General Physics IIfaculty.chas.uni.edu/~shand/GP2LectureNotes/GP2_Chapter...Quantum-Mechanical Hydrogen Atom • The Bohr model worked very well for one-electron atoms such as hydrogen,

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General Physics II

Atoms

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Spectra• Spectra present the best clue

about the true nature of the atom.

• A gas comprised of atoms of a single element emits light when it is subjected to a high voltage in a gas discharge tube. When this light is passed through a diffraction grating, a spectrum of discrete wavelengths (lines) is seen. This is to be contrasted with the continuousspectrum obtained from sunlight or a heated solid. The spectrum from emitted light is called an emission spectrum.

spectrum (sodium)

Continuous Spectrum

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Spectra• Light from a source emitting a

continuous spectrum is passed through a cool elemental gas. If this light is passed through a diffraction grating, the resulting continuous spectrum is crossed by dark lines at specific wavelengths that have been absorbed by the gas. These dark lines make up the absorption spectrum of the gas.

• The lines of the absorption spectrum coincide with lines in the emission spectrum of the same gas. However, the absorption spectrum does nothave all the lines of the emission spectrum.

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Spectra• The discrete emission (and absorption) spectrum of an

element is characteristic of and unique to that element. Thus, a line spectrum is an atomic fingerprint – it can be used to identify an element (and therefore the atom that comprises the element).

spectrum (sodium)

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Workbook: Chapter 29, Question 2

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Atoms• Based upon experiments in which positively charged particles

were used to bombard thin gold foil, Rutherford developed the “solar system” model of the atom.

• The atom consists of a tiny positively charged nucleus at the center surrounded by orbiting negatively charged electrons. The overwhelming majority (> 99.9%) of the mass of the atom is due to the nucleus, even though the diameter of the nucleus is about 10-5 times that of the atom. Overall, the atom is neutral.

• Additional experiments showed that the nucleus consists of positively charged protons and uncharged neutrons. A neutron has a slightly greater mass than a proton, and each one has a mass about 1800 times that of an electron. The charge of a proton is +e and that of an electron is –e. Thus, neutral atoms have the same number of protons and electrons

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Atoms

• Isotopes have the same number of protons (Z) but different numbers of neutrons (N). The total number of neutrons and protons is called the mass number (A).

• Because isotopes have the same number of electrons, their chemical properties are virtually identical.

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Atoms• Rutherford’s model of the atom was incomplete because such

an atom could not be stable. The charged electrons accelerate as they move in circular orbits. However, an accelerating charged particle emits electromagnetic radiation. Thus, the orbiting electrons would continuously lose energy and spiral into the nucleus like a satellite falling to Earth.

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Workbook: Chapter 29, Question 5

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Bohr Model of the Hydrogen AtomBased upon The Rutherford model of the atom and Planck’s ideas about quantization of energies, Niels Bohr proposed a model of the hydrogen atom (the simplest atom) based upon the following postulates:

• The centripetal force holding the electron in orbit around the nucleus is due to the attractive electric force between them.

• The electron in the atom can occupy only certain discreteorbits for which the angular momentum has the quantized values Ln = n(h/2π), with the quantum number n = 1, 2, 3, …

• When the electron is in one of the allowed orbits, it does not emit radiation. (Thus, the allowed orbits are called stationary states.)

• The electron can make a transition between allowed orbits if the atom absorbs energy or emits energy (as a photon) equal to the energy difference between the initial and final states.

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Stationary States of Hydrogen Atom • Using the Bohr model and Newton’s laws, one can obtain

expressions for the allowed radii and energies. Before writing down these expressions, we note that the quantity h/2π occurs very often in quantum physics, so let us give it a name: “h-bar”. Its symbol is

• The allowed radii for the Bohr hydrogen atom are given by

• The Bohr radius is defined as

• So,

( /2 ).h π=

22 0

24 , 1,2,3,...nr n n

meπε⎛ ⎞

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

= =

2110

2 5.29 10 0.0529 nm.4 mBa

meπε −= ×= =

2 , 1,2,3,....n Br n a n= =

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Stationary States of Hydrogen Atom• Note that the radii are not equally spaced because the radius is

proportional to the square of n. • The speed of the electron decreases as n increases because

the angular momentum (second postulate).n n nL mv r n= =

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Stationary States of Hydrogen Atom• The allowed energies of the Bohr hydrogen atom are given by

• Note that the energies are all negative. This is because the electron is bound to the nucleus, so energy has to be suppliedto free the electron, i.e., give it zero total energy.

