How long will it take 200 mg of carbon-14 to decay to the point where only 75 mg remain if the half-life is 5770 years?
k = 0.693/t1/2
solve for k
Then
ln(No/N) = kt
No = 200 mg
N = 75 mg
k from above
Solve for t = time in years.
you want t where
200*2^(-t/5770) = 75
2^(-t/5770) = 75/200 = 0.375
-t/5770 = log(.375)/log(2)
t = -5770*log(.375)/log(2)
t = 8164 years
Makes sense since 1/2 > .375 > 1/4
so in half lives, 1 < t < 2
To determine the time it takes for a specific amount of carbon-14 to decay, we can use the half-life formula:
N = N₀ * (1/2)^(t / T)
Where:
- N is the final amount of carbon-14 (75 mg in this case)
- N₀ is the initial amount of carbon-14 (200 mg in this case)
- t is the time it takes for decay to occur
- T is the half-life of carbon-14 (5770 years in this case)
Now, let's solve for t in the equation:
75 mg = 200 mg * (1/2)^(t / 5770)
Rearranging the equation, we have:
(1/2)^(t / 5770) = 75 / 200
Dividing both sides by 200, we get:
(1/2)^(t / 5770) = 0.375
Taking the natural logarithm (ln) of both sides:
ln((1/2)^(t / 5770)) = ln(0.375)
Applying the logarithm property, we can bring down the exponent:
(t / 5770) * ln(1/2) = ln(0.375)
Now, let's solve for t:
t / 5770 = ln(0.375) / ln(1/2)
Using a calculator, we can find the natural logarithms:
t / 5770 ≈ -0.9808 / -0.6931
t / 5770 ≈ 1.415
Cross-multiplying:
t ≈ 1.415 * 5770
t ≈ 8155 years
Therefore, it will take approximately 8155 years for 200 mg of carbon-14 to decay to the point where only 75 mg remain, given a half-life of 5770 years.
To find out how long it will take for 200 mg of carbon-14 to decay to 75 mg, we can use the concept of radioactive decay and the half-life of carbon-14.
The half-life of carbon-14 is given as 5770 years. This means that after every 5770 years, half of the original amount of carbon-14 will decay. So, we begin by finding out how many half-lives it will take for the amount of carbon-14 to reduce from 200 mg to 75 mg.
Let's calculate the number of half-lives:
(200 mg) / (half-life amount) = (200 mg) / (100 mg) = 2 half-lives
So, it will take 2 half-lives for the amount of carbon-14 to reduce from 200 mg to 75 mg.
Next, we need to calculate the time it takes for 2 half-lives to elapse:
(time for one half-life) * (number of half-lives) = (5770 years) * (2) = 11,540 years
Therefore, it will take approximately 11,540 years for 200 mg of carbon-14 to decay to the point where only 75 mg remains, given the half-life of 5770 years.