Question

A 1 Ci source of radiation is a significant source. 238 U is an alpha emitter. What mass of 238 U has an activity of 1 Ci?

Short answer

the mass of uranium is 2.96x10³kg

Step-by-step solution

Step 1

Number of atom is,

Here R₀ is initial and r is final decay rate

$$\begin{gathered} \mathrm{r}=\frac{\mathrm{In}2}{{\mathrm{t}}_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}} \\ \mathrm{Here}{\mathrm{t}}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{is}\mathrm{half}-\mathrm{life} \end{gathered}$$

Step 2

Convert Ci to Bq

$1Ci=3.7x{10}^{10}Bq$

So,

And final decay rate is calculated

Substitute 4.47x10⁹ year for t_1/2 from Appendix C in above final rate equation

$$\begin{gathered} r=\frac{In2}{4.47x{10}^{9}year} \\ =(1.55x{10}^{-10}decay/year)\left(\frac{1year}{31536x{10}^{3}s}\right) \\ =4.92x{10}^{-18}decay/s \end{gathered}$$

Step 3

Then the number of atom is calculated as follows

Substitute 3.7x10¹⁰ decay/sec for R₀ and 4.92x10^-18 decays for r in above eqution

$$\begin{gathered} {\mathrm{N}}_0=7.5\mathrm{x}{10}^{27}\mathrm{atoms} \\ \mathrm{Then}\mathrm{the}\mathrm{mass}\mathrm{of}7.5\mathrm{x}{10}^{27}\mathrm{atoms}\mathrm{is} \\ \mathrm{m}=2.96\mathrm{x}{10}^3\mathrm{kg} \end{gathered}$$

Hence the mass of uranium is 2.96x10³kg