A 130-mg sample of technetium-99 m is used for a diagnostic test. If technetium-99 m has a half-life of 6.0 h, how many milligrams of the technetium-99 m sample remains active 30 h after the test?
Express your answer to two significant figures and include the appropriate units.
To solve this problem, we can use the half-life formula:
N = N0 * (1/2)^(t/T)
Where:
N is the amount of the sample remaining after time t
N0 is the initial amount of the sample
t is the elapsed time
T is the half-life of the sample
In this case, N0 is 130 mg, t is 30 h, and T is 6.0 h. Plugging these values into the formula, we can find N:
N = 130 mg * (1/2)^(30/6.0)
N = 130 mg * (1/2)^5
N = 130 mg * 1/32
N = 4.0625 mg
So, approximately 4.1 mg of the technetium-99 m sample remains active 30 h after the test.
To solve this problem, we can use the formula for exponential decay:
N(t) = N(0) * (1/2)^(t / T)
Where:
N(t) is the amount of the sample remaining at time t,
N(0) is the initial amount of the sample,
t is the time elapsed,
T is the half-life of the sample.
In this case, N(0) is 130 mg, t is 30 h, and T is 6.0 h.
Plugging these values into the formula, we get:
N(30) = 130 mg * (1/2)^(30 / 6.0)
Calculating the exponent:
30 / 6.0 = 5
Substituting the exponent:
N(30) = 130 mg * (1/2)^5
Calculating the value inside the parentheses:
(1/2)^5 = 1/32
Substituting the value inside the parentheses:
N(30) = 130 mg * 1/32
Calculating the product:
N(30) = 4.0625 mg
Therefore, approximately 4.06 mg of the technetium-99 m sample remains active 30 h after the test.