Atomic spectra: Class 12 Physics Notes, Short Questions and Numericals

Rutherford's atomic model and its failure, the Bohr model of hydrogen, energy levels and orbit radii, the hydrogen spectral series, emission and absorption spectra, X-rays and their spectrum, and the laser.

Key points

Rutherford's alpha scattering experiment (done by Geiger and Marsden) fired alpha particles at a thin gold foil. Most passed straight through, a few were deflected through large angles, and about one in eight thousand bounced almost straight back. This showed that an atom has a tiny, dense, positively charged nucleus holding nearly all the mass, with electrons moving around it in mostly empty space.

Rutherford's model has a defect. An electron going round a nucleus is accelerating, and classical theory says it should radiate energy and spiral into the nucleus. Atoms would then be unstable and would give a continuous spectrum. In reality atoms are stable and give line spectra. This was solved by Bohr in 1913.

Postulates of the Bohr model of hydrogen:

An electron moves in certain circular orbits without radiating energy. These are stationary states.
Only orbits are allowed where the angular momentum is a whole number multiple of h/2π: mvr = nh/2π, with n = 1, 2, 3 and so on.
Radiation is emitted or absorbed only when an electron jumps between two orbits. The photon energy is hf = Ei − Ef.

From these ideas the radius of the nth orbit is rn = n² r₁, where the first Bohr radius is r₁ = 0.529 × 10⁻¹⁰ m (0.529 Å). The orbits are not equally spaced; they grow as n². The speed of the electron falls as n increases.

The energy of the electron in the nth orbit is En = −13.6/n² eV. The negative sign means the electron is bound. The ground state (n = 1) has E = −13.6 eV. The first excited state has −3.4 eV, and the second −1.51 eV. The ionisation energy of hydrogen is the energy needed to take the electron from n = 1 to infinity, which is 13.6 eV. As n increases the levels come closer together and approach 0.

When an electron falls from a higher level ni to a lower level nf, a photon is emitted. The wavelength is given by 1/λ = R(1/nf² − 1/ni²), where R = 1.097 × 10⁷ m⁻¹ is the Rydberg constant. This agrees with the measured hydrogen spectrum.

Hydrogen spectral series (final level nf):

Lyman series, nf = 1: ultraviolet region.
Balmer series, nf = 2: mostly visible region.
Paschen series, nf = 3: infrared region.
Brackett series, nf = 4, and Pfund series, nf = 5: far infrared region.

An emission spectrum is made by a hot gas or vapour emitting light. A line emission spectrum has bright lines on a dark background, and each element has its own pattern. An absorption spectrum is seen when white light passes through a cool gas. Dark lines appear at exactly the same wavelengths that the gas would emit. Dark lines in sunlight (Fraunhofer lines) tell us which elements are present in the Sun's atmosphere.

X-rays are produced when fast electrons strike a metal target in an X-ray tube. A hot cathode emits electrons that are accelerated by a high voltage of several kilovolts. Most of the energy becomes heat. The X-ray spectrum has two parts. The continuous spectrum comes from the slowing of electrons (braking radiation). It has a minimum wavelength λmin = hc/(eV), which depends only on the tube voltage. The characteristic lines come from electrons dropping into vacancies in the inner shells of the target atoms, and they identify the target element.

X-rays have wavelengths from about 10⁻¹¹ m to 10⁻⁹ m. They are electromagnetic waves that affect photographic plates, ionise gases and cause fluorescence. They are used in medicine for imaging and treatment, and in industry to find cracks. Too much exposure damages living cells.

A laser (light amplification by stimulated emission of radiation) gives a light beam that is monochromatic, coherent and highly directional. It works by stimulated emission: a photon makes an excited atom drop to a lower level and emit a second identical photon. A laser needs a population inversion, meaning more atoms in the excited level than in the lower level. This is achieved by pumping. Mirrors at the ends of the laser cavity make light travel back and forth. Uses include surgery, cutting, barcode readers and optical fibre communication.

Study tip: Memorise En = −13.6/n² eV and r₁ = 0.529 Å, since most numericals come from them. Draw an energy level diagram with the Lyman, Balmer and Paschen arrows. When using hc = 1240 eV nm, keep the wavelength in nm and convert at the end.

Definitions and flashcards

Say the meaning first, then open the card to check.

Nucleus

The small dense positive core of an atom that holds almost all its mass.

Stationary state

An orbit in which the electron does not radiate energy.

Ground state

The lowest energy level of an atom.

Ionisation energy

The energy needed to remove an electron from an atom in its ground state.

Line spectrum

A spectrum of separate bright lines emitted by excited atoms.

Absorption spectrum

A continuous spectrum crossed by dark lines where a gas absorbs light.

