the hydrogen spectrum and the bohr model

8/13/2019 The Hydrogen Spectrum and the Bohr Model

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The Hydrogen Spectrum and the Bohr Model

In formulating his quantum theory, Planck was influenced by the results of investigations of the

emission spectra produced by free (gas-phase) atoms, results that

led him to question whether any spectrum, even that of an incandescent lightbulb,

was truly continuous. Among these earlier results was a discovery made in 1885

by a Swiss mathematician and schoolteacher named Johann Balmer (18251898).

The Hydrogen Emission Spectrum

Balmer determined that the frequencies of the four brightest lines in the visible region of the

emission spectrum of hydrogen (Figure 7.9a) fit the simple

equation

(7.6) n=A3.2881*10

15

s

-1

Ba

1

2

2

-1

n

2

b

CONCEPT TEST

322| Chapter 7| Electrons in Atoms and Periodic Properties

where nis a whole number greater than 2specifically, 3 for the red line, 4 for the

green, 5 for the blue, and 6 for the violet. Without having seen any other lines in

the hydrogen emission spectrum, Balmer predicted there should be at least one

more (n 7) at the edge of the violet region, and indeed such a line was later


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discovered.

Balmer also predicted that hydrogen emission lines should exist in regions outside the visible range,

lines corresponding to frequencies calculated by replacing

1/2

2

in Equation 7.6 with 1/1

2

, 1/3

2

, 1/4

2

, and so forth. He was right. In 1908

German physicist Friedrich Paschen (18651947) discovered hydrogen emission

lines in the infrared region, corresponding to 1/3

2

instead of 1/2

2

in Balmers equation. A few years later, Theodore Lyman (18741954) at Harvard University

discovered hydrogen emission lines in the UV region corresponding to 1/1

2

. By the

1920s the 1/4

2

and 1/5

2

series of emission lines had been discovered. Like the 1/3

2

lines, they are in the infrared region.


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Later, Swedish physicist Johannes Robert Rydberg (18541919) revised

Balmers equation by changing frequency to wave number(1/l), which is the number of wavelengths

per unit of distance. Rydbergs equation is

(7.7)

where n1is a whole number that remains fixed for a series of emission lines and

where n2

is a whole number equal to n1 1,n1 2,...,for successive bright lines

in the spectrum.

When Balmer and Rydberg derived their equations describing the hydrogen

spectrum, they didnt know why the equations worked.The discrete frequencies of

hydrogens emission lines indicated that only certain levels of internal energy were

available in hydrogen atoms. However, classical (macroscale) physics could not

explain the existence of these internal energy levels. A new model was needed that

works at the atomic level.

1

l

=C1.097*10

-2

(nm)

-1

Da

1

n1

2

-1

n2

2

b


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SAMPLE EXERCISE 7.4 Calculating the Wavelength of a Line

in the Hydrogen Emission Spectrum

In the visible portion of the hydrogen emission spectrum (Figure 7.9a), what

is the wavelength of the bright line corresponding to n 3 in Equation 7.6?

Collect and OrganizeWe are to calculate the wavelength of a line in hydrogen emission spectrum

given the nvalues of its initial and final energy

levels. Equation 7.7 relates what we know (nvalues) to what we seek (wavelength of light).

AnalyzeWe know that the emission line is in the visible region, so we should

obtain a wavelength between 400 and 750 nanometeres. In Equation 7.7,n2

must be greater than n1

. Fitting this requirement to the given nvalues, we let

n2 3 and n1 2.

Solve

l=656 nm

=C1.097*10

-2

(nm)

-1

D(0.1389) =1.524*10

-3

(nm)

-1

1

l

=C1.097*10

-2

(nm)

-1


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Da

1

2

2

-1

3

2

b =C1.097*10

-2

(nm)

-1

Da

1

4

-1

9

b

7.4 | The Hydrogen Spectrum and the Bohr Model | 323

Think about ItThe calculated wavelength is in the visible region of the

electromagnetic spectrum, so the answer is reasonable.

