Define decay constant.

A sample of 1.6 mol of the radioactive nuclide radon-210 $\left( {{}_{86}^{210}{\rm{Rn}}} \right)$ decays into polonium-206 $\left( {{}_{84}^{206}{\rm{Po}}} \right)$ with the production of one other particle.

$${}_{86}^{210}{\rm{Rn}} \to {}_{84}^{206}{\rm{Po + X}}$$

(i) Identify particle X.
(ii) The radioactive decay constant of radon-210 is 8.0×10^–5s^–1. Determine the time required to produce 1.1 mol of polonium-206.

Particle X has an initial kinetic energy of 6.2MeV after the decay in (b). In a scattering experiment, particle X is aimed head-on at a stationary gold-197 $\left( {{}_{76}^{197}{\rm{Au}}} \right)$ nucleus.

Determine the distance of closest approach of particle X to the Au nucleus.

the probability of decay of a nucleus per unit time;

(i) alpha particle / helium nucleus;

(ii) number of Po nuclei produced=number of Rn nuclei decayed (seen or implied);
$0.5 = 1.6{e^{ - \lambda t}}$;
$t = \left( { - \frac{{\ln \frac{{0.5}}{{1.6}}}}{\lambda } = } \right)\frac{{1.163}}{{8.0 \times {{10}^{ - 5}}}}$;
1.5×10⁴(s);

initial kinetic energy=electric potential energy at closest distance;
kinetic energy $E = \left( {6.2 \times {{10}^6} \times 1.6 \times {{10}^{ - 19}} = } \right)9.9 \times {10^{ - 13}}\left( {\rm{J}} \right)$;
$d = k\frac{{{q_1}{q_2}}}{E} = 8.99 \times {10^9}\frac{{2 \times 79 \times {{\left[ {1.6 \times {{10}^{ - 19}}} \right]}^2}}}{{9.9 \times {{10}^{ - 13}}}}\left( {\rm{m}} \right)$ or $= 3.7 \times {10^{ - 14}}\left( {\rm{m}} \right)$;