A sample of argon-39 had an original mass of 1578 grams. After 538 years, the sample is 394.5 grams. What is the half-life of argon-39?
A.
135 years
B.
180 years
C.
269 years
D.
538 years
C. 269 years
Explanation:
To find the half-life of argon-39, we can use the half-life formula:
A = A0 * (1/2)^(t/t1/2)
Where:
A = final mass (394.5 grams)
A0 = initial mass (1578 grams)
t = time elapsed (538 years)
t1/2 = half-life
Substitute the given values into the formula:
394.5 = 1578 * (1/2)^(538/t1/2)
Divide both sides by 1578:
0.25 = (1/2)^(538/t1/2)
Substitute 0.25 with (1/2)^2:
(1/2)^2 = (1/2)^(538/t1/2)
Since the bases are the same, the exponents must be equal:
2 = 538/t1/2
t1/2 = 538/2
t1/2 = 269 years
Therefore, the half-life of argon-39 is 269 years.