The Balmer Series

The visible Balmer lines are produced when electrons fall from higher energy levels to:

n = 2

Important transitions include:

n = 3 → n = 2

n = 4 → n = 2

n = 5 → n = 2

n = 6 → n = 2

These produce characteristic wavelengths in the visible part of the electromagnetic spectrum.

The strongest familiar Balmer lines occur at approximately:

These values are often used when studying hydrogen spectra.

The Red Hydrogen Line

One of the most recognizable lines in the hydrogen spectrum is the red line at approximately:

656 nm

It is produced by the transition:

n = 3 → n = 2

This line is commonly called H-alpha.

Because this transition has a particular energy difference, it always produces the same characteristic photon wavelength under ordinary conditions.

H-alpha is especially important in astronomy and is widely used to study hydrogen-rich regions of space.

---

Other Hydrogen Spectral Series

The Balmer series is only one part of hydrogen's spectrum.

Different groups of transitions produce different spectral series.

Lyman Series

Electrons fall to:

n = 1

These transitions mainly produce ultraviolet radiation.

Balmer Series

Electrons fall to:

n = 2

Several of these transitions produce visible light.

Paschen Series

Electrons fall to:

n = 3

These transitions mainly produce infrared radiation.

There are additional hydrogen series at still longer wavelengths.

Why Some Hydrogen Lines Are Invisible

Human eyes detect only a small portion of the electromagnetic spectrum.

Visible wavelengths are approximately:

380 nm to 750 nm

Hydrogen emits photons outside this range as well.

Transitions producing ultraviolet or infrared radiation cannot be seen directly by the human eye, but they can be detected using scientific instruments.

Therefore, an emission spectrum is not limited to visible light.

It can include electromagnetic radiation across many regions of the spectrum.

---

The Hydrogen Energy Levels

The allowed energy levels of hydrogen become progressively closer together as n increases.

The lowest level is:

n = 1

This is the ground state.

Higher values such as:

n = 2

n = 3

n = 4

represent excited states.

As n increases, the levels become closer together and approach the ionization limit.

Worked Example: Finding the Energy Difference

Suppose an electron falls from:

E₃ = −1.51 eV

to:

E₂ = −3.40 eV

The magnitude of the energy released is:

E_photon = |−3.40 − (−1.51)|

E_photon = 1.89 eV

That energy is released as a photon.

This particular transition in hydrogen corresponds to the red H-alpha line at approximately 656 nm.

---

Worked Example: Comparing Two Spectral Lines

Suppose:

Transition A releases 3.0 eV.

Transition B releases 2.0 eV.

Transition A releases more energy.

Therefore, Transition A produces:

• a higher-energy photon
• a higher frequency
• a shorter wavelength

Transition B produces:

• a lower-energy photon
• a lower frequency
• a longer wavelength

Notice that higher energy does not mean longer wavelength.

It means the opposite.

---

Why Different Elements Have Different Spectra

Every element has its own atomic structure.

Different atoms have:

• different numbers of protons
• different numbers of electrons
• different electron configurations
• different interactions between electrons and the nucleus
• different allowed energy states

Therefore, the energy gaps between electron states differ from element to element.

Different energy gaps produce different photon energies.

Different photon energies produce different wavelengths.

Therefore:

each element has a characteristic emission spectrum.

Spectra as Atomic Fingerprints

Because each element produces a characteristic pattern of spectral lines, its spectrum can be used like a fingerprint.

Suppose an unknown gas produces bright lines at several wavelengths.

Scientists can compare those wavelengths with known laboratory spectra.

If the lines match the characteristic spectrum of helium, helium is likely present.

If another group matches hydrogen, hydrogen is likely present too.

A mixture can therefore contain spectral lines from several elements.

---

Identifying an Unknown Element

Suppose an unknown sample produces lines at:

447 nm

501 nm

588 nm

668 nm

Scientists do not simply look at one line.

Instead, they compare the pattern of lines with reference spectra.

The pattern includes:

• the wavelengths present
• their relative positions
• sometimes their relative intensities

A strong match between the observed pattern and a known element provides evidence that the element is present.

---

Why One Line Is Usually Not Enough

Different elements can sometimes have spectral lines at similar wavelengths.

Therefore, identifying an element using only one line can be unreliable.

A stronger identification uses multiple matching lines.

For example, if an unknown sample has five prominent lines and all five match the known spectrum of an element, the evidence is much stronger than matching only one line.

This is similar to comparing several features of a fingerprint rather than only one.

---

Flame Tests and Emission Spectra

A simple example of atomic emission occurs in a flame test.

When certain metal ions are heated in a flame, electrons gain energy.

As electrons return to lower energy states, photons are emitted.

Different substances can produce characteristic flame colours.

Examples commonly observed include:

• sodium — strong yellow
• potassium — lilac
• lithium — crimson red
• calcium — orange-red
• copper compounds — blue-green

The colour seen by eye is only a simplified observation of the much more detailed emission spectrum.

Why Spectroscopy Is Better Than Simply Looking at Colour

Two substances can produce colours that appear similar to the human eye.

A spectroscope separates the light into individual wavelengths.

This reveals much more information.

Instead of simply observing:

"the flame looks yellow"

scientists can determine:

which exact wavelengths are present.

This makes spectral analysis much more precise than visual colour identification alone.

---

The Spectroscope

A spectroscope is an instrument used to separate light into its component wavelengths.

A simple spectroscope may use:

• a narrow slit
• a diffraction grating or prism
• a viewing system or detector

The light enters through the slit and is separated according to wavelength.

For an excited atomic gas, separate bright lines become visible.

Modern instruments usually use electronic detectors to measure the wavelengths and intensities very accurately.

Diffraction Gratings

A diffraction grating contains a very large number of closely spaced lines or grooves.

Different wavelengths of light diffract at different angles.

This separates the emitted light into its component wavelengths.

As a result, a mixture of wavelengths that initially travels together can be observed as separate spectral lines.

---

Gas Discharge Tubes

A common classroom demonstration uses a gas discharge tube.

The tube contains a low-pressure gas such as:

• hydrogen
• helium
• neon
• mercury vapour

An electrical potential difference is applied.

The electrical energy excites atoms in the gas.

As their electrons return to lower energy states, photons are emitted.

Looking at the light through a spectroscope reveals the characteristic line spectrum of the gas.

---

Neon Signs

Neon lighting provides a familiar application of atomic emission.

Electrical energy excites atoms in the gas.

The atoms then emit characteristic wavelengths as their electrons return to lower-energy states.

Pure neon is particularly associated with a reddish-orange glow.

Other gases or coatings can be used to produce other colours.

The underlying process is still based on electronic excitation and emission.

---

Emission Spectra in Astronomy

Spectroscopy is one of the most powerful tools in astronomy.

Astronomers cannot normally collect samples directly from:

• stars
• nebulae
• distant galaxies

Instead, they analyse the light reaching Earth.

Spectral lines can reveal which elements are present.

Hydrogen, helium, sodium, calcium, iron and many other elements have been identified in astronomical objects using their characteristic spectra.

Hydrogen in Space

Hydrogen is the most abundant element in the universe.

Its spectral lines are therefore extremely important in astronomy.

The red H-alpha line is especially useful for observing regions containing excited hydrogen.

Many glowing nebulae contain large amounts of hydrogen gas.

When hydrogen atoms are excited and then release energy, strong H-alpha emission can be produced.

This helps astronomers study regions where stars are forming.

