Suppose that 10 grams of carbon ( 14C )is released in a nuclear energy plant accident. How long will it take for the 10 gramto decay to 1 gram?
look up the half-life of C14
1=10e^(-.693 t/thalf)
solve for t
Hint. Take Ln of each side.
To determine how long it will take for 10 grams of carbon-14 (14C) to decay to 1 gram, we need to use the concept of half-life. The half-life of carbon-14 is approximately 5730 years.
First, let's understand what half-life means. The half-life is the time it takes for half of the radioactive substance to decay. In this case, the half-life of carbon-14 is 5730 years, meaning that after 5730 years, half of the carbon-14 will have decayed.
To calculate the number of half-lives it takes for 10 grams to decay to 1 gram, we can use the equation:
N = (initial mass) / (final mass) = (initial mass) / (half-life mass)
N = 10 grams / 1 gram = 10
So, we need to find how many times the initial mass (10 grams) can be divided by the half-life mass:
10 = 2^n
where n is the number of half-lives.
Taking the logarithm base 2 of both sides of the equation:
log2(10) = log2(2^n)
n = log2(10)
Using a calculator, we find:
n ≈ 3.32
Since we cannot have fractions of half-lives, we can round up to the nearest whole number. Therefore, it will take approximately 4 half-lives for 10 grams of carbon-14 to decay to 1 gram.
To find the total time it takes, we need to multiply the half-life (5730 years) by the number of half-lives:
Total time = half-life × number of half-lives = 5730 years × 4 = 22,920 years
Therefore, it will take approximately 22,920 years for 10 grams of carbon-14 to decay to 1 gram.