Practice ExerciseWhat is the wavelength, in nanometers, of the line in

the hydrogen spectrum corresponding to n 4 in Equation 7.6?

The Bohr Model of Hydrogen

Scientists in the early 20th century faced yet another dilemma. Ernest Rutherford

had established a model of the atom that was mostly empty space occupied by negatively charged

electrons with a tiny nucleus containing virtually all the mass and

all the positive charge. What kept the electrons from falling into the nucleus?

Rutherford had suggested that the electrons had to be in motion and hypothesized


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that they might orbit the nucleus the way planets orbit the sun. However, classical physics predicted

that negative electrons orbiting a positive nucleus would emit

energy in the form of electromagnetic radiation and eventually spiral into the nucleus. This does not

happen!

The problem of explaining why a hydrogen atoms electron is not pulled into

its nucleus was addressed by Danish physicist Niels Bohr (18851962), who was

well acquainted with the issue because he had studied with Rutherford. Bohr

designed a theoretical model based on the electron in a hydrogen atom traveling around the nucleus

in one of an array of concentric orbits. Each orbit represents an allowed energy level and is

designated by the value of nas shown in

Equation 7.8:

(7.8)

where n 1,2,3,...,q. In the Bohr model an electron in the orbit closest to

the nucleus (n 1) has the lowest energy:

The next closest orbit has an nvalue of 2 and an electron in it has an energy of

Note that this value is less negative than the value for the electron in the n 1

orbit. As the value of nincreases, the radius of the orbit increases and so, too, does

the energy of an electron in the orbit; its value becomes less negative. As n

approaches q,Eapproaches zero:

Zero energy means that the electron is no longer part of the atom. In other words,

the H atom has become a H

ion and a free electron.

An important feature of the Bohr model is that it provides a theoretical

framework for explaining the experimental observations of Balmer, Rydberg,

and others. To see the connection, consider what happens when an electron

moves between two allowed energy levels in Bohrs model. If we label the initial

energy level (the level where the electron starts) ninitial, and we label the second

E=-2.178*10


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-18

J a

1

q2

b =0

E=-2.178*10

-18

J a

1

2

2

b =-5.445*10

-19

J

E=-2.178*10

-18

J a

1

1

2

b =-2.178*10

-18

J

E=-2.178*10

-18

J a

1


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n

2

b

CHEMTOUR Bohr Model of the Atom

CONNECTION We discussed

Rutherfords gold-foil experiment and the

development of the idea of the nuclear atom

in Section 2.1.

324| Chapter 7| Electrons in Atoms and Periodic Properties

level (the level where the electron ends up) nfinal, then the change in energy of the

electron is

(7.9)

If the electron moves to an orbit farther from the nucleus, then nfinal ninitial, and

the value of the terms inside the parentheses in Equation 7.9 is negative because

. This negative value multiplied by the negative coefficient gives us a

positive Eand represents an increase in electron energy. On the other hand, if an

electron moves from an outer orbit to one closer to the nucleus, then nfinal ninitial

,

and the sign of Eis negative. This means the electron loses energy.

When the electron in a hydrogen atom is in the lowest (n 1) energy level,

the atom is said to be in its ground state. If the electron in a hydrogen atom is in

an energy level above n 1, then the atom is said to be in an excited state.

According to the Bohr model the hydrogen electron can move from the n 1

(ground state) energy level to a higher level (for example,n 3) by

absorbing a quantity of energy ( E) that exactly matches the energy

difference between the two states. Similarly, an electron in an excited

state can move to an even higher energy level by absorbing a quantity of energythat exactly matchesthe energy difference between the


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two excited states. An electron in an excited state can also move to a

lower-energy excited state, or to the ground state, by emitting a quantity of energy that exactly

matches the energy difference between those

two states. This type of electron movement is called an electron

transition.