---

More Than Identifying Elements

Spectra can provide information beyond chemical composition.

Depending on the situation, scientists can use spectral information to investigate:

• temperature
• density
• motion
• chemical composition
• magnetic fields
• conditions inside plasmas
• properties of stars and galaxies

Changes in the positions of spectral lines can also provide information about motion through the Doppler effect.

The key point is that spectral lines provide a highly precise way of studying matter remotely.

---

Emission and Absorption Spectra

Emission and absorption spectra are closely related.

Emission Spectrum

Produced when excited atoms release photons.

Appears as:

bright lines against a dark background

Absorption Spectrum

Produced when atoms absorb particular wavelengths from a continuous source.

Appears as:

dark lines within a continuous spectrum

The characteristic wavelengths correspond to energy differences between allowed atomic states.

Why Emission and Absorption Lines Are Related

Suppose an atom can absorb a photon that moves an electron:

E₁ → E₂

The required photon has energy:

ΔE = E₂ − E₁

The reverse transition:

E₂ → E₁

can emit a photon with that same energy difference.

Therefore, the wavelengths associated with absorption and emission are connected to the same atomic energy-level structure.

---

From Energy Levels to a Spectrum

The entire process can be summarized:

Atoms receive energy

↓

Electrons move to excited states

↓

Electrons return to lower states

↓

Specific energy differences are released

↓

Photons with specific energies are emitted

↓

Specific frequencies and wavelengths are produced

↓

A spectroscope separates the wavelengths

↓

Bright spectral lines appear

↓

The pattern can identify the element

This is the central connection between quantum physics and spectroscopy.

---

Example: Identifying Elements in a Mixture

Suppose reference spectra show:

Element A: lines at 410, 434, 486 and 656 nm

Element B: lines at 447, 502, 588 and 668 nm

An unknown mixture contains strong lines near:

410, 434, 447, 486, 502, 588, 656 and 668 nm.

The unknown spectrum contains the characteristic lines of both reference patterns.

Therefore, there is evidence that:

Element A and Element B are both present.

In real spectroscopy, scientists may consider line intensity, measurement uncertainty and additional spectral lines before making a final identification.

---

Worked Example: Calculating Photon Frequency

Suppose a spectral line has wavelength:

λ = 656 nm

Convert to metres:

656 nm = 6.56 × 10⁻⁷ m

Use:

c = fλ

Rearrange:

f = c/λ

Substitute:

f = (3.00 × 10⁸)/(6.56 × 10⁻⁷)

f ≈ 4.57 × 10¹⁴ Hz

This is the approximate frequency of the red H-alpha photon.

---

Worked Example: Photon Energy

Using:

f = 4.57 × 10¹⁴ Hz

and:

h = 6.63 × 10⁻³⁴ J·s

Use:

E = hf

E = (6.63 × 10⁻³⁴)(4.57 × 10¹⁴)

E ≈ 3.03 × 10⁻¹⁹ J

Therefore, each photon associated with this spectral line carries approximately:

3.03 × 10⁻¹⁹ J

of energy.

---

Common Mistakes

Thinking a line spectrum is a continuous rainbow

A line spectrum contains only particular wavelengths separated by dark regions.

Thinking spectral lines come from electrons moving around the nucleus continuously

The lines result from transitions between specific allowed energy states.

Thinking electrons emit light while moving upward

Moving upward generally requires absorption of energy.

Light is emitted when an electron moves to a lower-energy state.

Thinking every transition produces visible light

Many transitions produce ultraviolet or infrared radiation.

Thinking all elements have the same spectrum

Each element has its own characteristic energy-level structure and therefore its own characteristic spectral pattern.

Identifying an element from only one weak line

A pattern of several matching lines provides much stronger evidence.

Thinking colour alone is the same as a spectrum

A visible colour is much less detailed than a wavelength-resolved spectrum.

Confusing emission and absorption spectra

Emission spectra contain bright lines from emitted radiation.

Absorption spectra contain missing wavelengths within a continuous spectrum.

Assuming higher energy means longer wavelength

Higher photon energy corresponds to shorter wavelength.

Forgetting That Spectra Extend Beyond Visible Light

Emission spectra can contain ultraviolet, infrared and other wavelengths that humans cannot see.

Thinking the Dark Gaps Mean "No Energy"

The gaps mean those particular wavelengths are not strongly emitted in the observed atomic transitions. They do not mean the atom contains no energy.

---

Key Terms

Emission spectrum — The pattern of electromagnetic radiation emitted by a substance.

Line spectrum — A spectrum consisting of separate discrete wavelengths.

Spectral line — A particular wavelength appearing in a spectrum.

Continuous spectrum — A spectrum containing an essentially continuous range of wavelengths.

Quantized energy level — One of the specific allowed energies of an electron in an atom.

Ground state — The lowest allowed energy state.

Excited state — An allowed state above the ground state.

Excitation — The process of giving an atom energy so an electron moves to a higher state.

Transition — A change between allowed electron energy states.

Photon — A discrete packet of electromagnetic energy.

Emission — The release of electromagnetic radiation.

Frequency — The number of wave cycles passing a point each second.

Wavelength — The distance between corresponding points on successive waves.

Planck's constant — The constant h relating photon energy to frequency.

Hydrogen spectrum — The characteristic collection of wavelengths emitted or absorbed by hydrogen atoms.

Balmer series — Hydrogen spectral lines produced by transitions ending at n = 2.

Lyman series — Hydrogen spectral lines produced by transitions ending at n = 1.

Paschen series — Hydrogen spectral lines produced by transitions ending at n = 3.

H-alpha — The prominent hydrogen spectral line near 656 nm produced by the n = 3 → 2 transition.

Spectroscope — An instrument used to separate light into its component wavelengths.

Spectrometer — An instrument that measures properties of a spectrum, usually with greater quantitative precision.

Spectroscopy — The study and measurement of interactions between matter and electromagnetic radiation.

Diffraction grating — A structure containing many closely spaced lines or grooves that separates light according to wavelength.

Gas discharge tube — A low-pressure gas tube in which electrical energy excites atoms and produces characteristic emission.

Atomic fingerprint — An informal description of the characteristic spectral pattern that can be used to identify an element.

Flame test — A qualitative test in which substances are heated and identified partly by characteristic emitted colours.

Electron volt (eV) — A unit of energy commonly used for atomic-scale energies.

Absorption spectrum — A spectrum containing wavelengths removed when atoms absorb particular photon energies.

---

Key Takeaways

• Excited atoms can produce emission spectra.
• Energy supplied to an atom can move electrons into excited states.
• Excited electrons can return to lower energy states.
• Downward electron transitions release energy as photons.
• The photon energy equals the energy difference between the states.
• Photon energy and frequency are related by:

E = hf

• Frequency and wavelength are related by:

c = fλ

• Larger energy gaps produce higher-energy photons.
• Higher-energy photons have higher frequencies.
• Higher-frequency photons have shorter wavelengths.
• Atomic electron energies are quantized.
• Because only particular energy levels exist, only particular photon energies are produced.
• This produces a line spectrum rather than a continuous spectrum.
• Each bright spectral line corresponds to a particular photon wavelength.
• Each photon wavelength corresponds to an allowed atomic energy transition.
• Hydrogen produces a characteristic line spectrum.
• The visible hydrogen lines mainly belong to the Balmer series.
• Balmer transitions end at n = 2.
• The H-alpha line occurs near 656 nm and corresponds to n = 3 → 2.
• Hydrogen also produces ultraviolet and infrared spectral lines.
• Different elements have different atomic energy structures.
• Different elements therefore produce different patterns of spectral lines.
• An element's spectrum can act like an atomic fingerprint.
• Scientists identify elements by comparing unknown spectra with known reference spectra.
• Several matching spectral lines provide stronger evidence than one matching line.
• Spectroscopy can identify elements in laboratory samples and distant astronomical objects.
• Emission spectra provide strong experimental evidence for quantized atomic energy levels.

