– Fundamental law of radioactive decay:
\[
N=N_{0} e^{-4 t} \text {, }
\]
– Relation betvees the decay constant $\mathrm{X}$, the meas liletime $\mathrm{x}$, and the hall-hife $r$ :
\[
\lambda=\frac{1}{r}=\frac{\ln 2}{T} \text {. }
\]
– Specific activity is the activity of a unit mass of a radiolsotepe.
6.214. Knowing the decay constant $\lambda$ of a nucleus, find:
(a) the probability of decay of the nueleus during the time from 0 to $t$ :
(b) the mean lifetime $\tau$ of the nucleas.
6.215. What fraction of the radioactive cobalt nuclei whose halflife is 71.3 days decays during a month?
6.216. How many beta-particles are emitted during ane hour by $1.0 \mu \mathrm{g}$ of $\mathrm{Na}^{\mathrm{at}}$ radionuelide whose half-life is 15 hours?
6.217. To in vestigate the beta-decay of $\mathrm{Mg}^{33}$ radionuclide, a counter was activated at the moment $t=0$. It registered $N_{1}$ beta-particles by a moment $t_{1}=2.0 \mathrm{~s}$, and by a moment $t_{2}=3 t_{1}$ the number
of registered beta-particles was 2.66 times greater. Find the mean lifetime of the given nuclei.
6.218. The activity of a certain preparation decreases 2.5 times after 7.0 days. Find its half-life.
6.219. At the initial moment the activity of a certain radionuclide totalled 650 particles per minute. What will be the activity of the preparation after half its half-life period?
6.220. Find the decay constant and the mean lifetime of $\mathrm{Co}^{\mathrm{at}}$ radionuclide if its activity is known to decrease $4.0 \%$ per hour. The decay product is nonradioactive.
6.221. A U U preparation of mass $1.0 \mathrm{~g}$ emits $1.24-10^{4}$ alphaparticles per second. Find the half-life of this nuclide and the activity of the preparation.
6.222. Determine the age of ancient wooden items if it is known that the specific activity of $\mathrm{Cu}^{4}$ nuclide in them amounts to $3 / 5$ of that in lately felled trees. The half-life of $\mathrm{C}^{4}$ nuclei is 5570 years.
6.223. In a uranium ore the ratio of $U^{a+}$ nuclei to $\mathrm{Pb}^{\text {aus }}$ nuclei is $\eta=2.8$. Evaluate the age of the ore, assuming all the lead $\mathrm{Pb}^{\text {aw }}$ to be a final decay product of the uranium series. The half-life of $U^{\text {ua }}$ nuclei is $4.5-10$ years.
6.224. Calculate the specific activities of $\mathrm{Na}^{24}$ and $U^{n a}$ nuelides whose half-lifes are 15 hours and $7.1 \cdot 10^{\circ}$ years respectively.
6.225. A small anoust of solution coniaising $\mathrm{Na}^{* 4}$ radionuelide with activity $A=2.0 \cdot 10^{\circ}$ disintegrations per second was injected in the bloodstream of a man. The activity of $1 \mathrm{~cm}^{3}$ of blood sample taken $t=5.0$ hours later turned out to be $A^{\prime}=16$ disintegrations per minute per $\mathrm{cm}^{\prime \prime}$. The half-life of the radionuclide is $T=15$ hours. Find the volume of the man’s blood.
6.226. The specific activity of a preparation consisting of radiosetive $\mathrm{Co}_{0}{ }^{\text {th }}$ and nonradioactive $\mathrm{Co}^{\mathrm{si}}$ is equal to $2.2 \cdot 10^{\mathrm{fl}} \mathrm{dis} /(\mathrm{s} \cdot \mathrm{g})$. The half-life of $\mathrm{Co}^{\mathrm{HH}}$ is 71.3 days. Find the ratio of the mass of radioactive cobalt in that preparation to the total mass of the preparation (in per ceat).
6.227. A certain preparation iscludes twe beta-active components with different half-lifes. The measurements resulted in the following dependence of the natural logarithm of preparation activity on time $t$ expresed in hours:
\begin{tabular}{lccccccccc}
$i$ & 0 & 1 & 2 & 3 & 5 & 7 & 10 & 14 & 20 \\
is 4 & 4.10 & 3.80 & 3.10 & 2.80 & 2.06 & 1.82 & 1.00 & 1.32 & 0.90
\end{tabular}

Find the half-lifes of both components and the ratio of radioactive nuclei of these components at the moment $t=0$.
6.228. A pu radionuelide with half-1ife $T=14.3$ days is produced in a reactor at a constant rate $q=2.7 \cdot 10^{\circ}$ nuclei per second. How soon after the beginsing of production of that radionuclide will its setivity be equal to $A=1.0-10^{\circ} \mathrm{dis} / \mathrm{s}$ ?
6.229. A radionuclide $A_{1}$ with decay constant $\lambda_{1}$ transforms into a radionvelide $A_{2}$ with decay constant $\lambda_{2}$. Assuming that at the initial moment the preparation contained only the radionuclide $\boldsymbol{A}_{\mathrm{k}}$. find:
(a) the equation describing accumulation of the radionuclide $A_{2}$ with time;
(b) the time interval after which the activity of radionuclide $A_{3}$ reaches the maximum value.
6.239. Solve the forezoing problem if $\lambda_{1}-\lambda_{1}-\lambda$.
6.231. A radionuelide $A_{1}$ goes through the transformation chain $A_{1} \rightarrow A_{2} \rightarrow A_{3}$ (stable) with respective decay constants $\lambda_{1}$ and $\lambda_{2}$ : Assuming that at the initial moment the preparation contained only the radionuclide $A_{1}$ equal in quastity to $N_{10}$ nuelei, find the equation describing accumulation of the stable isotope $A_{\text {, }}$.
