Nuclear Atom and Atomic Nuclear Atom and Atomic SpectraSpectra

PhysicsMontwood High School

R. Casao

Line SpectraLine Spectra Heated materials emit light and different

materials emit different kinds of light. Toaster coils glow red when in operation;

match flames have a yellow color; and the flame from a gas stove is blue.

To analyze this light, we can use a prism or a diffraction grating to separate the various wavelengths in a beam of light into a spectrum.

If the light source is a hot solid (such as the filament of a light bulb) or liquid, the spectrum is continuous; light of all wavelengths is present.

Continuous Spectrum

If the source is a heated gas, such as the neon in an advertising sign, the spectrum includes only a few colors in the form of isolated sharp parallel lines.

A spectrum of sharp parallel lines is called an emission line spectrum and the lines are called spectral lines.

Line Emission Spectrum of Hydrogen Atoms

Line spectrum

Each spectral line corresponds to a definite wavelength and frequency.

Each element in its gaseous state has a unique set of wavelengths in its line spectrum.

Scientists use the line spectra to identify elements and compounds. For example: astronomers use line spectra to identify molecules in interstellar space.

While a heated gas selectively emits only certain wavelengths, a cool gas selectively absorbs certain wavelengths.

If we pass white light (continuous spectrum) light through a gas and look at the transmitted light, we find a series of dark lines corresponding to the wavelengths that have been absorbed.

This is called an absorption line spectrum. A given atom or molecule absorbs the

same characteristic set of wavelengths when its cool as it emits when heated.

Scientists use the absorption line spectra to identify substances in the same way they can use emission line spectra.

Why does a given kind of atom emit and absorb only certain very specific wavelengths?

Atoms are much smaller than wavelengths of visible light, so we can not actually see an atom using the light.

In the early 1900’s, Thomson believed that if an atom collided with another atom, as in a heated gas, each electron would oscillate around its equilibrium position with a characteristic frequency and emit electromagnetic radiation with that frequency.

If the atom were illuminated with light of many frequencies, each electron would selectively absorb only light whose frequency matched the electron’s natural oscillation frequency (resonance).

Rutherford suggested that electrons revolve in orbits around the nucleus, just like the planets revolve around the Sun.

Classical electromagnetic theory states that any accelerating electric charge (either oscillating or revolving) radiates electromagnetic waves.

An electron orbiting inside an atom would always have a centripetal acceleration toward the nucleus, and so should be emitting radiation at all times.

The energy of an orbiting electron should therefore decrease continuously, its orbit should become smaller and smaller, and it should spiral into the nucleus within a fraction of a second.

According to classical theory, the frequency of the electromagnetic waves emitted should equal the frequency of revolution.

As the electrons radiated energy, their angular speeds would change continuously and they would emit a continuous spectrum, not the line spectrum actually observed.

Rutherford’s model of electrons orbiting the nucleus, based on Newtonian mechanics and classical electromagnetic theory, makes three entirely wrong predictions about atoms:

1. They should emit light continuously.2. They should be unstable.3. The light they emit should have a

continuous spectrum.

According to Classical Physics: An orbiting electron is accelerating, so it should radiate

electromagnetic waves. The waves should carry away energy, so the electron

should lose energy and spiral inward. The electron’s angular speed should increase as its orbit shrank, so the frequency of the radiated waves should increase. Classical physics says that atoms should collapse within a fraction of a second and should emit light with a continuous spectrum as they do so.

Energy Levels and the Bohr Model of the Energy Levels and the Bohr Model of the AtomAtom

Bohr’s hypothesized that the emission line spectrum of an element tells us that atoms of that element emit photons with only certain frequencies f and certain specific energies E = h·f.

During the emission of a photon, the internal energy of the atom changes by an amount equal to the energy of the photon.

Each atom must be able to exist with only certain specific values of internal energy.

Eac atom has a set of possible energy levels, but it cannot have an energy intermediate between two levels.

All atoms of a given element have the same set of energy levels, but atoms of different elements have different sets.

If an atom is raised, or excited, to a high energy level, the excited atom can make a transition from one energy level to a lower energy level by emitting a photon with energy equal to the energy difference between the initial and final levels.