The central idea is:

Unique energy levels → unique energy gaps → unique photon wavelengths → unique emission spectrum

---

Check Your Understanding

1. What is an emission spectrum?

2. Describe how an atom becomes excited.

3. What happens when an excited electron moves to a lower energy state?

4. Why is a photon emitted during a downward electron transition?

5. What determines the energy of the emitted photon?

6. State the relationship between photon energy and frequency.

7. State the relationship between frequency and wavelength.

8. What happens to wavelength as photon energy increases?

9. What is a line emission spectrum?

10. Why does an atomic line spectrum contain separate lines rather than a continuous range of colours?

11. Explain how line spectra provide evidence for quantized energy levels.

12. Describe the appearance of a typical atomic emission spectrum.

13. Why are dark spaces present between the bright spectral lines?

14. Why is hydrogen particularly useful when introducing atomic emission spectra?

15. What is the Balmer series?

16. To which energy level do Balmer-series transitions end?

17. Which hydrogen transition produces the H-alpha line?

18. Approximately what wavelength is H-alpha?

19. Why are some hydrogen spectral lines invisible to humans?

20. What regions of the electromagnetic spectrum contain the Lyman and Paschen series?

21. Explain why different elements have different emission spectra.

22. Why can an emission spectrum be described as an atomic fingerprint?

23. Why is matching several spectral lines more reliable than matching only one?

24. Describe how a spectroscope helps scientists analyse emitted light.

25. Explain how flame tests are related to atomic emission spectra.

26. What is the difference between an emission spectrum and an absorption spectrum?

27. Explain how astronomers can determine which elements are present in distant astronomical objects.

28. A photon has a wavelength of 500 nm. Calculate its frequency using:

c = 3.00 × 10⁸ m/s

29. Use your answer to Question 28 to calculate the photon energy using:

h = 6.63 × 10⁻³⁴ J·s

30. Challenge: An unknown gas produces bright spectral lines at approximately:

410 nm, 434 nm, 486 nm and 656 nm.

a. Compare these wavelengths with the visible Balmer lines of hydrogen.
b. Identify the likely element.
c. Explain why matching several lines gives stronger evidence than matching only one.
d. Which of the four wavelengths corresponds to the highest-energy photon?
e. Which has the lowest frequency?
f. Which has the shortest wavelength?
g. Which has the longest wavelength?
h. Identify the hydrogen transition responsible for the 656 nm line.
i. Explain what happens to an electron during this transition.
j. Explain why the electron releases a photon.
k. Explain why the photon has a specific wavelength rather than any wavelength.
l. Explain how this observation supports the idea of quantized atomic energy levels.
m. Describe how the same principle could be used to identify another element in a mixture.
n. Explain one way emission spectroscopy is useful outside the classroom.

Absorption Spectra

When light containing many wavelengths passes through a cool, low-density gas, atoms in the gas can absorb certain wavelengths of that light.

The atoms do not absorb every wavelength. They absorb photons only when the photon energy matches an allowed energy difference between electron energy levels.

The missing wavelengths appear as dark lines in an otherwise continuous spectrum.

This pattern is called an absorption spectrum.

Absorption spectra provide powerful evidence for quantized energy levels and allow scientists to determine the chemical composition of objects that may be millions or even billions of light-years away.

How an Absorption Spectrum Forms

Imagine white light travelling through a cloud of cooler gas.

White light contains a broad range of wavelengths.

As the light passes through the gas:

1. photons interact with atoms
2. some photons have exactly the right energies to excite electrons
3. those photons are absorbed
4. electrons move to higher allowed energy levels
5. those wavelengths are removed from the light travelling directly toward the observer

When the remaining light is separated by wavelength, we see a continuous spectrum crossed by dark absorption lines.

The basic process is:

Continuous light → cooler gas → selected wavelengths absorbed → dark spectral lines

Electron Transitions During Absorption

Electrons in atoms can occupy only certain allowed energy levels.

Suppose an electron begins at:

E₁

and there is a higher allowed state:

E₂

For the electron to move from E₁ to E₂, it must gain exactly the required energy:

ΔE = E₂ − E₁

A photon can provide this energy.

If:

E_photon = ΔE

the photon can be absorbed and the electron can move to the higher level.

This is called an upward electron transition.

---

Photons and Energy

The energy of a photon is related to its frequency by:

E = hf

where:

E = photon energy

h = Planck's constant

f = frequency

Frequency and wavelength are related by:

c = fλ

Therefore, each allowed energy transition corresponds to a particular:

• photon energy
• frequency
• wavelength

Only photons with suitable energies can produce particular transitions.

---

Why Doesn't the Atom Absorb Every Wavelength?

Suppose an electron requires:

3.0 eV

to move to the next allowed energy level.

A photon with energy:

3.0 eV

can cause that transition.

But a photon with:

2.5 eV

cannot simply move the electron partway between the two energy levels.

There is no allowed electron state there.

Similarly, in the simplified atomic model, a 2.9 eV photon cannot cause a transition requiring exactly 3.0 eV.

This is because atomic energy levels are quantized.

What Happens to the Absorbed Photon?

When an atom absorbs a photon, the photon's energy is transferred to the atom.

An electron moves to a higher-energy state.

The original photon is therefore no longer travelling along its original path toward the observer.

The excited electron may later return to a lower energy level and emit another photon.

However, that photon may be emitted in a different direction.

As a result, the original wavelength has reduced intensity along the observer's line of sight.

This produces a dark absorption line.

---

Dark Does Not Mean No Light

An absorption line is often described as a dark line, but this does not necessarily mean absolutely zero light exists at that wavelength.

It means that the intensity at that wavelength is lower than the surrounding continuous spectrum.

Modern instruments measure these changes in intensity very precisely.

A spectrum plotted as intensity against wavelength may therefore show absorption lines as dips.

---

Continuous Spectra

A hot, dense source can produce a broad, approximately continuous spectrum.

Examples include:

• stars
• hot solids
• dense glowing gases
• incandescent materials

If this light reaches us directly, we may observe a continuous range of wavelengths.

If it passes through cooler gas first, certain wavelengths can be absorbed.

The result is an absorption spectrum.

---

Three Important Types of Spectra

It is useful to distinguish three basic types.

Continuous Spectrum

A broad, uninterrupted range of wavelengths.

Emission Spectrum

Bright lines at particular wavelengths against a darker background.

Absorption Spectrum

Dark lines at particular wavelengths within a continuous spectrum.

Emission vs Absorption Spectra

Emission and absorption spectra are closely related.

The important connection is that both are determined by the same atomic energy levels.

---

The Same Energy Gaps

Suppose an atom can absorb a photon to move an electron:

E₁ → E₂

The required photon energy is:

ΔE = E₂ − E₁

Later, an electron making the reverse transition:

E₂ → E₁

can emit a photon with the same energy difference.

Therefore, an element's absorption and emission lines occur at characteristic corresponding wavelengths.

A Useful Analogy

Think of atomic energy levels as the floors of a building.

