Question

The radium isotope ${}^{223}Ra$, an alpha emitter, has a half-life of localid="1650483334573" $11.43days$. You happen to have a localid="1650483350108" $1.0g$cube of ${}^{223}Ra$, so you decide to use it to boil water for tea. You fill a well-insulated container with $100mL$of water at $18°C$and drop in the cube of radium. How long will it take the water to boil?

Short answer

Time taken to boil water is,$∆t=18.9s$

Step-by-step solution

Given information

We are given that,

Half life is, $t_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=11.43days=9.876\times {10}^{5}s$

And $m=100mL=0.10L$

And${}^{223}Ra\to {}^{219}Rn+{}^{4}He$

Explanation

Energy is,

$$\begin{gathered} E=\left[m\left({}^{223}Ra\right)-m\left({}^{219}Rn\right)-m\left({}^{4}He\right)\right]\times 931.49MeV/u \\ =\left[223.018499u-219.009477u-4.002602\right]\times 931.49MeV/u \\ =5.98MeV \end{gathered}$$

Energy needed to raise temperature of water at $18°Cto100°C$

localid="1650483848390" $$\begin{gathered} Q=mc∆t \\ =\left(0.10\right)\left(\frac{1kg}{1L}\right)\left(4190J/kgK\right)\left(100K-18K\right) \\ =34400J \end{gathered}$$

Now, value of $\alpha -particle$is,

localid="1650483872050" $5.98\times {10}^{6}eV\times \left(1.6\times {10}^{-19}J/eV\right)=9.57\times {10}^{-13}J$

No. of decays,

localid="1650483894424" $$\begin{gathered} N=\frac{Q}{\alpha -particle} \\ =\frac{34400J}{9.57\times {10}^{-13}J} \\ =3.59\times {10}^{16} \end{gathered}$$

Simplify

No. of radium atoms is,

$$\begin{gathered} N_0=\frac{1g}{223g/mol}\times 6.02\times {10}^{23}atoms/mol \\ =2.7\times {10}^{21}atoms \end{gathered}$$

Now, $∆t$is given as,

localid="1650483972186" $$\begin{gathered} \frac{dN}{dt}=\frac{∆N}{∆t}=-r{N}_0 \\ ∆t=-\frac{∆N}{r{N}_0} \\ =-\tau \frac{∆N}{N_0} \\ =\frac{t_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\ln \left(2\right)}\times \frac{∆N}{N_0} \\ =\frac{9.876\times {10}^{5}s}{\ln \left(2\right)}\times \frac{\left(-3.59\times {10}^{16}\right)}{2.7\times {10}^{21}} \\ =18.9s \end{gathered}$$