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Radon in the Basement

Nuclear decay and recoil

Radon-222 gas seeps into some houses from the soil beneath them. It decays by alpha emission:

^{222}_{86}\text{Rn} \rightarrow\ ^{218}_{84}\text{Po} + ^{4}_{2}\text{He}

Atomic masses:

^{222}\text{Rn}

222.017578\ \text{u}

;

^{218}\text{Po}

218.008973\ \text{u}

;

^{4}\text{He}

4.002603\ \text{u}

. Assume the radon nucleus is at rest when it decays, and that the speeds involved are small enough for non-relativistic mechanics. Let

Q

be the energy released,

m_\alpha

the mass of the alpha particle and

M

the mass of the daughter nucleus.

Data: c = 3.00 × 10⁸ m s⁻¹ · e = 1.60 × 10⁻¹⁹ C · 1 u = 1.661 × 10⁻²⁷ kg = 931.5 MeV/c²

(a)

calculate1 mark

Calculate the energy

Q

released in one decay, in MeV, and explain why the atomic masses given (which include the atoms' electrons) can be used without any correction for electron masses.

(b)

derive3 marks

Show that the kinetic energy of the alpha particle is

K_\alpha = \dfrac{M}{M + m_\alpha}Q

. Use the limiting case

M \to \infty

to comment on your result, and calculate

K_\alpha

and the kinetic energy of the recoiling polonium nucleus.

(c)

calculate2 marks

Calculate the speeds of the alpha particle and of the recoiling polonium nucleus, and use your answer to comment on the assumption that non-relativistic mechanics is adequate.

(d)

discuss2 marks

Typical chemical bond energies are a few eV. Discuss why the recoil of the polonium nucleus matters even though it takes less than 2% of the energy released. A student claims that if the radon atom were chemically bonded to a much heavier dust particle when it decayed, the dust particle would take the recoil and the alpha particle would leave with all of

Q

. Use your results from (b) to evaluate this claim.

Your score: 0 / 8(0 of 4 parts marked — open “Mark it” on each part)