half life carbon -14 is 5,730 years. If you start out 50 grams, how long will it take until it reaches 2 grams?
To determine how long it will take for a sample of carbon-14 to decay to 2 grams from an initial mass of 50 grams, we can use the concept of half-life.
The half-life of carbon-14 is 5,730 years, which means that after this time has passed, half of the original carbon-14 atoms will have decayed. We can use this information to calculate the number of half-lives needed to reach 2 grams.
1. Start by finding the number of half-lives. Since each half-life corresponds to a reduction in mass by half, we can calculate the number of half-lives required to reach 2 grams from 50 grams:
Number of Half-lives = log2 (initial mass / final mass)
= log2 (50g / 2g)
Using a logarithmic calculator, we find that the number of half-lives is approximately 4.3219.
2. Next, calculate the time it takes for 4.3219 half-lives to occur. Since each half-life is 5,730 years, we can multiply the number of half-lives by the length of a single half-life:
Total time = (Number of half-lives) * (Half-life duration)
= 4.3219 * 5,730 years
The total time is approximately 24,707.41 years.
Therefore, it will take approximately 24,707.41 years for a 50-gram sample of carbon-14 to decay to 2 grams.
x = Xi e^-kt
x/Xi = .5 = e^-5730 k
ln .5 = -5730 k
-.69315 = -5730 k
k = 1.21*10^-4
x = Xi e^-1.21*10^-4 t
2 = 50 e^-1.21*10^-4 t
.04 = e^-1.21*10^-4 t
ln .04 = -1.21*10^-4 t
-3.22 = -1.21*10^-4 t
2.66 * 10^4
= 26,600 years