If Ei is the initial energy of the atom before a transition, Ef is the final energy after the transition, and the photon’s energy is h·f = (h·c)/λ, then conservation of energy gives:

i f

h ch f E E

For example: an excited lithium atom emits red light with wavelength λ = 671 nm. The corresponding photon energy is:

34 8

9

19

6.63 10 3 10

671 10

2.96 10 1.85

mx J s xh c sEx m

E x J eV

Emission line spectra show that many different wavelengths are emitted by each atom.

Each kind of atom must have a number of energy levels, with different spacing between each energy level.

Each wavelength in the spectrum corresponds to a transition between two specific energy levels of the atom.

Caution: the lines of an emission line spectrum are not all produced by a single atom. The spectrum from a gas shows the light emitted from all the different transitions that occurred in different atoms of the sample.

The observation that atoms are stable means that each atom has a lowest energy level, called the ground level.

Levels with energies greater than the ground level are called excited levels.

An atom in the excited level, called an excited atom, can make a transition into the ground level by emitting a photon.

Collisions are not the only way that an atom’s energy can be raised from one level to a higher level.

If an atom initially in the lower energy level is struck by a photon with just the right amount of energy, the photon can be absorbed and the atom willend up in the higher level.

In a previous example, we mentioned two levels in the lithium atom with an energy difference of 1.85 eV.– For a photon to be absorbed and excite the atom

from the lower level to the higher one, the photon must have an energy of 1.85 eV and a wavelength of 671 nm.

– An atom absorbs the same wavelengths that it emits.

This explains the correspondence between an element’s emission line spectrum and its absorption line spectrum.

A lithium atom cannot absorb a photon of wavelength 670 nm or 672 nm, its all or nothing, which explains why absorption line spectra have distinct dark lines.

Atoms can only absorb photons with specific wavelengths.

An atom that has been excited into a high energy level does not stay there long.

After a short time, called the lifetime of the level (about 10-8 s), the excited atom will emit a photon and make the transition into a lower excited level or the ground level.

To keep a gas of atoms glowing, energy must continually be provided to re-excite atoms so tht they can emit more photons.

Example: Emission and Absorption Example: Emission and Absorption SpectraSpectra

A hypothetical atom has energy levels at 0.0 eV (ground level), 1.0 eV, and 3.0 eV. What are the frequencies and wavelengths of the spectral lines this atom can emit when excited?

1415

1415

1415

8

614

8

614

8

14

1.02.42 10

4.136 10

2.04.84 10

4.136 10

3.07.25 10

4.136 10

3 101.24 10

2.42 10

3 106.2 10

4.84 10

3 10

7.25 10

E eVf x Hz

h x eV s

E eVf x Hz

h x eV s

E eVf x Hz

h x eV s

mxc s x mf x Hz

mxc s x mf x Hz

mxc sf x H

64.14 10x mz

What wavelengths can this atom absorb if it is in its ground level?– From the ground level, only a 1 eV or 3 eV photon

can be absorbed.– A 2 eV photon cannot be absorbed because the

atom has no energy level 2 eV above the ground level.

– Passing light from a hot solid through a gas of these hypothetical atoms (almost all of which would be in the ground state if the gas were cool) would yield a continuous spectrum with dark absorption lines at 1240 nm and 414 nm.

– If a gas of these atoms were at a sufficiently high temperature, collisions would excite a number of atoms into the 1 eV energy level and these excited atoms can absorb 2 eV photons and an absorption line at 620 nm would appear in the spectrum.

Suppose we take our hypothetical gas and illuminate it with violet light of wavelength 414 nm.– Atoms in the ground level can absorb this

photon and make a transition to the 3.0 eV level.

– Some of these atoms will make a transition back to the ground level by emitting a 414 nm photon.

– Other atoms will return to the ground level in two steps, first emitting a 620 nm photon to transition to the 1.0 eV level, then a 1240 nm photon to transition back to the ground level.

– Thus this gas will emit longer-wavelength radiation than it absorbs, a phenomenon called fluorescence.

Franck-Hertz Experiment: Are Energy Levels Real?

Franck and Hertz studied the motion of electrons through mercury vapor under the action of an electric field.