An electron cannot stop between floors.

To move from the first floor to the third floor, it must receive exactly enough energy for that transition.

During absorption:

photon energy → electron moves upward

During emission:

electron moves downward → photon energy released

The analogy is useful, but remember that atomic energy levels are not literal physical floors or distances.

They represent allowed energies.

---

The Hydrogen Absorption Spectrum

Hydrogen has a characteristic set of allowed electron energy levels.

When continuous light passes through hydrogen gas, hydrogen atoms can absorb photons with energies corresponding to allowed upward transitions.

For visible wavelengths, important transitions include those beginning at:

n = 2

and moving to higher energy levels.

For example:

n = 2 → n = 3

n = 2 → n = 4

n = 2 → n = 5

n = 2 → n = 6

These correspond to the same characteristic wavelengths associated with visible Balmer emission lines, but the electron transitions occur in the opposite direction.

Hydrogen's Visible Absorption Lines

Important Balmer wavelengths include approximately:

In an absorption spectrum, these appear as reductions in intensity at those wavelengths.

---

Comparing Hydrogen Emission and Absorption

Consider the hydrogen transition associated with approximately:

656 nm

During absorption:

n = 2 → n = 3

The electron absorbs a photon.

During emission:

n = 3 → n = 2

The electron emits a photon.

The direction of the electron transition is reversed, but the energy difference between the two states is the same.

Therefore, the characteristic wavelength is also the same.

---

Why Every Element Has a Unique Absorption Spectrum

Different elements have different:

• nuclear charges
• numbers of electrons
• electron arrangements
• interactions between particles
• allowed atomic energy states

Therefore, different elements have different energy gaps.

Different energy gaps correspond to different photon energies.

Different photon energies correspond to different wavelengths.

Therefore:

each element produces a characteristic pattern of absorption lines.

This makes absorption spectra useful as atomic fingerprints.

Identifying an Element from an Absorption Spectrum

Suppose an unknown spectrum contains dark lines at:

410 nm

434 nm

486 nm

656 nm

These wavelengths closely match visible hydrogen Balmer wavelengths.

This provides evidence that hydrogen is present in the absorbing gas.

Scientists usually look for several matching lines, rather than relying on a single line.

Multiple matches provide much stronger evidence.

---

Interpreting a Simple Absorption Spectrum

When examining an absorption spectrum:

Look for Dark Lines

Each dark line represents a wavelength where the intensity has been reduced.

Determine the Wavelengths

Use the wavelength scale to determine where the lines occur.

Compare with Reference Spectra

Look for matching patterns from known elements.

Look for Several Matches

One matching line may occur by coincidence.

Several matching lines provide stronger evidence.

Consider Multiple Elements

A real sample can contain many elements, so the observed spectrum may contain several overlapping patterns.

---

Example: Identifying Two Elements

Suppose reference data show:

Element A: 410, 434, 486, 656 nm

Element B: 422, 527, 616 nm

An unknown spectrum contains absorption lines at:

410, 422, 434, 486, 527, 616 and 656 nm

The spectrum contains all four lines from Element A and all three from Element B.

Therefore, the evidence suggests that:

both Element A and Element B are present.

---

Absorption Spectra and the Sun

One of the most famous examples of an absorption spectrum is the spectrum of the Sun.

The Sun's hot, dense interior produces a broad range of wavelengths.

This radiation passes through cooler gases in the Sun's outer layers.

Atoms in these cooler layers absorb particular wavelengths.

When sunlight is separated into a spectrum, numerous dark lines are visible.

These are called Fraunhofer lines.

Fraunhofer Lines

In the early nineteenth century, Joseph von Fraunhofer carefully studied the dark lines in the solar spectrum.

Thousands of absorption features can now be measured in sunlight.

These lines contain information about substances in the Sun's atmosphere.

By comparing solar absorption lines with laboratory spectra, scientists can identify elements present in the Sun.

---

What Is the Sun Made Of?

Spectroscopy shows that the Sun contains large amounts of:

• hydrogen
• helium

and smaller quantities of many heavier elements.

Spectral features associated with elements such as:

• sodium
• calcium
• iron

can also be detected.

This is remarkable because scientists do not need to collect physical samples from the Sun.

They can analyse its light.

---

Absorption Spectra in Astronomy

Astronomy depends heavily on spectroscopy because most astronomical objects are far beyond our physical reach.

Astronomers analyse light from:

• stars
• planets
• stellar atmospheres
• galaxies
• gas clouds
• some exoplanet atmospheres

The positions and patterns of absorption lines can reveal chemical composition.

Reading the Chemical Fingerprint of a Star

Imagine that a star's spectrum contains dark lines at the same wavelengths as laboratory hydrogen absorption lines.

That provides evidence that hydrogen is present in material interacting with the star's light.

Suppose additional lines match sodium and calcium.

Scientists can infer that these elements are also present.

The overall process is:

Observe star

↓

Separate its light into wavelengths

↓

Measure absorption lines

↓

Compare lines with laboratory spectra

↓

Identify elements

---

Why Laboratory Spectra Are Important

Scientists can produce spectra of known substances in laboratories.

These become reference patterns.

When an astronomical spectrum is collected, its lines can be compared with these references.

A strong match between multiple lines provides evidence that the same element is present in the astronomical object.

This allows laboratory atomic physics to be applied to objects at enormous distances.

---

Spectra Can Tell Us More Than Composition

Absorption spectra can reveal much more than which elements are present.

Scientists can also obtain information about:

• temperature
• density
• pressure
• motion
• magnetic fields
• chemical abundances
• conditions in stellar atmospheres

The shapes, strengths, widths and positions of spectral lines can all contain information.

At an introductory level, however, the most important use is:

identifying elements from characteristic spectral lines.

---

Absorption Spectra and Motion

Suppose scientists know the normal laboratory wavelength of a particular absorption line.

If the same line appears at a slightly different wavelength in a star's spectrum, the shift can provide information about the star's motion.

This is related to the Doppler effect.

If spectral lines are shifted toward longer wavelengths, this is called a:

redshift

If they are shifted toward shorter wavelengths, this is called a:

blueshift

The important point is that scientists first need to know which spectral lines they are examining.

The characteristic atomic pattern provides that reference.

---

Exoplanet Atmospheres

Absorption spectroscopy can even help scientists investigate some planets orbiting other stars.

When an exoplanet passes in front of its star, a small amount of starlight may pass through the planet's atmosphere.

Atoms and molecules in that atmosphere can absorb particular wavelengths.

Scientists can compare the spectrum during the transit with the normal stellar spectrum.

This can provide evidence about substances in the planet's atmosphere.

Worked Example: Energy of an Absorbed Photon

Suppose an atom absorbs light with wavelength:

λ = 500 nm

Convert to metres:

500 nm = 5.00 × 10⁻⁷ m

Use:

c = fλ

Rearrange:

f = c/λ

Using:

c = 3.00 × 10⁸ m/s

f = (3.00 × 10⁸)/(5.00 × 10⁻⁷)

f = 6.00 × 10¹⁴ Hz

Now use:

E = hf

with:

h = 6.63 × 10⁻³⁴ J·s

E = (6.63 × 10⁻³⁴)(6.00 × 10¹⁴)

E ≈ 3.98 × 10⁻¹⁹ J

Therefore, the electron absorbs approximately:

3.98 × 10⁻¹⁹ J

of energy.

---

Worked Example: Comparing Two Absorption Lines

Suppose an atom absorbs photons at:

400 nm

and:

650 nm

Which transition requires more energy?

