Tritium (half-life = 12.3 y) is used to verify the age of expensive brandies. If an old brandy contains only the following fraction of the tritium present in new brandy, how long ago was it produced?
1/128
After one half life the ratio is 1/2
After two half lives the ratio is 1/4
After three the ratio is 1/8
so how many half lives to get to 1/128?
24.6
To determine how long ago the old brandy was produced, we need to calculate the number of half-lives it has gone through.
The fraction of tritium remaining after each half-life can be found using the formula:
Remaining fraction = (1/2)^(number of half-lives)
Given that the old brandy contains 1/128th of the tritium present in new brandy, we can set up the equation:
1/128 = (1/2)^(number of half-lives)
To solve for the number of half-lives, we can take the logarithm of both sides of the equation:
log(1/128) = log((1/2)^(number of half-lives))
Using the logarithmic property, we can bring down the exponent:
-7 = number of half-lives * log(1/2)
Now, we can solve for the number of half-lives:
number of half-lives = -7 / log(1/2)
Using the base-10 logarithm, we find:
number of half-lives = -7 / log10(1/2)
Calculating this value gives:
number of half-lives ≈ 47.699
Since each half-life of tritium is 12.3 years, we can find the time by multiplying the number of half-lives by the half-life duration:
time = number of half-lives * half-life duration
Plugging in the values, we get:
time ≈ 47.699 * 12.3
Therefore, the old brandy was produced approximately 586.8 years ago.
To determine how long ago the brandy was produced, we need to calculate the number of half-life periods that passed.
Given that the fraction of tritium in the old brandy is 1/128 of that in the new brandy, we can calculate the number of half-life periods using the equation:
Fraction = (1/2)^(number of half-life periods)
Setting the given fraction equal to 1/128:
1/128 = (1/2)^(number of half-life periods)
To solve for the number of half-life periods, we can take the logarithm of both sides of the equation. Since the base is 1/2, we'll use the base-2 logarithm:
log(base2)(1/128) = number of half-life periods
log(base2)(1/128) = -7
Therefore, the number of half-life periods that have passed is -7. Since half-life periods cannot be negative, we need to take the absolute value:
|number of half-life periods| = 7
Since the half-life of Tritium is 12.3 years, we can determine the age of the brandy by multiplying the number of half-life periods by the half-life:
age = 7 * 12.3
Thus, the brandy was produced about 86.1 years ago.