Answer
$ \approx \bf 207\rm \;million \;years$
Work Step by Step
This problem involves dating a lava rock using the potassium-argon dating method. The isotope $^{40}\text{K}$ has a half-life of 1.28 billion years and decays by two routes:
- 89% decays into $^{40}\text{Ca}$ (calcium) by beta-minus decay.
- 11% decays into $^{40}\text{Ar}$ (argon) by electron capture.
The ratio of $^{40}\text{Ar}$ to $^{40}\text{K}$ in the rock is 0.013.
We need to find the age of the rock.
- Let $ N_0 $ represent the initial number of $^{40}\text{K}$ atoms at the time of lava solidification.
- Let $ N_K $ represent the current number of $^{40}\text{K}$ atoms remaining.
- Let $ N_{\text{Ar}} $ represent the number of $^{40}\text{Ar}$ atoms formed by the decay of $^{40}\text{K}$.
Since 11% of $^{40}\text{K}$ decays into $^{40}\text{Ar}$, so
$$
N_{\text{Ar}} = 0.11 (N_0 - N_K)\tag 1
$$
The ratio of $^{40}\text{Ar}$ to $^{40}\text{K}$ atoms in the rock is given as:
$$
\frac{N_{\text{Ar}}}{N_K} = 0.013
$$
So,
$$
N_{\text{Ar}}=0.013 N_K\tag 2
$$
Plug into (1);
$$
0.013 N_K= 0.11 (N_0 - N_K)
$$
Rearrange to isolate $ \dfrac{N_0}{N_K} $:
$$
\frac{0.11 (N_0 - N_K)}{N_K} = 0.013
$$
$$
0.11 \left( \frac{N_0}{N_K} - 1 \right) = 0.013
$$
$$
\frac{N_0}{N_K} - 1 = \frac{0.013}{0.11}
$$
$$
\frac{N_0}{N_K} = 1 + \frac{0.013}{0.11}=\bf 1.11818
$$
Therefore:
$$
\frac{N_K}{N_0} = \frac{1}{1.11818} \approx \bf 0.89431\tag 3
$$
This means that 89.4% of the original $^{40}\text{K}$ atoms remain at the time when the $^{40}\text{Ar}/^{40}\text{K}$ ratio is 0.013.
Now we need to calculate the Age Using the Decay Formula:
$$
N_K = N_0 \left( \frac{1}{2} \right)^{t / t_{1/2}}
$$
$$
\frac{N_K}{N_0}= \left( \frac{1}{2} \right)^{t / t_{1/2}}
$$
Plug from (4);
$$
0.89431= \left( \frac{1}{2} \right)^{t / 1.28 }
$$
Take the Natural Logarithm of Both Sides:
$$
\ln(0.89431) = \frac{t}{1.28 } \ln\left( \frac{1}{2} \right)
$$
Solve for $ t $:
$$
t=\dfrac{ 1.28 \ln(0.89431)}{\ln\left( \frac{1}{2} \right)}
$$
$$
t=\bf 0.206276\;\rm bilion\;yesr=\color{red}{\bf 206,276,000}\;yr $$
