The Photoelectric Effect & Atomic Spectra
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The Photoelectric Effect
& Atomic Spectra
Big idea: Light doesn't just travel as a smooth wave — it also arrives in individual, indivisible "packets" of energy called photons. Whether it's knocking an electron clean out of a metal, or getting swallowed whole by an atom to boost an electron to a higher energy level, energy in these interactions always moves in one discrete lump at a time, never in a continuous trickle.
- The photoelectric effect is when electrons ("photoelectrons") are knocked out of a metal's surface by light — and it only works above a minimum ("threshold") frequency, no matter how bright the light is.
- This proves light is quantised — delivered in discrete packets called photons, each carrying energy
E = hf, and each photon can only be absorbed by one electron. - The photoelectric equation
hf = Φ + KEmaxis just energy conservation: photon energy in = energy needed to escape (work function) + leftover kinetic energy. - The electronvolt (eV) is a tiny, convenient energy unit for quantum-scale energies:
1 eV = 1.6 × 10⁻¹⁹ J. - The gold-leaf electroscope experiment gives real, observable evidence for photons — intensity changes speed of emission, frequency changes energy of electrons, and there's a hard threshold frequency cutoff.
- Atomic line spectra happen when excited electrons drop down between fixed energy levels inside atoms, releasing a photon whose energy exactly matches the energy gap — creating a unique "fingerprint" of coloured lines for each element.
Definition: The photoelectric effect is the phenomenon where electrons are emitted from the surface of a metal when it absorbs electromagnetic radiation (usually UV or visible light). These ejected electrons are called photoelectrons — same particle as a normal electron, just given a special name because of how it was released.
This is exactly why the photoelectric effect provides some of the best evidence that light is quantised — carried in discrete packets (photons) rather than as a smooth, continuous wave:
- Each photoelectron can only absorb one whole photon — never a little bit from many photons added together.
- This means only light above a certain threshold frequency f₀ will ever release an electron — dimmer or brighter doesn't matter, only the "size" of each individual packet does.
This classic setup lets you literally the photoelectric effect happen:
- A zinc plate is attached to a gold leaf, which is negatively charged — so it repels away from a central charged rod and stands up at an angle.
- UV light shines on the plate. If the UV frequency is above the threshold, electrons are knocked off the plate.
- As negative charge (electrons) leaves the system, the leaf becomes less negatively charged, repels less, and falls back down towards the rod.
| You change... | You observe... | What it tells us |
|---|---|---|
| UV lamp moved closer (↑ intensity) | Leaf falls faster | More photons/second → more photoelectrons/second. Intensity affects , not energy. |
| Use higher-frequency light | Leaf falls at the same rate | KEmax of each electron increases, but rate of emission is unaffected by frequency alone. |
| Use a filament (low-frequency) lamp instead of UV | No change at all — leaf stays up | Below threshold frequency, electrons are ever emitted, no matter how bright. |
| Charge the plate positively instead | No change — leaf stays down | Emitted electrons get pulled straight back by the positive charge before escaping. |
| Any moment UV (above threshold) hits the plate | Emission is instant | One photon → one electron, immediately. No "charging up" delay like wave theory predicted. |
This equation is just a statement of conservation of energy. The energy carried in by one photon has to go — some of it is "spent" pulling the electron free of the metal (the work function), and whatever energy is left over becomes the electron's kinetic energy as it flies off.
- h = Planck's constant = 6.63 × 10⁻³⁴ J s
- f = frequency of the incident radiation (Hz)
- Φ = work function of the metal (J) — minimum energy needed to free an electron
- ½mv²max (= KEmax) = maximum kinetic energy of the emitted photoelectron (J)
The threshold frequency f₀ is the minimum frequency of EM radiation needed to just barely release a photoelectron — at this exact frequency, the photon has energy to cover the work function, with nothing left over for kinetic energy:
- If
hf < Φ, no electrons are emitted at all — regardless of intensity. KEmaxdepends only on frequency of the incident light, on intensity.- Most emitted photoelectrons actually have less than KEmax — some energy is lost as electrons collide with atoms on their way out of the metal. KEmax is the energy of electrons released right at the very surface, with no energy lost on the way out.
Rearranging the equation into the familiar straight-line form y = mx + c is one of the most
useful things you can do with it:
- y-intercept = −Φ (the work function, read off as a negative value)
- x-intercept = f₀ (the threshold frequency)
- Gradient = h (Planck's constant) — this is actually how Planck's constant was first measured experimentally!
- Below f₀, the line simply doesn't exist — zero electrons are emitted, so the graph sits flat on the x-axis until f₀ is reached.
A graph of KEmax (eV) against frequency f for sodium metal shows a straight line crossing the x-axis at f₀ = 4 × 10¹⁴ Hz. Calculate the work function of sodium in eV.
KE_max = hf − Φ, matching y = mx + cΦ = hf₀Φ = (6.63 × 10⁻³⁴) × (4 × 10¹⁴) = 2.652 × 10⁻¹⁹ JΦ = 2.652 × 10⁻¹⁹ ÷ 1.6 × 10⁻¹⁹ = 1.66 eVQuantum-scale energies (like photon energies or work functions) come out as tiny numbers in Joules — around 10⁻¹⁹ J. That's awkward to write and compare, so physicists invented a more convenient unit built specifically for this scale: the electronvolt (eV).