The shorter wavelength corresponds to the higher frequency.

Higher frequency means higher photon energy.

Therefore:

400 nm corresponds to the larger energy transition.

So:

400 nm photon → larger ΔE

650 nm photon → smaller ΔE

---

Intensity and Absorption Lines

A spectrum can also be displayed as a graph of:

intensity vs wavelength

Instead of seeing dark vertical lines, absorption features appear as downward dips.

The horizontal axis represents wavelength.

The vertical axis represents the intensity of detected light.

A deeper dip means that less light was detected at that wavelength.

However, interpreting the exact depth of spectral lines can involve several physical factors, so line depth should not automatically be treated as a simple direct measurement of the amount of an element without further analysis.

---

A Simple Spectrum Interpretation Strategy

When given a spectrum in a question:

First: identify whether it is continuous, emission or absorption.

Second: locate the important spectral lines.

Third: read their wavelengths.

Fourth: compare them with reference spectra.

Fifth: look for patterns of several matching lines.

Sixth: identify the element or elements that best match.

If the question involves electron transitions, remember:

Absorption → electron moves upward

Emission → electron moves downward

---

Common Mistakes

Saying Dark Lines Are Emitted

Dark absorption lines represent wavelengths that have been removed or reduced from the observed continuous spectrum.

Thinking the Electron Moves Down During Absorption

Absorption supplies energy.

The electron moves to a higher-energy state.

Thinking Any Photon Can Be Absorbed

The photon must correspond to an allowed transition.

Thinking Electrons Can Stop Between Energy Levels

Atomic bound energy states are quantized.

Confusing Absorption with Emission

Absorption: photon absorbed, electron moves upward.

Emission: photon emitted, electron moves downward.

Thinking Emission and Absorption Lines Occur at Completely Different Wavelengths

For the same element and corresponding transitions, characteristic wavelengths are related to the same energy gaps.

Thinking a Dark Line Means Absolutely No Light

It generally indicates reduced intensity at that wavelength.

Identifying an Element from One Line Alone

Several matching spectral lines provide stronger evidence.

Thinking Every Absorption Line Is Visible

Many important lines occur in ultraviolet or infrared regions.

Assuming Every Dark Line Belongs to One Element

Real spectra can contain absorption features from many different atoms, ions and molecules.

Assuming Line Depth Alone Gives the Amount of an Element

Spectral line strength can depend on several physical conditions.

Forgetting to Convert Nanometres to Metres

For calculations:

1 nm = 1 × 10⁻⁹ m

---

Key Terms

Absorption spectrum — A spectrum containing reduced intensity at particular wavelengths because radiation has been absorbed.

Absorption line — A dark line or intensity dip corresponding to a wavelength that has been absorbed.

Continuous spectrum — A broad, uninterrupted range of wavelengths.

Emission spectrum — A spectrum containing radiation emitted at particular wavelengths.

Line spectrum — A spectrum containing distinct spectral features at particular wavelengths.

Photon — A discrete packet of electromagnetic energy.

Absorption — The transfer of photon energy to matter.

Electron transition — A change of an electron from one allowed energy state to another.

Upward transition — A transition to a higher-energy state caused by energy absorption.

Downward transition — A transition to a lower-energy state, often accompanied by photon emission.

Ground state — The lowest allowed atomic energy state.

Excited state — An allowed state with energy greater than the ground state.

Quantized energy — Energy restricted to particular discrete allowed values.

Energy gap — The difference in energy between two allowed states.

Wavelength — The distance between equivalent points on successive waves.

Frequency — The number of wave cycles passing a point each second.

Intensity — A measure related to the amount of radiation detected.

Spectral line — A feature appearing at a particular wavelength in a spectrum.

Atomic fingerprint — The characteristic pattern of spectral lines associated with an element.

Spectroscope — An instrument used to separate light according to wavelength.

Spectrometer — An instrument used to measure a spectrum quantitatively.

Spectroscopy — The study and measurement of interactions between electromagnetic radiation and matter.

Fraunhofer lines — Absorption features observed in the solar spectrum.

Balmer series — Hydrogen spectral lines associated with transitions involving n = 2.

Redshift — A displacement of spectral features toward longer wavelengths.

Blueshift — A displacement of spectral features toward shorter wavelengths.

Transmission spectroscopy — Analysis of wavelengths transmitted through a material, such as an atmosphere, to determine what has been absorbed.

Reference spectrum — A known spectrum used for comparison with an unknown spectrum.

---

Key Takeaways

• An absorption spectrum forms when radiation passes through matter that absorbs particular wavelengths.
• A common astronomical absorption spectrum occurs when continuous radiation passes through cooler gas.
• Absorbed photons transfer energy to atoms.
• During absorption, electrons move from lower to higher allowed energy states.
• The photon must have an appropriate energy for the transition.
• Atomic energy levels are quantized.
• Electrons cannot simply occupy arbitrary energies between allowed bound states.
• Photon energy is related to frequency by:

E = hf

• Frequency and wavelength are related by:

c = fλ

• Shorter wavelengths correspond to higher photon energies.
• An absorption spectrum usually appears as dark lines within a continuous spectrum.
• The dark lines represent wavelengths with reduced intensity.
• Emission involves downward electron transitions.
• Absorption involves upward electron transitions.
• Emission and absorption features of the same element are related to the same atomic energy differences.
• Different elements have different energy-level structures.
• Each element therefore produces a characteristic pattern of spectral lines.
• These patterns can act as atomic fingerprints.
• Several matching lines provide stronger identification than one line.
• Hydrogen produces characteristic absorption lines associated with its allowed energy transitions.
• The Sun's spectrum contains many absorption features known as Fraunhofer lines.
• Astronomers compare stellar spectra with laboratory reference spectra to identify elements.
• Absorption spectroscopy allows scientists to study objects without physically collecting samples.
• Spectra can provide information about composition and, with further analysis, temperature, motion and other physical conditions.
• Absorption spectroscopy can also help investigate some exoplanet atmospheres.

The central idea is:

Specific atomic energy levels → specific energy gaps → specific photons absorbed → specific dark spectral lines

And the simplest comparison is:

Absorption: photon in → electron up

Emission: electron down → photon out

Atomic Transitions

Electrons in atoms do not have just any possible energy. Instead, they occupy specific quantized energy levels.

An atomic transition occurs when an electron changes from one allowed energy state to another.

There are two main possibilities:

Electron moves upward → energy is absorbed

Electron moves downward → energy is released

These transitions explain why atoms absorb and emit only particular wavelengths of electromagnetic radiation and why every element has a characteristic spectrum.

Quantized Energy Levels

Electrons bound within atoms can occupy only particular allowed energy states.

These are called energy levels.

A simplified diagram might look like this:

E₄ ─────────────

E₃ ─────────────

E₂ ─────────────

E₁ ─────────────

Each horizontal line represents an allowed energy state.

An electron can occupy:

E₁, E₂, E₃ or E₄

but it cannot occupy an arbitrary energy between these levels.

This is what is meant by quantization.

---

Energy Level Diagrams

An energy level diagram is a convenient way of representing the allowed energies of electrons.

The vertical direction represents energy, not physical height or distance from the nucleus.

Higher lines represent higher energies.

Lower lines represent lower energies.

For example:

higher energy

E₄ ─────────────

E₃ ─────────────

E₂ ─────────────

E₁ ─────────────

lower energy

The electron's position on this diagram tells us its energy state.

