Chapter 22 (quanta to quarks chpt 1/5)
Discoveries from 1895 onwards saw the demise of classical physics and the rise of quantum theory o Quantum theory – objects can possess only certain discrete amounts of energy – packets/steps/levels rather than continuous Thomson proposed that electrons were constituents of the atom, Rutherford proposed a nuclear atom (an atom with a nucleus) and Bohr built on that by amalgamating quantum ideas into a classical model o Some quantum ideas were so strange that physicists didn’t want to apply it but found it was the only way to explain observations RUTHERFORD o Born a Kiwi, worked with J. J. Thomson in England who ID’d the electron as a component of the atom o Model had shifted from Dalton’s small indestructible sphere to Thomson’s ‘plum pudding’ model Neg electrons in a sphere of pos charge like the spots in a pudding o Alpha scattering experiment – showed that atoms were mostly space with a small nucleus Fire alpha particles at a thin gold foil, radon as source of alpha A very small fraction (1/8000) were deflected at greater than 90degrees Rutherford reaction to this large angle deflection ‘the most incredible event that has ever happened to me in my life. It was almost as incredible as if you had fired a fifteen inch shell at tissue paper and it came back and hit you’ Explained by proposing a nuclear atom Some alpha comes in and bounces around off a gold nucleus Rutherford showed that the charge that caused the deflection (as alpha particle is pos and so is nucleus therefore repulsion and deflection) was in a region 10,000 times smaller than the radius of the atom He concluded that most of the mass and pos charge was concentrated in a very small nucleus Ratio of proton to atom radius for hydrogen is about 2.5*10^5 If room is 10m as diameter of atom, nucleus would be 40microns o PROBLEM: is electrons in orbit as per Rutherford model, then they are accelerating and should be emitting em radiation. So the atom should be unstable Electrons should radiate and lose energy, spiral into centre and atom disintegrates BOHR o Danish, trained by Rutherford o Attempted to apply the quantum ideas of Planck and Einstein Einstein later showed that a problem of infinities will occur in any process where classical and quantum theories are linked o Photons are quanta of visible light
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Philip Zhu
o
In 1913 Bohr saw the Balmer Equation (an empirical equation – not theoretical but works) and this ‘made everything clear to him’ and he realised how electrons were arranged in the hydrogen atom (but not others) Balmer’s equation, Angstrom had measured the wavelengths of the four visible spectral lines of hydrogen (Balmer Series) and Balmer came up with an equation to relate them
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where b was empirically found to be 364.56nm Rydberg modified Balmer’s equation
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RH is Rydberg’s constant = 1.097X10^7 Spectra Continuous which comes from incandescent objects Bright line which come from excited gases Light emitted from an excited gas through a prism form a series of coloured lines – of those wavelengths of light emitted Absorption spectra of cool gases Cool gases absorb certain wavelengths and reemit it in all different directions, so dark lines on an otherwise continuous spectrum Aurora is the emission of light from excited air molecules (both atoms and ions) Excited by interactions with charged particles from the sun after intense solar activity Bohr’s Postulates Made when he received the Nobel prize in 1922 1. That electron exist in stationary states (when they do not emit any radiation) with an unexplained stability (later due to de Broglie et al) any permanent change in their motion must consist of a complete transition to another stationary state 2. No radiation is emitted from a stationary state. A transition between states will be accompanied by the emission or absorption of radiation Then made his quantisation condition An electron in a stationary state has an angular momentum that is an integral multiple of (h/2pi, Planck’s constant on 2pi) o And angular momentum, L =mvr Using these postulates (the quantum ideas of Planck and stationary states), together with the energy of electrons as per classical physics, we can derive a theoretical equation for the wavelengths of the spectral lines of hydrogen than agrees with Balmer’s empirical equation Bohr postulates + classical phys = theoretical eqn that matches Balmer’s empirical eqn = yay!
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Philip Zhu
Bohr: The maths – get energy for hydrogen per Rutherford and classical phys, impose bohr postulates, calc energies of stationary states and thus change in energy between stationary states and can then calc frequency of spectral lines o Sum of Ek and electrical potential energy is the total energy of the electron
Now the electrical force is
And this force provides the centripetal force so
where qe is the charge
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And the potential energy of the electron is
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Total energy is the sum of Ep +Ek
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this is the total classical energy of Rutherford’s
hydrogen Input Bohr’s quantisation condition, restricts electron to stationary states Angular momentum must be an integer multiple of Planck/2pi L=mvr=nh/2pi – n an integer – becomes the Principle Quantum Number Can rearrange for the radius of stationary states
mvr=nh/2pi is
Now from earlier
can get v2
And sub this into
So
So this is the radius of the nth shell Note that when n=1 o o o
Stable orbits at square multiples of the radius of the first
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Philip Zhu
o
Now back to the classical energy and impose this condition for the radii
and So
and So
Eg, since E1 for hydrogen = -13.6eV, E2=?
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=-3.4eV We can use this formula for the energy states along with the second postulate about the emission/absorption of photons as electrons move between shells, to get a formula for their energy and hence their wavelength, the emitted/absorbed photon carries the energy difference in the two levels Consider a jump from a higher to a lower state, eg fall from n=4 to n=1
, this is the energy of the emitted photon, E=hf So
take out a factor of -1 so the bracket switches
since c=fλ
, same form as Balmer’s
and when
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is calculated, it agrees with Rydberg’s RH
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Balmer’s equation is an empirical one, and not a theoretical one from Bohr agrees, this is a major achievement that is strong support of the Bohr model And the hydrogen atom is explained – we can calculate the wavelengths of the many spectral lines of hydrogen o Balmer series, visible spectrum from falls to n=2 (the first excited state) o Later series’ wavelengths all agreed with Bohr Lyman series, ultraviolet, transitions to ground state n=1 Paschen series, infrared, transitions to n=3 Brackett series, infrared, transitions to n=4 Pfund series, infrared, transitions to n=5 Limitations of Bohr model o Not possible to calculate the wavelengths of spectral lines for any other elements 4
Philip Zhu
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Only works for hydrogen Spectral lines are not of equal intensity, Bohr cannot explain Other lines called hyperfine lines, Bohr cannot explain When gas excited in mag field the spectral lines are split (Zeeman Effect), Bohr cannot explain Bohr model is a mix of quantum and classical physics and this itself is a problem
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Philip Zhu