Characteristic X-rays

X-rays of fixed wavelengths that identify the target element.

Population inversion

A state with more atoms in a higher energy level than in a lower one.

Stimulated emission

Emission of a photon caused by another photon of the same energy.

Formula sheet

Name	Formula	What it means
Bohr's quantum condition	mvr = nh / 2π	n is a whole number.
Orbit radius	rn = n² × 0.529 × 10⁻¹⁰ m	For hydrogen.
Energy levels	En = −13.6 / n² eV	For hydrogen.
Photon emitted	hf = Ei − Ef	Energy lost in a jump.
Spectral lines	1/λ = R(1/nf² − 1/ni²)	R = 1.097 × 10⁷ m⁻¹.
Minimum X-ray wavelength	λmin = hc / (eV)	V is the tube voltage.

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

What did Rutherford's scattering experiment show about the atom? (3 marks)

Most alpha particles went straight through the gold foil, so an atom is mostly empty space. A few were deflected by large angles and some returned backward, so the positive charge and nearly all the mass are packed in a very small nucleus.

State the postulates of Bohr's theory. (3 marks)

Electrons move in stationary circular orbits without radiating. Only orbits with angular momentum nh/2π are allowed. Radiation is emitted or absorbed only when an electron jumps from one orbit to another, with photon energy hf equal to the difference of the two energies.

What is the ionisation energy of hydrogen? (2 marks)

It is the energy needed to remove the electron from the ground state to infinity. It equals 13.6 eV.

Name the series of the hydrogen spectrum that lies in the ultraviolet and the visible regions. (2 marks)

The Lyman series (transitions to n = 1) is in the ultraviolet. The Balmer series (transitions to n = 2) is mainly in the visible region.

Explain how dark lines appear in an absorption spectrum. (3 marks)

When white light passes through a cool gas, atoms of the gas absorb photons whose energies match jumps between their energy levels. The atoms re-emit the energy in all directions, so those wavelengths are weak in the forward beam. Dark lines then appear on the continuous spectrum, at the same wavelengths that the gas emits when hot.

Describe how a laser works. (5 marks)

A laser has an active medium placed between two mirrors, one of them partly transparent. An energy source pumps atoms into an excited level until more atoms are in the excited level than in the lower level, which is called population inversion. A passing photon of the right energy causes an excited atom to emit another photon of the same frequency, phase and direction. This is stimulated emission. The photons travel back and forth between the mirrors and cause more emission, so the beam grows. Part of it leaves through the partly transparent mirror. The light is monochromatic, coherent and narrow.

Numericals with worked steps

Find the energy of the electron in the n = 4 level of hydrogen and the radius of that orbit.

E4 = −13.6 / 4² = −13.6 / 16 = −0.85 eV.
r4 = 4² × 0.529 = 16 × 0.529 = 8.464 Å.

Answer: E = −0.85 eV, r ≈ 8.46 × 10⁻¹⁰ m

Find the energy of the photon emitted when an electron falls from n = 3 to n = 2 in hydrogen, and its wavelength (hc = 1240 eV nm).

E3 = −13.6/9 = −1.511 eV. E2 = −13.6/4 = −3.4 eV.
Photon energy = −1.511 − (−3.4) = 1.889 eV.
λ = 1240 / 1.889 = 656.4 nm.

Answer: E ≈ 1.89 eV, λ ≈ 656 nm (red line of the Balmer series)

Use the Rydberg formula to find the longest wavelength of the Lyman series (R = 1.097 × 10⁷ m⁻¹).

Longest wavelength is for ni = 2, nf = 1.
1/λ = R(1/1 − 1/4) = R × 3/4 = 1.097 × 10⁷ × 0.75 = 8.2275 × 10⁶ m⁻¹.
λ = 1 / 8.2275 × 10⁶ = 1.215 × 10⁻⁷ m.

Answer: λ ≈ 1.22 × 10⁻⁷ m (122 nm)

Find the minimum wavelength of X-rays from a tube operated at 20 kV (hc = 1240 eV nm).

Electron energy = eV = 20 000 eV.
λmin = hc / (eV) = 1240 / 20 000 nm.
= 0.062 nm = 6.2 × 10⁻¹¹ m.

Answer: λmin = 6.2 × 10⁻¹¹ m

Common mistakes

Forgetting that Bohr energies are negative, so the emitted photon energy is the higher level minus the lower level.
Writing the radius as proportional to n instead of n².
Mixing up the series: Lyman is nf = 1 (UV), Balmer nf = 2 (visible), Paschen nf = 3 (infrared).
Saying the X-ray minimum wavelength depends on the target material. It depends only on the tube voltage.
Confusing absorption lines (dark) with emission lines (bright) for the same element.

Practise this chapter

Practice questions and timed test Printable worksheet