Chapter 21

Each statement that follows refers to a comparison between two radioisotopes, $A$ and $X.$ Indicate whether each of the following statements is true or false. (a) If the half-life for $\mathrm{A}$ is shorter than the half-life for $\mathrm{X},\mathrm{A}$ has a larger decay rate constant. (b) If $X$ is "not radioactive," its half-life is essentially zero. (c) If A has a half-life of 10 yr, and $X$ has a half-life of $10,000\mathrm{yr}$, A would be a more suitable radioisotope to measure processes occurring on the 40 -yr time scale.

It has been suggested that strontium-90 (generated by nuclear testing) deposited in the hot desert will undergo radioactive decay more rapidly because it will be exposed to much higher average temperatures. (a) Is this a reasonable suggestion? (b) Does the process of radioactive decay have an activation energy, like the Arrhenius behavior of many chemical reactions (Section 14.5$)?$

Some watch dials are coated with a phosphor, like ZnS, and a polymer in which some of the ${}^1\mathrm{H}$ atoms have been replaced by ${}^3\mathrm{H}$ atoms, tritium. The phosphor emits light when struck by the beta particle from the tritium decay, causing the dials to glow in the dark. The half-life of tritium is 12.3 yr. If the light given off is assumed to be directly proportional to the amount of tritium, by how much will a dial be dimmed in a watch that is 50 yr old?

It takes 180 minutes for a 200 -mg sample of an unknown radioactive substance to decay to $112\mathrm{mg}$. What is the halflife of this substance?

Cobalt-60 is a strong gamma emitter that has a half-life of $5.26\mathrm{yr}$. The cobalt- 60 in a radiotherapy unit must be replaced when its radioactivity falls to $75\mathrm{\%}$ of the original sample. If an original sample was purchased in June $2016,$ when will it be necessary to replace the cobalt- $60?$

How much time is required for a $5.00-g$ sample of ${}^{233}$ Pa to decay to $0.625\mathrm{g}$ if the half-life for the beta decay of ${}^{233}\mathrm{Pa}$ is 27.4 days?

A 10.00 -g plant fossil from an archaeological site is found to have a ${}^{14}\mathrm{C}$ activity of 3094 disintegrations over a period of ten hours. A living plant is found to have a ${}^{14}\mathrm{C}$ activity of 9207 disintegrations over the same period of time for an equivalent amount of sample with respect to the total contents of carbon. Given that the half-life of ${}^{14}\mathrm{C}$ is 5715 years, how old is the plant fossil?

A wooden artifact from an Indian temple nas a ' of 42 counts per minute as compared with an activity of 58.2 counts per minute for a standard zero age. From the half-life of ${}^{14}\mathrm{C}$ decay, 5715 years, calculate the age of the artifact.

Phosphorus- 32 is commonly used in nuclear medicine for the identification of malignant tumors. It decays to sulphur- 32 with a half-life of 14.29 days. If a patient is given 3.5 mg of phosphorus-32, how much phosphorus-32 will remain after 1 month (i.e. 30 days)?

Iodine-131 is used as a nuclear medicine to treat hyperthyroidism. The half- life of ${}^{131}\mathrm{I}$ is 8.04 days. How long will it take for a $500\mathrm{mg}$ sample of ${}^{131}\mathrm{I}$ to decay into $1\mathrm{\%}$ of its original mass?