They found that when the electron kinetic energy was 4.9 eV or greater, the vapor emitted ultraviolet light of wavelength 250 nm.

Suppose mercury atoms have an excited energy level 4.9eV above the ground level.

An atom can be raised to this level by collision with an electron; it later decays back to the ground level by emitting a photon.

The wavelength of the photon should be:

This is equal to the wavelength Franck & Hertz measured, which demonstrates that this energy level actually exists in the mercury atom.

Electron Waves and the Bohr Model of Hydrogen

Bohr’s hypothesis established the relationship between atomic spectra and energy levels.

It did not provide general principles for predicting the energy levels of a particular atom.

15 8

74.136 10 3 10

2.5 10 2504.9

mx eV s xh c s x m nmE eV

Bohr postulated that each energy level of a hydrogen atom corresponds to a specific stable circular orbit of the electron around the nucleus.

Bohr postulated that an electron in such an orbit does not radiate (a break from electromagnetic theory).

Instead, an atom radiates energy only when an electron makes a transition from an orbit of energy Ei to a different orbit with lower energy Ef, emitting a photon of energy hf = Ei – Ef.

Relating angular frequency of the light emitted to the angular speed of the electron in highly excited energy levels, Bohr found that the electron’s angular momentum is quantized.

The angular momentum of an electron is an integral multiple of .

Number the orbits by integer n, where n = 1, 2, 3, … and call the radius of orbit n rn and the speed of the electron in that orbit vn.

The value of n for each orbit is called the principal quantum number for the orbit.

The angular momentum of an electron of mass m in such an orbit is

2h

2n n n

hL m v r n

deBroglie’s picture of electron waves can be used to justify Bohr’s equation.

Instead of visualizing the orbiting electron as a particle moving around the nucleus in a circular path, think of the orbiting electron as a sinusoidal standing wave with wavelength λ that extends around the circle.

A standing wave on a string transmits no energy, and electrons in Bohr’s orbit radiate no energy.

For the wave to come out even and join onto itself smoothly, the circumference of this circle must include a whole number of wavelengths as shown.

For an orbit of radius rn and circumference 2·λ·rn, we must have 2·λ·rn = n·λ, where n is the wavelength and n = 1, 2, 3, …

From the deBroglie wavelength equation, the wavelength of a particle with mass m moving with speed vn is

Substituting into the circumference equation:

nn

hm v

2 nn

hr n

m v

This matched Bohr’s results; a wave picture of the atom also leads to the quantization of the electron’s angular momentum.

Consider the hydrogen atom consisting of a single electron with mass m and charge –e in a circular orbit around a single proton with charge +e.

The centripetal acceleration of the electron about the proton is and the centripetal force needed

to produce this acceleration is: The electrical attraction between the proton

and the electron provides the centripetal force on the electron.

2n

n

vr 2

nc

n

vF m

r

22

2

22

2

2

2 2

2 2 2

2 2 2

2

2

into

2

4

nn n

n n

nn n

n n

n n

n n

n n

m vk e n hm v r

r r

k e rn hv v

m r m r

substitute v v

n h k em r m r

n h k em r m r

The orbital radii in the Bohr model):

2 2

2 24n

n hr

k m e

2 2 2

2 2 2

2 22

2

4

4

nn n

n

n h k ean m and r cancel

m r m r

n hk e

m r

Orbital speeds in the Bohr model:

The orbital radius rn is proportional to n2, so the smallest orbital radius corresponds to n = 1.

This minimum radius is called the Bohr radius, ao:

The orbital radius in the Bohr model can be rewritten as:

Permitted orbits have radii ao, 4·ao, 9·ao, …

2

2no

ev

n h

2

2o o

ha

m e

2n or n a

The radius ao of the smallest Bohr orbit is:

Orbital speed vn is proportional to 1/n.– The greater the value of n, the larger the orbital

radius of the electron and the slower its orbital speed.

– The greatest possible speed of the electron in the hydrogen atom occurs for n = 1.

