Nuclear Physics: Physics 9702 (Cambridge International AS & A Level)

A hospital radiopharmacy stores a small sample of a short-lived radioactive tracer. The decay constant of this tracer is $\lambda = 4.62\times10^{-2}\text{ s}^{-1}$.

What is the half-life of the tracer?

A student studying nucleosynthesis models the oxygen-16 nuclide, $^{16}_{8}\text{O}$, whose nucleus contains 8 protons and 8 neutrons. The mass of a free proton is $m_p = 1.007276\text{ u}$, the mass of a free neutron is $m_n = 1.008665\text{ u}$, and the mass of the assembled oxygen-16 nucleus is $m_{nuc} = 15.990526\text{ u}$.

Take $1\text{ u} = 1.66\times10^{-27}\text{ kg} = 931.5\text{ MeV}$ and $c = 3.00\times10^8\text{ m s}^{-1}$.

(a) Calculate the mass defect $\Delta m$ of the oxygen-16 nucleus, in u. [2]

(b) (i) Convert this mass defect to kilograms, and hence use $E=c^2\Delta m$ to calculate the binding energy of the nucleus in joules. [2]

(b) (ii) Calculate the binding energy of the nucleus in MeV, using $1\text{ u}=931.5\text{ MeV}$. [2]

(c) Calculate the binding energy per nucleon of oxygen-16, in MeV. [1]

A nuclear medicine department receives a sample of a radioisotope for a diagnostic scan. When the sample is first measured, its activity is $A_0=8.00\times10^{10}\text{ Bq}$. The decay constant of the isotope is $\lambda=1.54\times10^{-5}\text{ s}^{-1}$.

(a) Show that the half-life of the isotope is approximately $12.5$ hours. [2]

(b) Calculate the number of undecayed nuclei, $N_0$, present in the sample at the moment of the first measurement. [2]

(c) Calculate the activity of the sample $24.0$ hours after the first measurement. [3]

In an experimental fusion reactor, deuterium and tritium nuclei fuse according to the equation: $^2_1\text{H}+\,^3_1\text{H}\rightarrow\,^4_2\text{He}+\,^1_0\text{n}$

The nuclear masses involved are $m(^2_1\text{H})=2.013553\text{ u}$, $m(^3_1\text{H})=3.015500\text{ u}$, $m(^4_2\text{He})=4.001506\text{ u}$ and $m(^1_0\text{n})=1.008665\text{ u}$.

Take $1\text{ u}=1.66\times10^{-27}\text{ kg}=931.5\text{ MeV}$ and $c=3.00\times10^8\text{ m s}^{-1}$.

(a) With reference to the graph of binding energy per nucleon against nucleon number, explain why this fusion reaction releases energy. [2]

(b) Calculate the total mass before the reaction and the total mass after the reaction, and hence find the mass defect $\Delta m$ for this reaction, in u. [2]

(c) Calculate the energy released in this fusion reaction, in MeV. [2]

(d) Show that this energy is equivalent to about $2.82\times10^{-12}\text{ J}$. [2]

A fission event splits a heavy nucleus X, of nucleon number $236$ and binding energy per nucleon $7.6\text{ MeV}$, into two identical fragment nuclei Y, each of nucleon number $118$ and binding energy per nucleon $8.5\text{ MeV}$.

What is the total energy released in this fission event?

A smoke detector contains a small americium-241 source with $3.00\times10^{14}$ undecayed nuclei. The decay constant of americium-241 is $\lambda=5.08\times10^{-11}\text{ s}^{-1}$.

What is the activity of the source?

Iron-56, $^{56}_{26}\text{Fe}$, is one of the most tightly bound nuclides found in nature. Its nucleus contains 26 protons and 30 neutrons. The mass of a free proton is $m_p=1.007276\text{ u}$, the mass of a free neutron is $m_n=1.008665\text{ u}$, and the mass of the assembled iron-56 nucleus is $m_{nuc}=55.920800\text{ u}$.

Take $1\text{ u} = 931.5\text{ MeV}$.

(a) Calculate the mass defect $\Delta m$ of the iron-56 nucleus, in u. [2]

(b) Calculate the binding energy of the nucleus, in MeV. [2]

(c) Calculate the binding energy per nucleon of iron-56, in MeV. [1]

(d) With reference to the shape of the graph of binding energy per nucleon against nucleon number, explain why iron-56 is one of the most stable nuclides, and why both the fission of very heavy nuclei and the fusion of very light nuclei can release energy. [3]

A fission event studied at a research reactor sees a uranium-235 nucleus absorb a slow neutron and split according to the equation: $^1_0\text{n}+\,^{235}_{92}\text{U}\rightarrow\,^{140}_{54}\text{Xe}+\,^{94}_{z}\text{Sr}+x\,^1_0\text{n}$ where $x$ additional neutrons are also released and $z$ is the proton number of the strontium fragment.

The nuclear masses involved are $m(^{235}_{92}\text{U})=235.043930\text{ u}$, $m(^1_0\text{n})=1.008665\text{ u}$, $m(^{140}_{54}\text{Xe})=139.921646\text{ u}$ and $m(^{94}_{z}\text{Sr})=93.915356\text{ u}$.

Take $1\text{ u} = 1.66\times10^{-27}\text{ kg} = 931.5\text{ MeV}$ and $c = 3.00\times10^8\text{ m s}^{-1}$.

(a) By conserving nucleon number and proton number, find the value of $x$ and the value of $z$. [2]

(b) Calculate the total mass of the particles before the reaction and the total mass of the particles after the reaction, and hence find the mass defect $\Delta m$ for this reaction, in u. [2]

(c) Calculate the energy released in this fission reaction, in MeV. [2]

(d) Calculate the energy released in this fission reaction, in joules. [2]

A student uses a Geiger-Muller tube to investigate a radioactive source, X. Because of naturally-occurring background radiation, the tube registers a small constant count rate even when the source is not present. With no source in the laboratory, the tube registers a background count rate of $20$ counts per minute.

With the source in place, the tube's measured (uncorrected) count rate is $620$ counts per minute at time $t=0$, falling to $170$ counts per minute at $t=15.0$ minutes.

(a) Explain why the background count rate should be subtracted from each measured reading before the decay of X is analysed. [1]

(b) Calculate the corrected count rate due to X alone at $t=0$ and at $t=15.0$ minutes. [2]

(c) Show that the half-life of X is $7.50$ minutes. [2]

(d) Calculate the decay constant of X, in $\text{s}^{-1}$. [2]

A nuclear power station's reactor generates a steady thermal power output of $3.00\text{ GW}$. Assuming this energy originates entirely from the conversion of mass to energy via $E=mc^2$, and taking $c=3.00\times10^8\text{ m s}^{-1}$, calculate the rate at which mass is being converted to energy inside the reactor.