Atoms and Nuclei — Practice

Question 1

What did each of the three main observations in the alpha scattering experiment tell Rutherford?

Accepted Answer

Most alpha particles passing straight through showed the atom is mostly empty space. A small fraction deflected through large angles showed there is a concentrated positive charge. The rare near-backward scattering showed that charge must also be very massive, since a light target could not reverse a fast alpha particle.

Question 2

Which of Bohr's three postulates was genuinely new, and what did it state?

Accepted Answer

The second, that angular momentum is quantised in units of $h/2\pi$. The first merely asserts that stationary orbits do not radiate, and the third applies Einstein's photon idea to atoms. Everything computable in the model follows from the quantisation condition plus classical mechanics.

Question 3

Why do all nuclei have essentially the same density?

Accepted Answer

Because the radius goes as $A^{1/3}$, so volume is proportional to $A$ while mass is also proportional to $A$. The ratio is therefore constant at about $2.3	imes10^{17}$ kg m$^{-3}$. This constancy shows the nuclear force saturates, with each nucleon binding only its immediate neighbours.

Question 4

Find the wavelength emitted when a hydrogen electron falls from $n = 3$ to $n = 2$.

Accepted Answer

The levels are at $-1.51$ eV and $-3.40$ eV, so 1.89 eV is released. Using $hc = 1240$ eV nm gives 656 nm — the red H-alpha line and the first line of the Balmer series.

Question 5

Hydrogen atoms are excited to the $n = 4$ level. How many distinct spectral lines can appear?

Accepted Answer

Every pair of levels gives a transition, so the count is $4	imes3/2 = 6$ lines: three Lyman, two Balmer and one Paschen. The general result is $n(n-1)/2$.

Question 6

Find the binding energy per nucleon of helium-4, taking $m_p = 1.00728$, $m_n = 1.00867$ and the nuclear mass as 4.00151 u.

Accepted Answer

The parts sum to 4.03190 u, so the mass defect is 0.03039 u. Multiplying by 931.5 gives 28.3 MeV, which is 7.07 MeV per nucleon. Helium-4 is unusually tightly bound for so light a nucleus, which is why alpha particles are emitted as intact units.

Question 7

A sample has a half-life of 20 minutes. What fraction remains after one hour, and what is the mean life?

Accepted Answer

One hour is three half-lives, so one eighth remains, and the activity has also fallen to one eighth since $A = \lambda N$. The mean life is $20/0.693 = 28.9$ minutes — longer than the half-life, not shorter.

Question 8

A 5 MeV alpha particle is fired head-on at a gold nucleus with $Z = 79$. Find the distance of closest approach and comment on its size.

Accepted Answer

Using $r_0 = 2kZe^{2}/K$ with $ke^{2} = 1.44$ MeV fm gives 45.5 fm. This is far larger than the gold nucleus itself, which is about 7 fm across, so the alpha particle never reaches the nucleus. That is exactly why Rutherford's purely Coulombic analysis worked without any knowledge of nuclear forces.

Question 9

Explain using the binding energy curve why both splitting uranium and fusing hydrogen release energy, and compare the yields.

Accepted Answer

Uranium sits near 7.6 MeV per nucleon and its fragments near 8.5, a gain of about 0.9 MeV per nucleon which over 236 nucleons gives roughly 200 MeV per fission. Hydrogen sits at essentially zero and helium-4 at 7.07, a gain of about 7 MeV per nucleon giving 28 MeV for four nucleons. Both move their nuclei toward the iron peak, and fusion wins comfortably per nucleon but requires overcoming the Coulomb barrier at temperatures of order $10^{7}$ K.

Question 10

Beta particles emerge with a continuous range of energies while alpha and gamma emissions are sharply defined. What did this imply?

Accepted Answer

A nucleus dropping between two definite states should release a definite energy, so a continuous beta spectrum appeared to violate energy conservation, and Bohr was prepared to abandon the conservation law at nuclear scale. Pauli instead proposed in 1930 that an unseen neutral, nearly massless particle carries away the balance and shares the energy randomly with the electron. That particle is the neutrino, detected only in 1956, so the continuous beta spectrum is direct evidence for it.

Question 11

Hydrogen atoms in a discharge tube are excited to $n = 4$. How many distinct spectral lines can the sample emit, and how many of them lie in the visible region?

Accepted Answer

The number of distinct lines is $n(n-1)/2 = 4(3)/2 = 6$. The visible ones are the Balmer lines ending on $n = 2$, namely $4	o2$ and $3	o2$, so only **two** of the six are visible. Three end on $n = 1$ and are ultraviolet, and the remaining one, $4	o3$, is infrared. Grouping by the landing level checks the total: $3+2+1 = 6$.

Question 12

Phosphorus-32 decays by $\beta^-$ emission to sulphur-32. Given atomic masses 31.973907 u and 31.972071 u, find the $Q$-value, and explain why no electron mass has to be added or subtracted here but would have to be for $\beta^+$ decay.

Accepted Answer

$Q = (31.973907 - 31.972071)(931.5) = (0.001836)(931.5) = 1.71$ MeV. Atomic masses work unadjusted for $\beta^-$ because the parent atom carries $Z$ electrons while the daughter atom carries $Z+1$, and the extra one is precisely the emitted beta particle, so the counts balance. In $\beta^+$ decay the daughter atom needs one electron fewer while a positron is also created, leaving two electron masses unaccounted for, so $2m_ec^{2} = 1.022$ MeV must be subtracted.

Practice — Atoms and Nuclei

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