It does not show the literal location of the electron within the atom.

The Ground State

The ground state is the lowest available energy state of an atom.

For a simple diagram:

E₄ ─────────────

E₃ ─────────────

E₂ ─────────────

E₁ ─────●───────

the electron is in E₁.

If E₁ is the lowest allowed level, the electron is in its ground state.

The ground state is generally the most stable state.

---

Excited States

An excited state is an allowed energy state above the ground state.

For example:

E₄ ─────────────

E₃ ─────●───────

E₂ ─────────────

E₁ ─────────────

The electron is now in an excited state.

For this to happen, the atom must have gained energy.

Possible sources include:

• electromagnetic radiation
• electrical energy
• thermal energy
• collisions with particles

Excited states are generally less stable than the ground state.

---

Absorbing Energy

An electron can move from a lower energy state to a higher energy state by absorbing energy.

For example:

E₁ → E₃

is an upward transition.

The electron must gain an amount of energy equal to the difference between the two levels.

The energy change is:

ΔE = E_final − E_initial

For an upward transition:

ΔE > 0

because the electron gains energy.

Worked Example: Energy Absorbed

Suppose:

E₁ = −10 eV

E₂ = −6 eV

An electron moves:

E₁ → E₂

Calculate the energy change.

ΔE = E_final − E_initial

ΔE = −6 − (−10)

ΔE = +4 eV

The positive sign tells us that the electron has gained energy.

Therefore:

4 eV must be absorbed.

---

Absorption of Photons

One way an electron can gain energy is by absorbing a photon.

The photon carries electromagnetic energy.

Photon energy is:

E = hf

where:

E = photon energy

h = Planck's constant

f = photon frequency

For an atomic transition to occur:

E_photon = ΔE

The photon must supply the appropriate amount of energy for the allowed transition.

---

Why the Energy Must Match

Suppose an atom has:

E₁ = −10 eV

E₂ = −6 eV

The energy difference is:

4 eV

A photon with the appropriate 4 eV energy can cause:

E₁ → E₂

But a 2 eV photon cannot simply move the electron halfway between E₁ and E₂.

There is no allowed bound state there.

Likewise, in a simplified energy-level model, a 3 eV photon cannot produce this particular 4 eV transition.

This is a direct consequence of quantized energy levels.

---

Electrons Do Not Stay Excited Forever

After absorbing energy, an electron may occupy an excited state.

However, many excited states are unstable.

Eventually, the electron can return to a lower-energy state.

For example:

E₃ → E₂

or:

E₃ → E₁

When this happens, energy is released.

The energy released corresponds to the difference between the two energy states.

Emission of a Photon

When an electron makes a downward transition, energy can be emitted as a photon.

For example:

E₃ → E₁

The photon energy is equal to the magnitude of the energy difference:

E_photon = |E_final − E_initial|

If:

E₃ = −2 eV

and:

E₁ = −10 eV

then:

E_photon = |−10 − (−2)|

E_photon = 8 eV

Therefore, an 8 eV photon is emitted.

---

Positive and Negative Energy Changes

Using:

ΔE = E_final − E_initial

the sign of ΔE tells us whether the electron gains or loses energy.

Positive ΔE

If:

ΔE > 0

the electron has moved to a higher-energy state.

Energy has been absorbed.

Negative ΔE

If:

ΔE < 0

the electron has moved to a lower-energy state.

Energy has been released.

For example:

E_initial = −2 eV

E_final = −10 eV

ΔE = −10 − (−2)

ΔE = −8 eV

The electron loses 8 eV.

The emitted photon therefore has energy:

8 eV

Photon energy itself is normally reported as a positive magnitude.

---

A Simple Rule to Remember

Moving Up

electron gains energy

ΔE > 0

energy absorbed

Moving Down

electron loses energy

ΔE < 0

energy released

For photon emission, use the magnitude of the energy difference:

E_photon = |ΔE|

---

One Excited State, Several Possible Transitions

Suppose an atom has:

E₄ = −1 eV

E₃ = −3 eV

E₂ = −6 eV

E₁ = −10 eV

An electron in E₄ could potentially return through several transitions.

For example:

E₄ → E₃

Energy released:

|−3 − (−1)| = 2 eV

E₄ → E₂

Energy released:

|−6 − (−1)| = 5 eV

E₄ → E₁

Energy released:

|−10 − (−1)| = 9 eV

Each transition produces a different photon energy.

Direct Transitions and Cascades

An excited electron does not necessarily return directly to the ground state.

Suppose an electron is at E₄.

It might transition directly:

E₄ → E₁

Or it could return in stages:

E₄ → E₃ → E₂ → E₁

This sequence of transitions is sometimes called a cascade.

Each downward transition can release a photon with an energy corresponding to that particular energy gap.

---

Worked Example: A Cascade

Suppose:

E₄ = −1 eV

E₃ = −3 eV

E₂ = −6 eV

E₁ = −10 eV

An electron follows:

E₄ → E₃ → E₁

First transition:

E₄ → E₃

Energy released:

|−3 − (−1)| = 2 eV

Second transition:

E₃ → E₁

Energy released:

|−10 − (−3)| = 7 eV

So two photons can be emitted:

2 eV photon

and:

7 eV photon

The total energy released is:

2 + 7 = 9 eV

Notice that this equals the overall difference:

E₄ → E₁

|−10 − (−1)| = 9 eV

Energy is conserved.

---

Atomic Transitions and Conservation of Energy

Atomic transitions obey the law of conservation of energy.

If an electron gains 5 eV during an upward transition, that energy must come from somewhere.

If an electron loses 5 eV during a downward transition, that energy must go somewhere.

For photon interactions:

energy gained by electron = energy lost by photon

or:

energy lost by electron = energy carried by emitted photon

Energy is transferred, not created or destroyed.

---

Photon Frequency and Atomic Transitions

The energy of an emitted photon is related to frequency:

E = hf

Therefore:

f = E/h

A larger atomic energy difference produces a higher-energy photon.

A higher-energy photon has a higher frequency.

So:

larger ΔE → higher photon energy → higher frequency

---

Photon Wavelength and Atomic Transitions

Frequency and wavelength are related by:

c = fλ

Therefore:

λ = c/f

Combining this with:

E = hf

gives:

E = hc/λ

This means:

larger energy gap → shorter wavelength

and:

smaller energy gap → longer wavelength

This connection explains the colours and wavelengths found in atomic spectra.

Worked Example: Frequency of an Emitted Photon

Suppose an atomic transition releases:

E = 3.98 × 10⁻¹⁹ J

Use:

E = hf

Rearrange:

f = E/h

Using:

h = 6.63 × 10⁻³⁴ J·s

f = (3.98 × 10⁻¹⁹)/(6.63 × 10⁻³⁴)

f ≈ 6.00 × 10¹⁴ Hz

Therefore, the emitted photon has a frequency of approximately:

6.00 × 10¹⁴ Hz

---

Worked Example: Wavelength of an Emitted Photon

Using:

f = 6.00 × 10¹⁴ Hz

and:

c = 3.00 × 10⁸ m/s

Use:

λ = c/f

λ = (3.00 × 10⁸)/(6.00 × 10¹⁴)

λ = 5.00 × 10⁻⁷ m

Convert to nanometres:

λ = 500 nm

So this atomic transition produces a photon with a wavelength of approximately 500 nm.

---

Energy in Joules and Electron Volts

Atomic energies are commonly expressed in electron volts (eV).

