Nuclei Set-5

Q1. A 10 g sample of a radioactive isotope has half-life $T_{1/2}=6\ \mathrm{h}$. Using the decay law $N(t)=N_0\left(\tfrac{1}{2}\right)^{t/T_{1/2}}$, what mass remains after $t=15\ \mathrm{h}$?

Q2. Consider the fusion reaction $^2_1\mathrm{H}+^2_1\mathrm{H}\to\;^3_2\mathrm{He}+n$. Given atomic masses $m(^2\mathrm{H})=2.014102\ \mathrm{u}$, $m(^3\mathrm{He})=3.016029\ \mathrm{u}$, $m(n)=1.008665\ \mathrm{u}$ and $1\ \mathrm{u}=931.5\ \mathrm{MeV}/c^2$, the Q-value of the reaction is approximately:

Q3. A nucleus with mass number $A=200$ splits symmetrically into two fragments each of mass number $100$. If the binding energy per nucleon is $7.5\ \mathrm{MeV}$ for the parent and $8.5\ \mathrm{MeV}$ for each daughter, the energy released per fission (approx.) is:

Q4. $^{238}\mathrm{U}$ decays by $\alpha$ emission to $^{234}\mathrm{Th}$ with $Q=4.27\ \mathrm{MeV}$. Treating the decay as a two-body process with the parent initially at rest, the kinetic energy of the recoiling $^{234}\mathrm{Th}$ nucleus is approximately (use mass numbers as proportional to masses, $m_\alpha\approx4$, $m_{\mathrm{Th}}\approx234$):

Q5. An unstable parent can decay to a daughter by either positron emission ($\beta^+$) or by orbital electron capture (EC). For atomic masses (including electrons) the mass difference $\Delta m=M_{\rm parent}-M_{\rm daughter}$ is $\Delta m=0.000858\ \mathrm{u}\approx0.80\ \mathrm{MeV}/c^2$. Using $m_e c^2=0.511\ \mathrm{MeV}$ and $1\ \mathrm{u}=931.5\ \mathrm{MeV}/c^2$, which decay mode is energetically allowed and what are the Q-values?

Q6. A radioactive isotope has half-life $T_{1/2}=5\ \mathrm{h}$. If the number of nuclei at time $t$ is given by $N=N_0\left(\tfrac{1}{2}\right)^{t/T_{1/2}}$, what fraction of the original sample remains after $t=15\ \mathrm{h}$?

Q7. An alpha-emitting nucleus (initially at rest) has decay $Q=5.00\ \mathrm{MeV}$. Let $m_\alpha\approx4\,u$ and the daughter nucleus mass $M_d\approx208\,u$. Using conservation of momentum and energy for two-body decay, the kinetic energy of the alpha is $K_\alpha = Q\displaystyle\frac{M_d}{M_d+m_\alpha}$. Approximately what is $K_\alpha$?

Q8. A proton of kinetic energy $E=1.60\ \mathrm{MeV}$ is directed head-on towards a stationary copper nucleus ($Z=29,\ A=63$). The distance of closest approach for a head-on Coulomb collision is $r=\dfrac{1.44\,Z_1Z_2}{E}\ \mathrm{fm}$ (with $E$ in MeV) and the nuclear radius is $R=r_0A^{1/3}$ with $r_0=1.2\ \mathrm{fm}$. Which statement is correct?

Q9. A parent nuclide P with half-life $T_P=5\ \mathrm{d}$ decays to daughter D with half-life $T_D=1\ \mathrm{d}$. For the chain P→D (initially only P present) the activities satisfy, for $t\gg T_D$,

Numerically, what is the ratio $A_D/A_P$ for this case when $t\gg T_D$?

Q10. The neutral atomic masses of a parent and daughter are $M_p=40.00000\,u$ and $M_d=39.99900\,u$. Using $1\,u=931.5\ \mathrm{MeV}/c^2$ and $m_e c^2=0.511\ \mathrm{MeV}$, decide which decay mode(s) are energetically allowed. Use
$Q_{\beta^+}=(M_p-M_d-2m_e)c^2$ and $Q_{EC}=(M_p-M_d)c^2$.

Q11. A radioactive sample decays to $12.5\%$ of its initial activity in $60\ \text{days}$. Using $A(t)=A_0\left(\tfrac{1}{2}\right)^{t/T_{1/2}}$, the half‑life $T_{1/2}$ is:

Q12. Two isotopes A and B are present initially with equal numbers of nuclei. Their half‑lives are $T_{1/2,A}=10\ \text{d}$ and $T_{1/2,B}=20\ \text{d}$. After $30\ \text{d}$, the ratio of their activities is $A_A/A_B$, where $A(t)=\lambda N(t)$ and $\lambda=\ln2/T_{1/2}$. The value of $A_A/A_B$ is:

Q13. A parent nucleus P (half‑life $20\ \text{h}$) decays to a radioactive daughter D (half‑life $5\ \text{h}$). Initially only P is present. At what time $t$ (in hours) will the activities of parent and daughter be equal? Use $\lambda=\ln2/T_{1/2}$ and the standard growth–decay relation for the daughter.

Q14. A neutral atom X has atomic mass $M_X=100.0000\ \mathrm{u}$. Possible daughter atomic masses are: for $\beta^-$ decay $M_{Y}=100.0030\ \mathrm{u}$, for $\beta^+$/electron capture $M_{Z}=99.9950\ \mathrm{u}$, and for $\alpha$ decay the residual daughter's atomic mass would be $95.9990\ \mathrm{u}$. Take $m_e=0.0005486\ \mathrm{u}$ and $1\ \mathrm{u}=931.5\ \mathrm{MeV}/c^2$. Using $Q_{\beta^-}=(M_X-M_Y)c^2$, $Q_{\beta^+}=(M_X-M_Z-2m_e)c^2$ and $Q_{\alpha}=M_Xc^2-(M_{\rm daughter}+M_\alpha)c^2$ (with $M_\alpha=4.001506\ \mathrm{u}$), which decay modes are energetically allowed?

Q15. A neutral atom has atomic mass $M_{\rm parent}=100.0000\ \mathrm{u}$ and decays only by electron capture (EC) to a daughter with atomic mass $M_{\rm daughter}=99.9995\ \mathrm{u}$. Using $Q_{\rm EC}=(M_{\rm parent}-M_{\rm daughter})c^2$ and $Q_{\beta^+}=Q_{\rm EC}-2m_ec^2$ with $m_e=0.0005486\ \mathrm{u}$ and $1\ \mathrm{u}=931.5\ \mathrm{MeV}/c^2$, what happens if every atom is fully ionized (no bound electrons present)?