Question

A standard curve of radioactive isotope decay is shown at the top of the right column. The graph line shows the fraction of the radioactive isotope over time (before present) in units of half lives. Recall that a half-life is the amount of time it takes for half of the radioactive isotope to decay. Labeling each data point with the corresponding fractions will help orient you to this graph. Draw an arrow to the data point for half-life = 1 and write the fraction of ¹⁴C that will remain after one half life. Calculate the fraction of ¹⁴C remaining at each half-life and write the fractions on the graph near arrows pointing to the data points. Convert each fraction to a decimal number and round off to a maximum of three significant digits (zero at the beginning of the number do not count as significant digits). Also write each decimal number in scientific notation.

Short answer

The fraction of ¹⁴C remaining after one half-life is $1\times \frac{1}{2}=\frac{1}{2}$.

In decimal and scientific notation, $\frac{1}{2}=0.5=5\times {10}^{-1}$.

Step-by-step solution

Step 1:14C remaining after one half-life

The fraction of ¹⁴C remaining after one half-life is $1\times \frac{1}{2}=\frac{1}{2}$.

In decimal and scientific notation, $\frac{1}{2}=0.5=5\times {10}^{-1}$.

Step 2: 14C remaining after two half-lives

The fraction of ¹⁴C remaining after two half-lives is $\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$.

In decimal and scientific notation, $\frac{1}{4}=0.25=2.5\times {10}^{-1}$.

Step 3: 14C remaining after three half-lives

The fraction of ¹⁴C remaining after three half-lives is $\frac{1}{4}\times \frac{1}{2}=\frac{1}{8}$.

In decimal and scientific notation, $\frac{1}{8}=0.125=1.25\times {10}^{-1}$.

Step 4: 14C remaining after four half-lives

The fraction of ¹⁴C remaining after four half-lives is $\frac{1}{8}\times \frac{1}{2}=\frac{1}{16}$.

In decimal and scientific notation, $\frac{1}{16}=0.0625=6.25\times {10}^{-2}$.

Step 5: 14C remaining after five half-lives

The fraction of ¹⁴C remaining after five half-lives is $\frac{1}{16}\times \frac{1}{2}=\frac{1}{32}$.

In decimal and scientific notation, $\frac{1}{32}=0.0313=3.13\times {10}^{-2}$.

Step 6: 14C remaining after six half-lives

The fraction of ¹⁴C remaining after six half-lives is $\frac{1}{32}\times \frac{1}{2}=\frac{1}{64}$.

In decimal and scientific notation, $\frac{1}{64}=0.0156=1.56\times {10}^{-2}$.

Step 7: 14C remaining after seven half-lives

The fraction of ¹⁴C remaining after seven half-lives is $\frac{1}{64}\times \frac{1}{2}=\frac{1}{128}$.

In decimal and scientific notation, $\frac{1}{128}=0.00781=7.81\times {10}^{-3}$.

Step 8: 14C remaining after eight half-lives

The fraction of ¹⁴C remaining after eight half-lives is $\frac{1}{128}\times \frac{1}{2}=\frac{1}{256}$.

In decimal and scientific notation, $\frac{1}{256}=0.00391=3.91\times {10}^{-3}$.

Step 9: 14C remaining after nine half-lives

The fraction of ¹⁴C remaining after nine half-lives is $\frac{1}{256}\times \frac{1}{2}=\frac{1}{512}$.

In decimal and scientific notation, $\frac{1}{512}=0.00195=1.95\times {10}^{-3}$.

Step 10: 14C remaining after ten half-lives

The fraction of ¹⁴C remaining after ten half-lives is $\frac{1}{512}\times \frac{1}{2}=\frac{1}{1024}$.

In decimal and scientific notation, $\frac{1}{1024}=0.000977=9.77\times {10}^{-4}$.