The conversion is:

1 eV = 1.60 × 10⁻¹⁹ J

Suppose a transition releases:

3.0 eV

Convert to joules:

E = (3.0)(1.60 × 10⁻¹⁹)

E = 4.80 × 10⁻¹⁹ J

This value can then be used in:

E = hf

to calculate the photon's frequency.

---

Worked Example: From Energy Levels to Wavelength

Suppose:

E_initial = −2.0 eV

E_final = −5.0 eV

Calculate the energy change

ΔE = −5.0 − (−2.0)

ΔE = −3.0 eV

Therefore, a photon with energy:

3.0 eV

is emitted.

Convert to joules

E = (3.0)(1.60 × 10⁻¹⁹)

E = 4.80 × 10⁻¹⁹ J

Calculate frequency

f = E/h

f = (4.80 × 10⁻¹⁹)/(6.63 × 10⁻³⁴)

f ≈ 7.24 × 10¹⁴ Hz

Calculate wavelength

λ = c/f

λ = (3.00 × 10⁸)/(7.24 × 10¹⁴)

λ ≈ 4.14 × 10⁻⁷ m

or:

414 nm

Therefore, the transition produces a photon with wavelength approximately:

414 nm

---

Reading an Energy Level Diagram

When given an energy-level diagram, use a systematic approach.

Identify the Initial Level

Where does the electron start?

Identify the Final Level

Where does the electron finish?

Determine the Direction

Upward arrow:

absorption

Downward arrow:

emission

Calculate the Energy Change

Use:

ΔE = E_final − E_initial

Interpret the Sign

Positive:

energy absorbed

Negative:

energy released

If a Photon Is Emitted

Use:

E_photon = |ΔE|

Then, if required, calculate:

f = E/h

and:

λ = c/f

---

Interpreting Arrow Lengths

On a properly scaled energy-level diagram, a longer vertical transition represents a larger energy difference.

Therefore:

larger vertical transition → larger |ΔE|

For emitted photons:

larger |ΔE| → higher photon energy → higher frequency → shorter wavelength

For smaller transitions:

smaller |ΔE| → lower photon energy → lower frequency → longer wavelength

This allows many questions to be answered without performing calculations.

---

Example: Comparing Three Transitions

Suppose three downward transitions are shown:

A: E₄ → E₃

B: E₄ → E₂

C: E₄ → E₁

If E₁ is the lowest state, Transition C has the largest energy difference.

Therefore, C produces:

• the highest-energy photon
• the highest-frequency photon
• the shortest-wavelength photon

Transition A has the smallest energy difference.

Therefore, A produces:

• the lowest-energy photon
• the lowest frequency
• the longest wavelength

---

Atomic Transitions and Emission Spectra

Each possible downward atomic transition can produce a photon with a particular energy.

Since:

E = hf

each photon has a particular frequency.

Since:

c = fλ

each has a particular wavelength.

When these emitted wavelengths are separated, they form an emission spectrum.

Why Emission Spectra Have Lines

Atomic energy levels are discrete.

Therefore, the differences between those levels are also specific.

This means atoms cannot emit every possible photon energy during bound-bound transitions.

Instead, they emit particular photon energies.

Therefore:

quantized energy levels

↓

specific energy differences

↓

specific photon energies

↓

specific frequencies

↓

specific wavelengths

↓

spectral lines

This is why atomic emission spectra contain lines rather than a continuous rainbow.

---

Atomic Transitions and Absorption Spectra

The same atomic energy levels also explain absorption spectra.

Suppose an atom can make:

E₂ → E₄

if it absorbs the correct photon.

The required photon energy is:

E₄ − E₂

The reverse transition:

E₄ → E₂

can emit a photon corresponding to the same energy difference.

Therefore, absorption and emission lines are closely related.

---

The Hydrogen Atom

Hydrogen is particularly useful for studying atomic transitions because it contains only one electron.

Its energy levels are commonly labelled:

n = 1, 2, 3, 4...

The ground state is:

n = 1

Higher values represent excited states.

Transitions between these states produce hydrogen's characteristic spectrum.

Hydrogen's Balmer Transitions

Some of hydrogen's most familiar visible emission lines occur when electrons fall to:

n = 2

Examples include:

n = 3 → 2

n = 4 → 2

n = 5 → 2

n = 6 → 2

These transitions form part of the Balmer series.

Because each transition has a different energy difference, each produces a different wavelength.

---

Why Different Elements Produce Different Spectra

Different elements have different:

• numbers of protons
• nuclear charges
• numbers of electrons
• electron arrangements
• allowed energy states

Therefore, their possible atomic transitions are different.

Different transitions produce different photon energies.

As a result, each element has a characteristic spectral pattern.

This allows scientists to identify elements using spectroscopy.

---

Atomic Transitions Are Not Literal Jumps Through Space

Energy-level diagrams can create a misleading mental picture.

When an electron moves from:

E₁ → E₂

it should not be imagined as a tiny ball physically travelling upward through the diagram.

The vertical axis represents energy.

An atomic transition is a change in the electron's quantum state.

The electron does not occupy all of the intermediate energies during the transition.

---

Excitation vs Ionization

It is important to distinguish between excitation and ionization.

Excitation

An electron moves from one bound energy state to a higher bound energy state.

The electron remains associated with the atom.

Ionization

Enough energy is supplied to remove the electron from the atom.

The atom becomes an ion.

The Ionization Limit

Energy-level diagrams often show an ionization limit.

For many diagrams, the reference:

E = 0

represents a free electron separated from the atom.

Bound electron states are shown with negative energies.

For example:

E = 0 eV — ionization limit

E₃ = −1.5 eV

E₂ = −3.4 eV

E₁ = −13.6 eV

To ionize an electron in E₁:

ΔE = 0 − (−13.6)

ΔE = 13.6 eV

So at least 13.6 eV is required in this simplified hydrogen example.

---

Real-World Importance of Atomic Transitions

Atomic transitions are important in many areas of science and technology.

Spectroscopy

Spectral lines reveal which elements are present in a substance.

Astronomy

Scientists identify elements in stars and gas clouds by analysing spectral lines.

Lasers

Controlled atomic or molecular transitions produce photons.

Fluorescent Lamps

Electrical energy excites atoms, which then emit radiation as they return to lower-energy states.

Neon Lighting

Excited gas atoms emit characteristic wavelengths.

LEDs

Although semiconductor physics differs from isolated atomic transitions, photons are also produced through transitions between allowed electronic energy states.

Atomic Clocks

Extremely precise transition frequencies are used to measure time.

---

A Complete Transition Problem

Suppose an atom has:

E₄ = −1 eV

E₃ = −3 eV

E₂ = −6 eV

E₁ = −10 eV

An electron begins in E₄ and moves directly to E₁.

Determine the Energy Change

ΔE = E_final − E_initial

ΔE = −10 − (−1)

ΔE = −9 eV

Interpret the Sign

The negative sign means the electron loses energy.

Therefore, energy is emitted.

Determine Photon Energy

E_photon = 9 eV

Convert to Joules

E = (9)(1.60 × 10⁻¹⁹)

E = 1.44 × 10⁻¹⁸ J

Calculate Frequency

f = E/h

f = (1.44 × 10⁻¹⁸)/(6.63 × 10⁻³⁴)

f ≈ 2.17 × 10¹⁵ Hz

Calculate Wavelength

λ = c/f

λ = (3.00 × 10⁸)/(2.17 × 10¹⁵)

λ ≈ 1.38 × 10⁻⁷ m

or:

138 nm

This wavelength lies in the ultraviolet region.

