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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

Pls guys am having problem on these topics: latent heat of fusion, specific heat capacity and the sub topics under them.Pls who can help?
hamidat Reply
Thanks George,I appreciate.
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Most especially it is the calculatory aspects that is giving me issue, but with these new strength that you guys have given me,I will put in my best to understand it again.
you can bring up a question and let's see what we can do to it
the distance between two suasive crests of water wave traveling of 3.6ms1 is 0.45m calculate the frequency of the wave
Idris Reply
v=f×lemda where the velocity is given and lends also given so simply u can calculate the frequency
You are right my brother, make frequency the subject of formula and equate the values of velocity and lamda into the equation, that all.
lExplain what happens to the energy carried by light that it is dimmed by passing it through two crossed polarizing filters.
Christoper Reply
When light is reflected at Brewster's angle from a smooth surface, it is 100% polarizedparallel to the surface. Part of the light will be refracted into the surface.
What is specific heat capacity?
hamidat Reply
Specific heat capacity is the amount of heat required to raise the temperature of one (Kg) of a substance through one Kelvin
formula for measuring Joules
Rowshan Reply
I don't understand, do you mean the S.I unit of work and energy?
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What limits the Magnification of an optical instrument?
Naeem Reply
Lithography is 2 micron
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Prabesh Reply
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Mark Reply
An hydraulic press is a type of machine that is operated by different pressure of water on pistons.
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Patrock Reply
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sharon Reply
What is Boyles law
Pascal Reply
it can simple defined as constant temperature
Boyles law states that the volume of a fixed amount of a gas is inversely proportional to the pressure acting on in provided that the temperature is constant.that is V=k(1/p) or V=k/p
what is motion
Mua Reply
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Shuaibu Reply
the minimum thrust that an object must have in oder yo escape the gravitational pull
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Deena Reply
what are the laws of motion
Practice Key Terms 7

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