and direction. since velocity is a part of acceleration that makes acceleration a vector quantity. an example of this is centripetal acceleration. when you're moving in a circular patter at a constant speed, you are still accelerating because your direction is constantly changing.
Josh
acceleration is a vector quantity. As explained by Josh Thompson, even in circular motion, bodies undergoing circular motion only accelerate because on the constantly changing direction of their constant speed. also retardation and acceleration are differentiated by virtue of their direction in
fitzgerald
respect to prevailing force
fitzgerald
What is the difference between impulse and momentum?
Manyo
Momentum is the product of the mass of a body and the change in velocity of its motion. ie P=m(v-u)/t (SI unit is kgm/s). it is literally the impact of collision from a moving body. While Impulse is the product of momentum and time. I = Pt (SI unit is kgm) or it is literally the change in momentum
fitzgerald
Or I = m(v-u)
fitzgerald
the tendency of a body to maintain it's inertia motion is called momentum( I believe you know what inertia means) so for a body to be in momentum it will be really hard to stop such body or object..... this is where impulse comes in.. the force applied to stop the momentum of such body is impulse..
Pelumi
Calculation of kinetic and potential energy
dion Reply
K.e=mv² P.e=mgh
Malia
K is actually 1/2 mv^2
Josh
what impulse is given to an a-particle of mass 6.7*10^-27 kg if it is ejected from a stationary nucleus at a speed of 3.2*10^-6ms²? what average force is needed if it is ejected in approximately 10^-8 s?
John
speed=velocity÷time velocity=speed×time=3.2×10^-6×10^-8=32×10^-14m/s impulse [I]=∆momentum[P]=mass×velocity=6.7×10^-27×32×10^-14=214.4×10^-41kg/ms force=impulse÷time=214.4×10^-41÷10^-8=214.4×10^-33N. dats how I solved it.if wrong pls correct me.
Melody
what is sound wave
Nworu Reply
sound wave is a mechanical longitudinal wave that transfers energy from one point to another
Ogor
its a longitudnal wave which is associted wth compresion nad rearfractions
bhat
what is power
PROMISE Reply
it's also a capability to do something or act in a particular way.
Kayode
Newton laws of motion
Mike
power also known as the rate of ability to do work
Slim
power means capabilty to do work p=w/t its unit is watt or j/s it also represents how much work is done fr evry second
bhat
what does fluorine do?
Cheyanne Reply
strengthen and whiten teeth.
Gia
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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