Question

Radioactive Decay

The half-life of radium is 1690 years.

If 10 grams is present now, how much will be present in 50 years?

Short answer

The amount of radium present in 50 years is 9.797 grams.

Step-by-step solution

Step 1. Given information

The half-life of radium is 1690 years.

If 10 grams is present now, how much will be present in 50 years

Step 2. Radium present in 50 years 

The amount A of radioactive material present at time t is expressed by the function $N\left(t\right)=N_0{e}^{kt}$where $k<0$represents the rate of decay.

The half-life of radium is 1690 years.

Substitute $N_0=10,N\left(t\right)=\frac{10}{2}=5,t=1690$in the function $N\left(t\right)=N_0{e}^{kt}$

role="math" localid="1647290301897" $$\begin{gathered} N\left(t\right)=N_0{e}^{kt} \\ 5=10e^{k\times 1690} \\ \frac{1}{2}=e^{k\times 1690} \\ \ln \left(0.5\right)=\ln \left(e^{k\times 1690}\right) \\ \ln \left(0.5\right)=k\times 1690 \\ k=\frac{\ln \left(0.5\right)}{1690} \\ k=-0.00041 \end{gathered}$$

Now radium present in 50 years can be calculated by,

substituting role="math" localid="1647290364834" $N_0=10,k=-0.00041,t=50$in the function $N\left(t\right)=N_0{e}^{kt}$

$$\begin{gathered} N\left(t\right)=N_0{e}^{kt} \\ N\left(50\right)=10e^{-0.00041\times 50} \\ N\left(50\right)=9.797 \end{gathered}$$