Question

uranium - 238 decays to thorium - 234 with a half - life of 4.5×10^9 years. how many years will it take for 75% of the uranium - 238 to decay?
0.8×10^9 years
4.5×10^9 years
4.5×10^10 years
9.0×10^9 years
3.8×10^9 years

Explanation:

Step1: Recall the radioactive - decay formula

The radioactive - decay formula is $N = N_0(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, where $N$ is the amount of the remaining radioactive substance at time $t$, $N_0$ is the initial amount of the radioactive substance, $T_{1/2}$ is the half - life of the radioactive substance.

Step2: Determine the remaining fraction of uranium - 238

If 75% of uranium - 238 has decayed, then the remaining fraction of uranium - 238 is $N/N_0=1 - 0.75 = 0.25$.

Step3: Substitute into the decay formula

Substitute $N/N_0 = 0.25$ and $T_{1/2}=4.5\times10^{9}$ years into the formula $N = N_0(\frac{1}{2})^{\frac{t}{T_{1/2}}}$. We get $0.25=(\frac{1}{2})^{\frac{t}{4.5\times10^{9}}}$. Since $0.25=\frac{1}{4}=(\frac{1}{2})^2$, then $\frac{t}{4.5\times10^{9}} = 2$.

Step4: Solve for $t$

Multiply both sides of the equation $\frac{t}{4.5\times10^{9}} = 2$ by $4.5\times10^{9}$ to find $t$. So $t = 2\times4.5\times10^{9}=9.0\times10^{9}$ years.

Answer:

$9.0\times10^{9}$ years