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The Helium Impostor at Mt John

Bohr model and spectra

Astronomers at Mt John University Observatory, Takapō, record the spectrum of a very hot O-type star. Among its absorption lines are some that appear to belong to hydrogen, but they are produced by singly ionised helium, He⁺ — a hydrogen-like ion with one electron orbiting a nucleus of charge

+Ze

, where

Z = 2

. Treat the electron (mass

m_e

, charge

-e

) as moving in a circular orbit of radius

r

around a stationary nucleus. Bohr's quantum condition is that the electron's angular momentum is

m_e v r = \dfrac{nh}{2\pi}

n = 1, 2, 3, \ldots

The energy needed to remove one electron from a neutral helium atom (its first ionisation energy) is

24.6\ \text{eV}

Data: h = 6.63 × 10⁻³⁴ J s · c = 3.00 × 10⁸ m s⁻¹ · e = 1.60 × 10⁻¹⁹ C · mₑ = 9.11 × 10⁻³¹ kg · k = 8.99 × 10⁹ N m² C⁻²

(a)

derive2 marks

By considering the force acting on the electron, together with Bohr's quantum condition, show that the allowed orbit radii are

r_n = \dfrac{n^2 h^2}{4\pi^2 Z k m_e e^2}

and that the total energy is

E_n = -\dfrac{2\pi^2 k^2 m_e e^4}{h^2} \cdot \dfrac{Z^2}{n^2}

(b)

show that2 marks

Evaluate the constant

\dfrac{2\pi^2 k^2 m_e e^4}{h^2}

in eV. Hence show that, in the Bohr model, the He⁺ transition from

n = 6

n = 4

has exactly the same wavelength as the hydrogen transition from

n = 3

n = 2

, and explain why every He⁺ transition between two even-numbered levels mimics a hydrogen line.

(c)

calculate1 mark

Calculate the wavelength of the He⁺ transition from

n = 7

n = 4

, which has no hydrogen counterpart.

(d)

discuss3 marks

For many years the He⁺ lines ending on

n = 4

in hot stars were attributed to hydrogen. Discuss how the pattern of lines in this series shows that they cannot all come from hydrogen, and explain why the series is seen in very hot O-type stars but not in cooler stars such as the Sun. Your answer should include the longest wavelength of a photon able to ionise a neutral helium atom.

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