Tritium (half-life = 12.3 yr) is used to verify the age of expensive brandies. If an old brandy contains only 1/64 of the tritium present in new brandy, then how long ago (in yr) was it produced?
Since tritium undergoes radioactive decay with a half-life of 12.3 years, we can use the exponential decay formula to calculate the age of the old brandy.
The fraction of remaining tritium after time t is given by:
\[ \frac{N(t)}{N_0} = \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
Where:
\(N(t)\) = amount of tritium remaining after time t
\(N_0\) = initial amount of tritium
\(t\) = time
\(t_{1/2}\) = half-life of tritium
Given that the old brandy contains only 1/64 of the tritium present in new brandy, we can rewrite the fraction of remaining tritium as:
\[ \frac{1}{64} = \left( \frac{1}{2} \right)^{\frac{t}{12.3}} \]
Taking the natural logarithm of both sides, we get:
\[ \ln{\left( \frac{1}{64} \right)} = \ln{\left( \left( \frac{1}{2} \right)^{\frac{t}{12.3}} \right)} \]
\[ \ln{\left( \frac{1}{64} \right)} = \frac{t}{12.3} \ln{\left( \frac{1}{2} \right)} \]
\[ -3.8712 = \frac{t}{12.3} \times (-0.6931) \]
Solving for t, we get:
\[ t = \frac{-3.8712 \times 12.3}{-0.6931} \]
\[ t = \frac{47.62016}{0.6931} \]
\[ t ≈ 68.74 \]
Therefore, the old brandy was produced approximately 68.74 years ago.