Practise › Questions › Nuclear radius and density
Nuclear radius and density questions
How do you measure something ten thousand times smaller than the atom it sits in? Nuclear radii can be estimated from the closest approach of an alpha particle and measured by diffracting electrons off the nucleus. The two agree on a striking result, that nuclei light and heavy are packed to very nearly the same density.
18 original questions · 49 marks · the nuclear radius and density notes · Nuclear physics
On this topic the library is tagged for AQA, Edexcel, OCR A.
Every question here is written for this library rather than taken from a past paper. Write your answer out before opening a mark scheme: the schemes award marks point by point, and the marks are easier to see when you have something of your own to compare against.
State the equation relating nuclear radius to nucleon number, and give the approximate value of the constant R0.
Mark scheme
R = R0A1/3, where A is the nucleon number (1); R0 ≈ 1.2 × 10−15 m (1.2 fm) (1).Calculate the radius of a nucleus with nucleon number A = 64.
R0 = 1.2 × 10−15 mMark scheme
R = R0A1/3 = 1.2 × 10−15 × 641/3 = 1.2 × 10−15 × 4 (1)
R = 4.8 × 10−15 m (1)State what the relationship R = R0A1/3 implies about the density of nuclear matter.
Mark scheme
Volume ∝ R3 ∝ A, and mass ∝ A (1); so the density (mass/volume) is approximately the same for all nuclei (1).State typical values for the radius of an atom and for the radius of a nucleus.
Mark scheme
Atomic radius: about 1 × 10−10 m (1). Nuclear radius: a few femtometres, of order 10−15 m to 10−14 m (1).A student has measured nuclear radii R for nuclides covering a wide range of nucleon numbers A. State the graph the student should plot to obtain a straight line through the origin, and state what the gradient of that line represents.
Mark scheme
Plot R against A1/3 (1); the gradient of the line is the constant R0, about 1.2 fm (1).Calculate the radius of an aluminium nucleus, for which A = 27.
R0 = 1.2 × 10−15 mMark scheme
R = R0A1/3 = 1.2 × 10−15 × 271/3 = 1.2 × 10−15 × 3 (1)
R = 3.6 × 10−15 m (1)Taking the mass of a nucleon as 1.66 × 10−27 kg and the volume per nucleon as that of a sphere of radius R0, estimate the density of nuclear matter.
R0 = 1.2 × 10−15 mMark scheme
ρ = u/((4/3)πR03) (1)
= (1.66 × 10−27)/((4/3)π × (1.2 × 10−15)3) (1)
ρ = 2.29 × 1017 kg m−3 (1)Calculate the ratio of the radius of a nucleus with A = 125 to the radius of a nucleus with A = 8.
Mark scheme
R125/R8 = (125/8)1/3 (1)
ratio = 2.5 (1)Explain how the near-constant density of nuclei supports the idea that nucleons are closely packed.
Electron diffraction shows that a nuclide has a nuclear radius of 6.0 × 10−15 m. Determine the nucleon number of the nuclide.
R0 = 1.2 × 10−15 mThe radius of a gold atom is 1.35 × 10−10 m and the radius of a gold nucleus is 7.0 × 10−15 m. Calculate the fraction of the volume of the atom occupied by the nucleus, and comment on your answer.
Explain why high-energy electrons are a suitable probe for measuring the radius of a nucleus.
An alpha particle with kinetic energy 5.0 MeV is fired head-on at a gold nucleus (Z = 79). Estimate the distance of closest approach.
k = 8.99 × 109 N m2 C−2, e = 1.60 × 10−19 CCalculate the density of a carbon-12 nucleus (A = 12), taking the nucleon mass as 1.66 × 10−27 kg and R0 = 1.2 × 10−15 m. Compare your value with the density of nuclear matter in general, 2.3 × 1017 kg m−3.
Describe how electron diffraction can be used to determine the radius of a nucleus.
Determine the minimum initial kinetic energy, in MeV, that an alpha particle needs in order to approach within 4.0 × 10−14 m of the centre of a gold nucleus (Z = 79) in a head-on collision.
k = 8.99 × 109 N m2 C−2, e = 1.60 × 10−19 C, 1 MeV = 1.60 × 10−13 JFor one nuclide, the closest-approach method with 5 MeV alpha particles gives a nuclear radius estimate of 4.5 × 10−14 m, while electron diffraction gives 7.2 × 10−15 m. Explain why the two values differ, and deduce which is the better estimate of the nuclear radius.
A neutron star can be modelled as a sphere with the density of nuclear matter, 2.3 × 1017 kg m−3. Estimate the radius of a neutron star of mass 4.0 × 1030 kg.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
Practise nuclear radius and density one question at a time
The player marks nothing for you. It shows one question, waits, then shows the scheme so you can mark yourself, and brings a question back sooner when it went badly.