Question

Find the mass of \(^{239}{\rm{Pu}}\) that has an activity of\(1.00\,\mu {\rm{Ci}}\).

Short answer

The mass of \(^{239}{\rm{Pu}}\) that has an activity of\(1.00\,\mu {\rm{Ci}}\) is \(17.2\,\mu {\rm{g}}\).

Step-by-step solution

Definition of mass

Mass is a unit of measurement for the amount of matter in a body.

use the radioactive decay law

Consider the given problem and solve.

From appendix A, read the atomic mass number of \({\;^{259}}{\rm{Pu}}\) :

\({A_{{\rm{Pu}}}} = 259\)

From appendix B, read the half life time of \(^{259}{\rm{Pu}}\):

\({t_{1/2}} = 2.41 \cdot {10^4}{\rm{y}}\)

Using the radioactive decay law,

\(R = \frac{{N\ln 2}}{{{t_{1/2}}}}\)

Hence, the fact: \(1\,{\rm{Ci}} = 3.70 \cdot {10^{10}}\,\;{\rm{Bq}}\).

Find the mass

Let us solve the given problem.

We obtain,

\(\begin{aligned} N &= \frac{{R{t_{1/2}}}}{{\ln 2}}\\ &= \frac{{1.00 \times {{10}^6} \times 3.70 \times {{10}^{10}}\;{{\rm{s}}^1} \times 2.41 \times {{10}^4} \times 360 \times 24 \times 3600\;{\rm{s}}}}{{0.693}}\\ &= 4.00 \times {10^{16}}\end{aligned}\)

Mass of Plutonium:

\(\begin{aligned} m &= NAu\\ &= 4.00 \times {10^{16}} \times 259 \times 1.66 \times {10^{27}}\;{\rm{kg}}\\ &= 17.2\,\mu {\rm{g}}\end{aligned}\)

Therefore, mass of Plutonium is \(17.2\,\mu {\rm{g}}\).