Definition: One electronvolt is the energy gained by a single electron when it is accelerated, from rest, through a potential difference of exactly 1 volt.
This comes directly from the definition of potential difference, V = E/Q, rearranged to
E = QV. Since an electron's charge is 1.6 × 10⁻¹⁹ C, pushing it through 1 V transfers exactly
1.6 × 10⁻¹⁹ J of energy to it — and we just call that amount "1 eV".
- eV → J: multiply by 1.6 × 10⁻¹⁹
- J → eV: divide by 1.6 × 10⁻¹⁹
When a charged particle accelerates from rest through a potential difference V, all the electrical potential energy it loses is converted into kinetic energy:
Show that the photon energy of light with wavelength 700 nm is about 1.8 eV.
c = fλ → f = c/λ, and E = hf → E = hc/λE = (6.63×10⁻³⁴ × 3.0×10⁸) / (700×10⁻⁹) = 2.84 × 10⁻¹⁹ JE = 2.84×10⁻¹⁹ / 1.6×10⁻¹⁹ = 1.78 eV ≈ 1.8 eV ✓Here's the deep conflict this whole chapter is built around: classical physics treated light purely as
a wave — and that model works great for explaining diffraction and interference (light bending
round obstacles, and beams interfering to make patterns of light and dark). But the photoelectric effect
be explained by a pure wave model. It only makes sense if light also behaves as a stream of
individual particles — photons — each carrying a fixed, discrete amount of energy E = hf.
- Wave behaviour: diffraction, interference (light spreading and combining continuously)
- Particle behaviour: photoelectric effect (instant emission, threshold frequency, one photon = one electron)
This is the famous wave-particle duality of light — it isn't "really" one or the other; it shows whichever behaviour the experiment is set up to reveal.
This is why the photoelectric effect is considered such powerful evidence: the observations from the gold-leaf electroscope experiment (see Section 1) simply don't add up under wave theory, but every single one falls out naturally once you assume light arrives in discrete photon packets that interact one-to-one with individual electrons.
Electrons inside an atom aren't free to have amount of energy — they can only occupy specific, fixed energy levels (like rungs on a ladder — you can stand on any rung, but never hover in between them). When an atom is heated or otherwise "excited", electrons absorb energy and jump up to higher energy levels. But electrons don't like staying excited — they quickly fall back down to lower levels, and every time they do, they release the energy difference as a single photon.
Definition: An emission line spectrum is produced when an excited electron moves from a higher to a lower energy level and emits a photon with energy exactly equal to the difference between those two levels.
Because E = hf and c = fλ, a bigger energy jump between levels always
means a higher frequency and therefore shorter wavelength photon — and vice versa. This is a
really common exam trap, so let's nail it down clearly:
| Bigger energy gap ΔE | Smaller energy gap ΔE |
|---|---|
| Higher frequency f | Lower frequency f |
| Shorter wavelength λ | Longer wavelength λ |
| e.g. drops to ground state (n=1) — often UV | e.g. drops between high levels — often infrared |
In hydrogen specifically, transitions ending at n = 2 happen to fall in the visible range — which is why hydrogen's famous rainbow-line spectrum (violet, blue, light blue, red) is such a classic textbook image. Transitions ending at n = 1 are higher energy (ultraviolet), and those ending at n = 3 or higher are lower energy (infrared).
Every element has a different arrangement of electrons and therefore a different, unique set of energy levels. Since the photon energies (and thus wavelengths/colours) emitted depend entirely on the gaps between levels, no two elements ever produce the same pattern of spectral lines — this is how astronomers identify which elements exist in distant stars, just from analysing the light that reaches us!
An electron in a hydrogen atom drops from the n = 2 level (E₂ = −3.40 eV) to the n = 1 level (E₁ = −13.6 eV). Calculate the wavelength of the emitted photon.
ΔE = E(n=2) − E(n=1) = −3.40 − (−13.6) = 10.2 eVΔE = 10.2 × 1.6×10⁻¹⁹ = 1.632 × 10⁻¹⁸ JΔE = hc/λ, rearranged to λ = hc/ΔE:λ = (6.63×10⁻³⁴ × 3.0×10⁸) / 1.632×10⁻¹⁸λ ≈ 1.22 × 10⁻⁷ m = 122 nm (this is in the ultraviolet range — makes sense, since it's a transition down to the ground state!)| Term / Formula | Meaning |
|---|---|
| Photoelectric effect | Electrons emitted from a metal surface when EM radiation is absorbed |
| Photoelectron | An electron released via the photoelectric effect |
| Work function, Φ | Minimum energy needed to release an electron from a metal surface |
| Threshold frequency, f₀ | Minimum frequency of radiation that can cause photoelectric emission |
E = hf = Φ + ½mv²max | The photoelectric equation (energy conservation) |
hf₀ = Φ | At threshold frequency, KE_max = 0 |
| 1 eV | = 1.6 × 10⁻¹⁹ J — energy gained by an electron across a 1 V p.d. |
eV = ½mv² | Kinetic energy gained accelerating through p.d. V |
| Emission line spectrum | Series of bright lines produced when excited electrons drop energy levels |
ΔE = E₁ − E₂ = hf = hc/λ | Energy of photon emitted = energy level difference |
| h (Planck's constant) | 6.63 × 10⁻³⁴ J s |
| c (speed of light) | 3.0 × 10⁸ m s⁻¹ |
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