• The n = 1 state has the lowest (most negative) energy. This is the ground state. Unless excited by an external influence, the atom will be in the ground state. The ground-state energy is

So,

22

0

1 1 , 1,2,3,...4 2nB

eE nan πε

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

=− =

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0

1 13.6 eV.4 2 B

eE aπε

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

=− =−

12 2

13.6 eV, 1,2,3,...nEE nn n

= =− =

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Stationary States of Hydrogen Atom

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Workbook: Chapter 29, Question 10

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Spectrum of Hydrogen• For hydrogen to emit light, the

atoms must first be excited from the ground state. According to the fourth postulate, an atom can be excited to a higher-energy state if it absorbs an amount of energy equal to the difference between the energy of the initial state and that of the final state. The energy absorbed can be from a photon or due to a collision with another particle.

• For example, for a hydrogen atom to make a transition from the ground state (n = 1) to the n= 4 state, the energy required is

4 1 ( 0.85 eV) ( 13.6 eV) =12.75 eV.E E EΔ = − = − − −

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Spectrum of Hydrogen• This absorbed energy corresponds to one line in the absorption

spectrum of hydrogen. Note that the atoms of a relatively cool gas are overwhelmingly in the ground state. Thus, in the production of an absorption spectrum, the initial state is almost always the ground state.

• When an atom is in an excited state, it will rapidly return to the ground state by making one or more transitions to the ground state. (E.g., 4→1; 4→2 then 2→1, etc.) Each transition to a lower energy state is accompanied by the emission of a photon. Each photon has an energy equal to the difference between the energy of the initial state and that of the final state. Each emitted photon corresponds to a line in the emission spectrum of hydrogen.

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Workbook: Chapter 29, Question 11

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Calculation of Emitted Wavelengths

2 2 2 2

2 2

. ( for emission)

.

1 13.6 eV 13.6 eV 13.6 e

91.18

V 1 1 .

( /13.6 eV .1 1

)

n m

n mtransitionphoton

photon n mphoton

photon

n mphoton

E E E E n mE E Ef h h

f h hn m m nhcc

fm n

λ

λ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

→

→

=Δ = − >

−= =

− −= −

=

= −

= =−

2 2

(emissionn ; , 1,2,3..m. 1

)1

.

m n

n m m⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

> =−

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Spectral Series of Hydrogen

• For the Balmer series of lines, m = 2 (n > 2). The visible lines of the hydrogen spectrum are members of the Balmer series.

• For the Lyman series, m = 1. All these lines lie in the ultraviolet region.

2 2

91.18 nm. 1 1n m

m n

λ⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

→ =−

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Ionization of Hydrogen• The ionization energy of a hydrogen atom is the energy that

must be supplied to just free the electron when it is initially in the ground state. Thus, it is the energy required to boost the electron from the n = 1 level to the n = ∞ level.Thus,

1.0 ( 13.6 eV) 13.6 eV.

ionization

ionization

E E EE

∞Δ = −Δ = − − =

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Workbook: Chapter 29, Question 12

Textbook: Chapter 29, Problem 57

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Quantum-Mechanical Hydrogen Atom• The Bohr model worked very well for one-electron atoms

such as hydrogen, singly ionized helium, etc. It did not explain every property of these atoms, however. Worse still, the Bohr model completely failed for multielectron atoms.

• To satisfactorily explain the behavior of multielectron atoms, quantum mechanics is required.

• Quantum particles (matter waves) obey a wave equation called the Schrödinger equation. Solving this equation for the electrons in an atom gives the probable positions of each electron. Precise Bohr orbits are incorrect. (They violate the Uncertainty Principle.) The angular momenta of electron orbits obtained from the solution of the Schrödinger equation give values consistent with experiment. Bohr’s rule did not and so they must be incorrect.

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Quantum-Mechanical Hydrogen Atom• The Schrödinger atom is described by four quantum

numbers: n, l, m, and ms. The principal quantum number nroughly corresponds to Bohr’s quantum number. The quantum numbers l and m dictate the shape (electron position probabilities) of the orbital (no orbits anymore!). The quantum number ms determines magnetic properties.