# 30.3 Bohr’s theory of the hydrogen atom  (Page 4/14)

 Page 4 / 14
$L={m}_{e}{\text{vr}}_{n}=n\frac{h}{2\pi }\left(n=1, 2, 3,\dots \right)\text{,}$

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 $L=\mathrm{I\omega }$ . For a small object at a radius $r,\phantom{\rule{0.25em}{0ex}}I={\text{mr}}^{2}$ and $\omega =v/r$ , so that $L=\left({\text{mr}}^{2}\right)\left(v/r\right)=\text{mvr}$ . Quantization says that this value of $\text{mvr}$ can only be equal to $h/2,\phantom{\rule{0.25em}{0ex}}2h/2,\phantom{\rule{0.25em}{0ex}}3h/2$ , 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 $Z$ protons ( $Z=1$ 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 ${m}_{e}{v}^{2}/{r}_{n}$ , while the Coulomb force is $k\left({\text{Zq}}_{e}\right)\left({q}_{e}\right)/{r}_{n}^{2}$ . 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,

$k\frac{{\text{Zq}}_{e}^{2}}{{r}_{n}^{2}}=\frac{{m}_{e}{v}^{2}}{{r}_{n}}\left(Coulomb = centripetal\right).$

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

${r}_{n}=\frac{{n}^{2}}{Z}{a}_{\text{B}}\text{, for allowed orbits}\left(n=1,2,3,\dots \right)\text{,}$

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

${a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}.$

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 ${n}^{2}$ , as illustrated in [link] .

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

${E}_{n}=\text{KE}+\text{PE}.$

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

#### Questions & Answers

Electric current is the flow of electrons
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each of two or more forms of the same element that contain equal numbers of protons but different numbers of neutrons in their nuclei, and hence differ in relative atomic mass but not in chemical properties; in particular, a radioactive form of an element. "some elements have only one stable isotope
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particles of light are like small packets of energy called photons, and flow or motion of photons is wave like
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light is just the energy of which photons emit
Matthew
the wave is how they travel
Matthew
photons do not emitt energy, they are energy. They are massless particles.
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Lalita
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Matthew
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a wave is a disturbance of which energy travels
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effective mass of photons only comes into picture when we consider it accelerating in gravitational field, mass of photon has no meaning as it is always travelling with speed of light and is never at rest. with that high speed, Energy and momentum are equivalent. and darkness is absense of photons.
Lalita
darkness is absense of light. not the presence of 'resting photons'. photons are never at rest.
Lalita
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this theory is presented in Einsteins theory of special relativity
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A.The velocity Vo for the streamline flow of liquid in a small tube depends on the radius r of the tube,the density and the viscosity iter of the liquid .use the dimensional analysis to obtain an expression for the velocity . B.Given that Vo =r square ×p all over 4×iter ×l
True
A.The velocity Vo for the streamline flow of liquid in a small tube depends on the radius r of the tube,the density (rho)and the viscosity (iter)of the liquid. Use the method of dimensional analysis to obtain an expression for the velocity . B.Given that Vo =r square x p all over 4 x iter x l
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Matthew, photons ARE light. there is no such thing as a photon that isn't moving. in fact the speed they move at is called C (for constant) in physics. through a vacuum they always travel at this speed no matter what. they can not slow down; except in another medium.
Brad
The reason why a photon can go at this speed is BECAUSE it had no mass. nothing can go this speed or faster because it needs to have no mass or negative mass. that's why it's called the constant.
Brad
when a photon hits something that is opaque, this is the only way to "stop"it. it isn't merely stopped but absorbed and turned into heat energy, then the remaining energy is reflected in different wavelengths. that reflection is what we call color. the darker something is, the less photons are ther
Brad
e. complete blackness is the absolute absence of photons altogether. I believe what you're referring to is not speed, but wavelength, which is indirectly proportional to the amount of energy a particular photon is made up of.
Brad
in order for a photon to have zero wavelength, it must (at least theoretically) have infinite energy.
Brad
about mass: you may have photons confused with electrons. elections have a mass so small that people say they are without mass, but they do. it is called electron mass or Me-.
Brad
you may also be getting electrons and photons confused because of the cherenkov effect. that is what happens when a particle travels faster than light IN THAT PARTICULAR MEDIUM. I emphasize that because no other particle besides photons can go the speed of c.
Brad
when a particle goes faster than light in a particular medium, a blue light is emitted, called cherenkov radiation. this is why nuclear reactors glow blue.
Brad
nuclear reactors release so much energy that when they emit electrons, those electrons are given enough energy to go faster than light in that medium (in this case water), releasing blue light. if you put the reactor in air or a vacuum, this effect wouldn't happen because the speed of light in air
Brad
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Brad
*if
Brad
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light waves can travel through a vacuum, and do not require a medium. In empty space, the wave does not dissipate (grow smaller) no matter how far it travels, because the wave is not interacting with anything else.
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linear motion
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