---

Common Mistakes

Thinking Upward Transitions Release Energy

Upward transitions require energy to be absorbed.

Thinking Downward Transitions Require Energy

Downward transitions release energy.

Ignoring Negative Energy Values

Remember to subtract carefully:

−3 − (−10) = +7

Reporting Negative Photon Energy

ΔE for the electron may be negative during emission, but the photon energy is reported as a positive magnitude.

Thinking Electrons Can Have Any Energy

Bound electrons occupy allowed quantized states.

Thinking an Electron Can Stop Between Levels

The spaces between levels do not represent allowed bound electron energies.

Assuming Every Excited Electron Goes Directly to the Ground State

Electrons may return through several transitions.

Forgetting Conservation of Energy in a Cascade

The total energy released through several transitions equals the overall energy difference between the starting and ending states.

Thinking Energy-Level Diagrams Show Physical Height

They show energy, not physical position.

Confusing Excitation with Ionization

Excitation keeps the electron bound.

Ionization removes the electron.

Forgetting the Frequency Relationship

Larger energy gap means higher photon frequency.

Forgetting the Wavelength Relationship

Larger energy gap means shorter photon wavelength.

---

Key Terms

Atomic transition — A change of an electron from one allowed atomic energy state to another.

Energy level — An allowed energy state of an electron.

Quantized — Restricted to particular discrete values.

Ground state — The lowest allowed energy state.

Excited state — An allowed energy state above the ground state.

Excitation — The process of moving an electron to a higher bound energy state.

De-excitation — A transition from a higher energy state to a lower one.

Absorption — The transfer of energy to an atom, potentially causing an upward transition.

Emission — The release of energy from an atom, often as a photon.

Photon — A discrete packet of electromagnetic energy.

Energy gap — The energy difference between two allowed states.

Energy-level diagram — A representation showing the allowed electron energies in an atom.

Initial state — The energy state occupied before a transition.

Final state — The energy state occupied after a transition.

Energy change (ΔE) — The difference between final and initial energies.

Planck's constant — The constant relating photon energy to frequency.

Frequency (f) — The number of wave cycles passing a point each second.

Wavelength (λ) — The distance between equivalent points on successive waves.

Electron volt (eV) — A unit of energy commonly used for atomic-scale processes.

Cascade — A sequence of transitions through intermediate energy states.

Ionization — The removal of an electron from an atom.

Ionization energy — The minimum energy required to remove an electron from a specified atomic state.

Ionization limit — The energy at which an electron is no longer bound to the atom.

Emission spectrum — The characteristic wavelengths emitted by excited matter.

Absorption spectrum — A spectrum showing wavelengths reduced by absorption.

Spectral line — A feature corresponding to a particular photon wavelength and atomic transition.

Conservation of energy — The principle that energy cannot be created or destroyed, only transferred or transformed.

---

Key Takeaways

• Electrons in atoms occupy quantized energy levels.
• An atomic transition occurs when an electron changes from one allowed energy state to another.
• The ground state is the lowest allowed energy state.
• Higher allowed states are excited states.
• Electrons move upward by absorbing energy.
• Electrons move downward by releasing energy.
• The electron energy change is:

ΔE = E_final − E_initial

• Positive ΔE means the electron gained energy.
• Negative ΔE means the electron lost energy.
• During photon emission, the photon carries the magnitude of the lost energy:

E_photon = |ΔE|

• Photon energy and frequency are related by:

E = hf

• Frequency and wavelength are related by:

c = fλ

• Larger energy gaps produce higher-energy photons.
• Higher-energy photons have higher frequencies.
• Higher-frequency photons have shorter wavelengths.
• Electrons cannot occupy arbitrary energies between allowed bound states.
• A photon must have an appropriate energy to cause a particular upward transition.
• Excited electrons can return directly to a lower level or through a cascade of intermediate levels.
• Energy is conserved during atomic transitions.
• Different transitions produce different photon energies.
• Different photon energies produce different spectral lines.
• Atomic emission spectra therefore provide evidence for quantized energy levels.
• Absorption spectra are produced when photons cause allowed upward transitions.
• Energy-level diagrams show energy states, not literal electron positions.
• Excitation and ionization are different: excitation leaves the electron bound, while ionization removes it.
• Different elements have different energy-level structures and therefore different characteristic spectra.

The simplest rules to remember are:

Electron up → energy absorbed

Electron down → energy emitted

Bigger energy gap → higher frequency → shorter wavelength

---

Check Your Understanding

1. What is an atomic transition?

2. What does it mean for atomic energy levels to be quantized?

3. Define the ground state.

4. Define an excited state.

5. What must happen for an electron to move to a higher energy level?

6. What happens when an excited electron returns to a lower energy level?

7. What does an upward arrow represent on an energy-level diagram?

8. What does a downward arrow represent?

9. State the equation for electron energy change.

10. An electron moves from −10 eV to −6 eV. Calculate ΔE and state whether energy is absorbed or released.

11. An electron moves from −2 eV to −8 eV. Calculate ΔE.

12. How much photon energy is emitted in Question 11?

13. Why is the emitted photon's energy reported as positive even though the electron's ΔE is negative?

14. Explain why a 3 eV photon cannot cause a transition requiring 5 eV in a simplified atomic model.

15. What is meant by a cascade?

16. An electron drops 2 eV and then 5 eV in two transitions. How much total energy has it lost?

17. How does this demonstrate conservation of energy?

18. What happens to photon frequency as the energy difference between levels increases?

19. What happens to photon wavelength as the energy difference increases?

20. A transition releases 4.80 × 10⁻¹⁹ J. Calculate the photon frequency using h = 6.63 × 10⁻³⁴ J·s.

21. Calculate the wavelength of the photon in Question 20.

22. Explain how atomic transitions produce an emission spectrum.

23. Explain how atomic transitions produce an absorption spectrum.

24. Why does each element have a characteristic spectral pattern?

25. Explain why an energy-level diagram should not be interpreted as showing an electron's physical height.

26. Explain the difference between excitation and ionization.

27. Why can an excited electron produce more than one photon while returning to the ground state?

28. Convert 5.0 eV into joules.

29. An electron falls through an energy difference of 6.0 eV. Would its photon have a longer or shorter wavelength than a photon produced by a 3.0 eV transition? Explain.

30. Challenge: An atom has the following energy levels:

E₄ = −1 eV
E₃ = −3 eV
E₂ = −6 eV
E₁ = −10 eV

a. Identify the ground state.
b. Identify the excited states shown.
c. Calculate ΔE for E₁ → E₂.
d. Is energy absorbed or emitted in part c?
e. Calculate ΔE for E₂ → E₄.
f. Calculate ΔE for E₄ → E₁.
g. What is the energy of the photon emitted in part f?
h. Calculate the energy released during E₄ → E₃.
i. Calculate the energy released during E₃ → E₁.
j. Show that the total energy released in E₄ → E₃ → E₁ equals the energy released in E₄ → E₁.
k. Which possible downward transition produces the highest-energy photon?
l. Which produces the highest-frequency photon?
m. Which produces the shortest-wavelength photon?
n. Could a 4 eV photon be absorbed by an electron initially in E₁? Identify the resulting state if possible.
o. Could a 5 eV photon produce an upward transition from E₁ using the levels shown? Explain.
p. Explain how these energy levels could produce several different spectral lines.