Hydrogen Energy Levels in the Bohr Hydrogen Energy Levels in the Bohr ModelModel

The equations for the kinetic and potential energies when the electron is in the orbit with quantum number n:

2212 342

231 19

11

8.854 10 6.626 10

9.11 10 1.602 10

5.29 10

o

o

Cx x J sN ma

x kg x C

a x m

Potential energy has a negative sign because we have taken the electric potential energy to be zero when the electron is infinitely far from the nucleus.

We are interested only in the differences in energy between orbits, so the reference position doesn’t matter.

The total energy En is the sum of the kinetic and potential energies:

42

2 2 2

4 4

2 2 2

12 8

4

no

nn o

m eK m v

n h

k e m eU

r n h

Since En has a different value for each n, this equation gives the energy levels of the hydrogen atom in the Bohr model.

Each distinct orbit corresponds to a distinct energy level.

The figure depicts the orbits and energy levels.

4

2 28n n no

m eE K U

n h

For each value of n there is a corresponding value for the orbital radius rn, orbital speed vn, angular momentum , and total energy En.

The energy of the atom is least when n = 1 and En has its most negative value (the ground level of the hydrogen atom with radius ao).

For n = 2, 3, 4, … the absolute value of En is smaller and the energy is progressively larger (less negative).

The figure on the next slide shows some of the possible transitions from one electron orbit to an orbit of lower energy.

2n h

L

Transition from orbit nU (upper) to nL (lower) results in the energy

of the emitted photon of wavelength λ equal to

EnU - EnL

h cE

From the energy equation:

We rewrite the equation:

R is the Rydberg constant; R = 1.097 x 107/m.

Equation for the hydrogen wavelengths in the Bohr model, nL < nU:

4

2 28n n no

m eE K U

n h

4

2 2 38no

h c R m eE where R

n h c

2 2

1 1 1

L U

Rn n

When a hydrogen atom absorbs a photon, an electron makes a transition from a level nL to a higher level nU.– This can happen only if the photon energy is

equal to EnU – EnL.

– The wavelength equation also predicts the wavelengths found in the absorption line spectrum of hydrogen.

the Bohr model predicts the values of the observed wavelengths for electron transitions within the hydrogen atom.– Lyman series involves electron transitions

from nU = 2, 3, 4, … to nL = 1.

– Balmer series involves electron transitions from nU = 3, 4, 5, … to nL = 2.

Paschen series involves electron transitions from nU = 4, 5, 6, 7 to nL = 3.

Brackett series involves electron transitions from nU = 5, 6, 7 to nL = 4.

Pfund series involves electron transitions from nU = 6, 7 to nL = 5.

The figure shows only transitions in which a hydrogen atom loses energy and a photon is emitted.

The wavelengths of those photons that an atom can absorb are the same as those that it can emit.

The Bohr model also predicted the value of the ionization energy of the hydrogen atom.– Ionization energy is the energy required to

remove the electron completely from the atom.

– Ionization corresponds to the transition from the ground level (n = 1) to an infinitely large orbit (n = ), so the energy that must be added to the atom is E - E1 = 0- E1 = -E1 = -(-13.6 eV) = 13.6 eV.

Example: Exploring the Bohr Model Find the kinetic energy, potential energy,

and total energies of the hydrogen atom in the first excited level.

4 4

2 2 2 2 2

2

8 8no o

n

m e m eK but h c R

n h h

h c RK

n

4 4

2 2 2 2 2

2

4 8

2

no o

n

m e m eU but h c R

n h h

h c RU

n

For the first excited energy level (n = 2), and expressed in terms of h·c·R:

4 4

2 2 2 2 2

2

8 8no o

n

m e m eE but h c R

n h h

h c RE

n

34 8 7

18

6.626 10 3 10 1.097 10 /

2.179 10 13.6

mh c R x J s x x msh c R x J eV

2 2

2 2

2 2

13.6 13.63.4

22 13.6 2 13.6

6.82

13.6 13.63.4

2

n

n

n

ev evK eV

nev ev

U eVn

ev evE eV

n

Find the wavelength of the photon emitted in a transition from n = 2 to the ground level.– The energy of the emitted photon is E2 – E1.

15 8

2 1

7

4.136 10 3 10

3.4 13.6

1.22 10 122

mx eV s xh c sE E eV eV

x m nm