24.125 CP - Chemistry: Molecular Nature Of Matter And Change

Question

Tritium $\left({}^{3}H;t_{1/2}=12.26yr\right)$ is continually formed in the upper troposphere by interaction of solar particles with nitrogen. As a result, natural waters contain a small amount of tritium. Two samples of wine are analyzed, one known to be made in 1941 and another made earlier. The water in the 1941 wine has 2.23 times as much tritium as the water in the other. When was the other wine produced?

Step-by-Step Solution

Answer

The other wine was produced in 1926

Step 1: Definition of Tritium

A hydrogen radioactive isotope with one proton and two neutrons in its nucleus and three times the mass of ordinary hydrogen.

Step 2: Calculating when the wine was produced

Using the half-life, calculate the rate constant for the rate law.

$t_{\frac{1}{2}} = \frac{\ln 2}{k}\to k = \frac{\ln 2}{t_{\frac{1}{2}}}$

$k = \frac{\ln 2}{12.26yr} = 0.0565$

Using the rate law $\ln \frac{A_o}{A_t} = kt$, calculate the time.

$\ln \frac{2.23}{1} = 0.0565t\to t = \ln \frac{2.32}{1}\times \frac{1}{0.0565}$

t = 15 years

To get the year of production, subtract the time from the initial year, 1941.

1941-15 = 1926