K A Tsokos Solutions for Chapter: Quantum and Nuclear Physics (HL), Exercise 2: Test yourself

An alpha particle is fired head-on at a stationary gold nucleus from far away. The distance of the closest approach is $8.5\times {10}^{-15} m$.  The initial speed of the particle $v_i = \_\_\_\times {10}^7 m/s$. (Take the mass of the alpha particle to be $6.64\times {10}^{-27} kg.$)

Radium's first excited nuclear level is $0.0678 \mathrm{MeV}$ above the ground state.The wavelength of the photon emitted is $1.83\times {10}^{-x} \mathrm{m}$. Find the value of x.

Two unstable isotopes are present in equal numbers(initially). Isotopes A has a half-life of 4min and isotope B has a half-life of 3min. Calculate the ratio of the activity of A to that of B After 0 min.

Two unstable isotopes are present in equal numbers(initially). Isotopes A has a half-life of 4min and isotope B has a half-life of 3min. Calculate the ratio of the activity of A to that of B After 4 min.

Two unstable isotopes are present in equal numbers(initially). Isotopes A has a half-life of 4min and isotope B has a half-life of 3min. Calculate the ratio of the activity of A to that of B After 12 min.

A sample contains two unstable isotopes. A counter placed near it is used to record the decays. Discuss how you would determine each of the half-lives of the isotopes from the data.

The half-life of an isotope with a very long half-life cannot be measured by observing its activity as a function of time since the variation in activity over any reasonable time interval would be too small to be observed. Let m be the mass in grams of a given isotope of the long half-life.

Show that the number of nuclei present in this quantity is $N_0 = \frac{m}{\mu }{N}_A$ where $\mu$is the molar mass of the isotope in $g mo{l}^{-1}$ and $N_A$ is the Avogadro constant.

The half-life of an isotope with a very long half-life cannot be measured by observing its activity as a function of time since the variation in activity over any reasonable time interval would be too small to be observed. Let m be the mass in grams of a given isotope of the long half-life.

From $A = -\frac{dN}{dt} = N_0\lambda e^{-\lambda t}$. Show that the initial activity is $A_0 \frac{m{N}_A}{\mu }\lambda$ And hence that the half-life can be determined by measuring the initial activity(in Bq) and the mass of the sample(in grams).