22.4.3Quantum and nuclear physics (A Level)

Calculating Discrete Energies

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Lesson summary

22.4.3 Calculating Discrete Energies — AS PHYSICS 9702 notes: Calculate photon energies, frequencies and wavelengths for transitions between atomic energy levels.Photon energy — ΔE = E2 − E1 = hf = hc/λ.E1 is the lower level and E2 is the higher level.Wavelength — λ = hc/(E2 − E1…

Dependence
Useful values
Reverse conversion
Sense check
Three levels
Shortest wavelength

Section 1 of 8The key equation

Photon energy

ΔE = E₂ − E₁ = hf = hc/λ.
E₁ is the lower level and E₂ is the higher level.

Wavelength

λ = hc/(E₂ − E₁).

Dependence

A larger energy difference gives a shorter wavelength, and a smaller difference gives a longer wavelength.

Useful values

Useful values hc = 1.99 × 10⁻²⁵ J m, or 1240 eV nm, so λ (nm) = 1240 / E (eV).

Section 2 of 8Working with units

Take the difference between the two levels in eV.

Multiply by 1.60 × 10⁻¹⁹ to get joules.

Use f = E/h or λ = hc/E.

Convert the wavelength to nm if sensible by dividing by 10⁻⁹.

Reverse conversion

Use E = hc/λ in joules and divide by 1.60 × 10⁻¹⁹ to get eV.

Sense check

Visible photons have energies between about 1.7 eV (red) and 3.3 eV (violet).

Section 3 of 8Multiple transitions

Three levels

Levels 1 (ground), 2 and 3 allow downward transitions 3 → 2, 3 → 1 and 2 → 1, giving three lines.

Shortest wavelength

The largest gap, 3 → 1, gives the shortest wavelength.

Energies add

E₃₁ = E₃₂ + E₂₁, and the frequencies add in the same way.

Wavelengths do not add

Do not add wavelengths of two transitions to get the combined transition.

Number of lines

For N levels the maximum number of emission lines is N(N − 1)/2.
Not all may be seen in a given experiment, so this is an upper limit.

Section 4 of 8Excitation by electron collision

Excitation by electron collision

Electron energy

An electron accelerated through a p.d.
V has kinetic energy eV.

Condition

It can excite an atom only if eV is at least as large as the energy gap.

Partial transfer

The electron need not give all its energy and keeps the remainder.

Photon difference

A photon must be fully absorbed with an exact match, but a colliding electron can keep the excess.

Minimum voltage

The minimum accelerating voltage for the first excitation is the gap in eV divided by e, numerically the gap in volts.

Section 5 of 8Finding an unknown level

Find the photon energy from the measured wavelength using E = hc/λ.

Decide from the context whether the transition is emission (down) or absorption (up).

For emission from an unknown upper level, add the photon energy to the lower level.

For emission to an unknown lower level, subtract the photon energy from the upper level.

Check that the new level lies below zero for a bound state.

Diagram aid

A quick energy level sketch shows the direction and prevents sign mistakes.

Ratios

If photon A has twice the frequency of photon B, it has twice the energy and half the wavelength.

Section 6 of 8More from the specification

Discrete atomic energy levels

In isolated atoms, electrons can only occupy certain quantised energy levels,
- (such as hydrogen)
meaning electron transitions between these levels result in the emission or absorption of photons with discrete energies.

Photon emission equation

When an excited electron transitions from a higher energy level E2 to a lower energy level E1, the energy change corresponds to the energy of the emitted photon: delta_E = E2 - E1 = h f = h c / lambda.

Line emission spectra

An emission spectrum consists of sharp, coloured lines of light on a dark background,
where each spectral line corresponds to a specific electronic transition and wavelength characteristic of the element.

Hydrogen line spectrum series

For hydrogen atoms, transitions ending at n=1 produce the ultraviolet Lyman series,
while transitions to n=2 produce visible light lines in the Balmer series of the hydrogen spectrum.

Spectrum

Range of colours in white light.

Series

Components in one loop, sharing the e.m.f. between them.

Section 7 of 8Key terms

Term	Meaning
Energy difference (ΔE)	The difference between two energy levels, equal to the energy of the photon emitted or absorbed.
Photon energy relation	ΔE = E₂ − E₁ = hf = hc/λ.
Excitation energy	The energy needed to raise an electron from one level to a higher level.
Planck constant	6.63 × 10⁻³⁴ J s, relating photon energy to frequency.
Electronvolt	A unit of energy equal to 1.60 × 10⁻¹⁹ J.

Section 8 of 8Exam technique

Not converting eV to joules before using h in J s — multiply by 1.60 × 10⁻¹⁹ first.

Adding wavelengths of two transitions — add the energies or frequencies, not the wavelengths.

Using the energies themselves rather than their difference in E = hf — use E₂ − E₁.

Giving a wavelength with the wrong power of ten — visible light is about 4 × 10⁻⁷ to 7 × 10⁻⁷ m.

Assuming an electron must lose all its energy to excite an atom — a photon must be absorbed fully but a colliding electron keeps the remainder.