6.232. A Bine radionuclide decays via the chain
\[
\text { Bine } \overrightarrow{i_{i}} \text { Pollt } \overrightarrow{r_{i}} \text { PLant (otable). }
\]
where the decay constants are $\lambda_{1}=1.60 \cdot 10^{-4} s^{-1}, \lambda_{2}=$ $=5.80 \cdot 10^{-4} \mathrm{~s}^{-1}$. Calculate alpha- and beta-activities of the $\mathrm{Bi}^{10}$ preparation of mass $1.00 \mathrm{mg}$ a month after its manufacture.
6.233. (a) What isotepe is produced from the alpha-radieactive $\mathrm{Ra}^{\mathrm{nin}}$ as a result of five alpha-disintegrations aad four $\beta^{- \text {-disintegra- }}$ tions?
(b) How many alpha- and $\beta^{- \text {-decays does }} \mathbf{U}^{\text {mo }}$ experience before turning finally into the stable $\mathrm{Pb}^{206}$ isotope?
6.234. A stationary $\mathrm{Pb}^{\text {ain }}$ nueleus emits an alpha-particle with kinetic energy $T_{\mathrm{a}}=5.77 \mathrm{MeV}$. Find the recoil velocity of a daughtor nucleus. What fraction of the total energy liberated in this decay is accounted for by the recoil energy of the daughter nucleus?
6.235. Find the amount of heat generated by $1.00 \mathrm{mg}$ of a Po $\mathrm{Pe}^{\mathrm{m}}$ preparation during the mean lifetime peried of these nuelei if the emitted alpha-particles are known to possess the kinetic energy $5.3 \mathrm{MeV}$ and practically all daughter nuelei are formed directly in the ground state.
6.236. The alpha-decay of $\mathrm{P}_{0}{ }^{\text {nut }}$ suelei (in the ground state) is accompanied by emission of two groups of alpha-particles with kinetic energies 5.30 and $4.50 \mathrm{MeV}$. Following the emission of these particles the daughter nuclei are found in the ground and excited states. Find the energy of gamma-quanta emitted by the excited nuclei.
6.237. The mean path length of alpha-particles in air under standard conditions is defined by the formula $R=0.98 \cdot 10^{-47} v_{0}^{\prime} \mathrm{cm}$, where $v_{0}(\mathrm{~cm} / \mathrm{s})$ is the initial velocity of an alpha-particle. Ûsing this formula, find for an alpha-particle with initial kinetie energy 7.0 MeV:
(a) its mean path length;
(b) the average number of ion pairs formed by the given alphaparticle over the whole path $R$ as well as over its first half, assuming the ion pair formation energy to be equal to $34 \mathrm{eV}$.
272
6.238. Find the energy $Q$ liberated in $\beta^{*}$, and $\beta^{*}$-decays and in $K$-capture if the masses of the parent atom $M_{p}$, the daughter atom $M_{4}$ and an electron $m$ are known.
6.239. Taking the values of atomic masses from the tables, find the maximum kinetic energy of beta-particles emitted by Bed nuclei and the corresponding kinetic energy of recoiling daughter nuclei formed directly in the ground state.
6.240. Evaluate the amount of heat produced during a day by a $\beta^{-}$-active $\mathrm{Na}^{24}$ preparation of mass $m=1.0 \mathrm{mg}$. The beta-particles are assumed to possess an average kinetic energy equal to $1 / 3$ of the highest possible energy of the given decay. The hall-life of $\mathrm{Na}^{* 4}$ is $\boldsymbol{T}=15$ hours.
6.241. Taking the values of atomic masses from the tables, calculate the kinetic energies of a positron and a neutrino emitted by $\mathrm{Ci}$ nucleus for the case when the daughter nucleus does not recoil.
6.242. Find the kinetic energy of the recoil nucleus in the positronic decay of a $\mathrm{N}^{2}$ nucleus for the case when the energy of positrons is maximum.
6.243. From the tables of atomic masses determine the velocity of a nucleus appearing as a result of $K$-capture in a Be’ atom provided the daughter nueleus turns out to be is the ground state.
6.24. Passing down to the ground state, excited $\mathrm{Ag}^{104}$ nuclei emit either gamma quanta with energy $87^{7} \mathrm{keV}$ or $\boldsymbol{K}$ conversion electrons whose binding energy is $26 \mathrm{keV}$. Find the velocity of these electrons.
6.245. A free stationary Ir ${ }^{101}$ neleus with exeitation energy $\boldsymbol{E}=129 \mathrm{keV}$ passes to the ground state, emitting a gamma quantum. Caleulate the fractional change of gamma quants energy due to recoil of the nucleus.
6.246. What must be the relative velocity of a source and an absorber consisting of free Irin nuclei to observe the maximum absorption of gamma quanta with energy $\varepsilon=129 \mathrm{keV}$ ?
6.247. A source of gamma quasts is placed at a height $h=20 \mathrm{~m}$ above an absorber. With what velocity should the source be displaced upward to counterbalance completely the gravitational variation of gamma quasta energy due to the Earth’s gravity at the point where the absorber is located?
6.248. What is the minimum height to which a gamma quanta source containing excited $\mathrm{Za}^{\mathrm{e}}$ nuclei has to be raised for the gravitational displacement of the Mössbauer line to exceed the line width itself, when registered on the Earth’s surface? The registered gamma quanta are known to have as energy $\varepsilon-93 \mathrm{keV}$ and appear on transition of $\mathrm{Zn}^{* \prime \prime}$ nuelei to the ground state, and the mean lifetime of the exciled state is $\mathrm{\tau}=14 \mu \mathrm{s}$.