Energy-level diagrams show the transitions from one energy level

to another that electrons in atoms can make. Figure 7.16 is such a diagram for the hydrogen atom.

The black arrow pointing upward represents absorption of sufficient energy to completely remove

the

electron from a hydrogen atom (ionization).The downward-pointing

colored arrows represent decreases in the internal energy of the hydrogen atom that occur when

photons are emitted as the electron moves

from a higher-energy level to a lower-energy level. If the colored

arrows pointed up, they would represent absorptionof photons leading to increases in the internal

energy of the atom. In every case the

energy of the photon matches the absolute value of E.

If you compare Equation 7.9 with Equation 7.7, you will see that

they are much alike. The coefficients differ only because of the different units used to express wave

number and energy.The key point is that

the equation developed to fit the absorption and emission spectra of

hydrogen has the same form as the theoretical equation developed by

Bohr to explain the internal structure of the hydrogen atom. Thus, atomic emission

and absorption spectra reveal the energies of electrons inside atoms.

Based on the lengths of the arrows in Figure 7.16, rank the following transitions

in order of greatest change in electron energy to the smallest change:

a. n 4 n 2b.n 3 n 2

c. n2 n1d.n4 n3

CONCEPT TEST

1


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nfinal

2 6

1

ninitial

2

E=-2.178*10

-18

J a

1

nfinal

2

-1

ninitial

2

b

ground statethe most stable, lowest

energy state available to an atom or ion.

excited state any energy state in an atom

or ion above the ground state.

electron transitionmovement of an electron between energy levels.

Lyman

(ultraviolet

wavelengths)

n= 1

n= 2

n= 3

n= 4


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n= 5

n= 6

n=

Balmer

(visible

wavelengths)

Paschen

(infrared

wavelengths)

Ionization

FIGURE 7.16An energy-level diagram

showing electron transitions for the electron in

the hydrogen atom. The arrow pointing up

represents ionization. Arrows pointing down

represent the electron emitting energy and

falling to a lower energy level. This diagram

shows several possible transitions for the single

electron in hydrogen; each arrow does not

represent a different electron in the atom.

7.4 | The Hydrogen Spectrum and the Bohr Model | 325

SAMPLE EXERCISE 7.5 Calculating the Energy Needed

for an Electron Transition

How much energy is required to ionize a ground-state hydrogen atom?

Collect and OrganizeWe are asked to determine the energy required to

remove the electron from a hydrogen atom in its ground state. Equation 7.9

enables us to calculate the energy change associated with any electron

transition.


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AnalyzeTo use Equation 7.9, we need to identify the initial (ninitial

) and final

(nfinal) energy levels of the electron. The ground state of a H atom corresponds

to the n 1 energy level. If the atom is ionized,n qand the electron is no

longer associated with the nucleus.

Solve

Dividing by q2

yields zero, so the difference in parentheses simplifies to 1,

which gives

E 2.178 10

18

J

Think about ItThis is a small amount of energy, but it is comparable to the

work function values (see Sample Exercise 7.3) which involved removing an

electron from a metal surface. The sign of Eis positive because energy must

be added to remove an electron from the atom and away from its positive

nucleus.

Practice ExerciseCalculate the energy, in joules, required to ionize a hydrogen atom when its

electron is initially in the n 3 energy level. Before doing

the calculation, predict whether this energy is greater than or less than the

2.178 10

18

J needed to ionize a ground-state hydrogen atom.

=-2.178*10

-18

J a

1

q2


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-1

1

2

b

E=-2.178*10

-18

J a

1

nfinal

2

-1

ninitial

2

b

One of the strengths of the Bohr model of the hydrogen atom is that it

accurately predicts the energy needed to remove the electron. This energy is

called the ionization energy of the hydrogen atom. We examine the ionization

energies of other elements in Section 7.11. However, the Bohr model applies only

to hydrogen atoms and to ions that have only a single elect

the hydrogen spectrum and the bohr model

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