How to figure out half-lives
Half-live of Indium-116 is 14 seconds. How long will it take for 32 grams of this istotopw to decay to 1.0 grams. I know the answer is 70seconds but do not know how to get it!
Every 14 seconds, the isotope's mass is halved.
You can iterate as TutorCat suggests, or you can follow this formula
k = 0.693/t1/2
Solve for k and substitute it into the equation below.
ln(No/N) = kt
No = 32
N = 1
k from above.
solve for t.
To figure out the half-life and time it takes for a radioactive isotope to decay, you can use the formula:
N = N₀ * (1/2)^(t/t₁/₂)
Where:
N is the final amount of the isotope,
N₀ is the initial amount of the isotope,
t is the time it takes for the isotope to decay,
t₁/₂ is the half-life of the isotope.
In this case, you know that the half-life of Indium-116 is 14 seconds, and you want to find out how long it will take for 32 grams of the isotope to decay to 1.0 grams.
1. Start by substituting the given values into the formula:
1.0g = 32g * (1/2)^(t/14s)
2. Divide both sides of the equation by 32g to isolate the exponential term:
(1/32) = (1/2)^(t/14s)
3. Take the logarithm (base 1/2) of both sides to eliminate the exponential:
log₂(1/32) = log₂((1/2)^(t/14s))
4. Use the logarithmic property to move the exponent to the front:
log₂(1/32) = (t/14s) * log₂(1/2)
5. Simplify the logarithmic expressions:
-5 = (t/14s) * (-1)
6. Multiply both sides by -14s to solve for t (time):
-5 * (-14s) = t
70s = t
Therefore, it will take 70 seconds for 32 grams of Indium-116 to decay to 1.0 grams.