To find out how long it will take for a 221 mg sample of West Virginium to decay to 44.2 mg, we need to use the exponential decay formula:
\[ N(t) = N_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]
Where:
- \( N(t) \) is the remaining amount of substance at time \( t \)
- \( N_0 \) is the initial amount of substance
- \( T_{1/2} \) is the half-life of the substance
- \( t \) is the time that has passed
Given:
- \( N_0 = 221 \) mg (initial mass)
- \( N(t) = 44.2 \) mg (final mass)
- \( T_{1/2} = 236 \) years (half-life)
We want to solve for \( t \). Let's rearrange the formula to solve for \( t \):
First, divide both sides by \( N_0 \):
\[ \frac{N(t)}{N_0} = \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]
Let's plug in the given values and take the natural logarithm of both sides to remove the exponent:
\[ \frac{44.2}{221} = \left(\frac{1}{2}\right)^{\frac{t}{236}} \]
\[ \ln\left(\frac{44.2}{221}\right) = \ln\left(\left(\frac{1}{2}\right)^{\frac{t}{236}}\right) \]
Using the power rule for logarithms on the right side, we get:
\[ \ln\left(\frac{44.2}{221}\right) = \frac{t}{236} \cdot \ln\left(\frac{1}{2}\right) \]
Now we need to solve for \( t \):
\[ t = \frac{\ln\left(\frac{44.2}{221}\right)}{\ln\left(\frac{1}{2}\right)} \cdot 236 \]
First, calculate the value of the natural log of the fraction:
\[ \ln\left(\frac{44.2}{221}\right) = \ln\left(\frac{1}{5}\right) \]
\[ \ln\left(\frac{44.2}{221}\right) \approx \ln(0.2) \]
Now calculate the natural log of 1/2, which is the natural log of 0.5:
\[ \ln\left(\frac{1}{2}\right) = \ln(0.5) \]
Finally, calculate \( t \) using these natural logarithms:
\[ t \approx \frac{\ln(0.2)}{\ln(0.5)} \cdot 236 \]
Let's plug in approximate values for natural logarithms:
\[ \ln(0.2) \approx -1.6094 \]
\[ \ln(0.5) \approx -0.6931 \]
So the calculation for \( t \) is:
\[ t \approx \frac{-1.6094}{-0.6931} \cdot 236 \]
\[ t \approx \frac{1.6094}{0.6931} \cdot 236 \]
\[ t \approx 2.322 \cdot 236 \]
\[ t \approx 547.992 \]
Finally, round to one decimal place:
\[ t \approx 548.0 \] years
Therefore, it will take approximately 548.0 years for a 221 mg sample of West Virginium to decay to 44.2 mg.
