determine half life of krypton-92 if only 6.0 millograms of an original 96.0 sample remains unchanged after 7.36 seconds
1.84 seconds
ln(No/N) = kt
No = 96
N = 6
k = unknown
t = 7.36 seconds.
Then substitute k into the below equation.
k = 0.693/t1/2 and solve for t1/2.
To determine the half-life of Krypton-92, we can use the formula for exponential decay:
N = N₀ * (1/2)^(t / t₁/₂)
where:
N is the remaining amount of the sample after time t
N₀ is the initial amount of the sample
t₁/₂ is the half-life
t is the time that has passed
In this case, we know the initial amount of the sample (N₀ = 96.0 milligrams), the remaining amount (N = 6.0 milligrams), and the time that has passed (t = 7.36 seconds).
Plugging in these values, we get:
6.0 = 96.0 * (1/2)^(7.36 / t₁/₂)
Divide both sides of the equation by 96.0:
(6.0 / 96.0) = (1/2)^(7.36 / t₁/₂)
Simplify the fraction on the left side:
1/16 = (1/2)^(7.36 / t₁/₂)
Now, we need to solve for t₁/₂. To do this, take the logarithm of both sides (base 1/2):
log₂(1/16) = log₂((1/2)^(7.36 / t₁/₂))
Using the logarithm property log_b(a^c) = c * log_b(a), we can simplify the equation:
-4 = (7.36 / t₁/₂) * log₂(1/2)
Since log₂(1/2) is -1, we can further simplify:
-4 = (7.36 / t₁/₂) * -1
Multiply both sides by -1:
4 = 7.36 / t₁/₂
Multiply both sides by t₁/₂:
4 * t₁/₂ = 7.36
Divide both sides by 4:
t₁/₂ = 7.36 / 4
Simplifying the right side:
t₁/₂ ≈ 1.84 seconds
Therefore, the half-life of Krypton-92 is approximately 1.84 seconds.
To determine the half-life of krypton-92 in this scenario, we can use the formula for exponential decay:
N = N0 * (1/2)^(t / t½)
Where:
N = Final amount of substance
N0 = Initial amount of substance
t = Time that has passed
t½ = Half-life of substance
In this case, we know that the initial amount (N0) is 96.0 milligrams, and the final amount (N) is 6.0 milligrams. The time that has passed (t) is 7.36 seconds. We need to solve for the half-life (t½).
Rearranging the formula, we can isolate the half-life:
(1/2)^(t / t½) = N / N0
Taking the logarithm of both sides will allow us to solve for t½:
log[(1/2)^(t / t½)] = log(N / N0)
Using the properties of logarithms, this simplifies to:
(t / t½) * log(1/2) = log(N / N0)
Now, we can substitute the given values and solve for t½:
(7.36 / t½) * log(1/2) = log(6.0 / 96.0)
To find the value of log(1/2), you can use a scientific calculator or the logarithm function on your phone or computer.
Solving the equation provides the value of the half-life (t½) for krypton-92 in this scenario.