Iodine 131 is a radioactive isotope. After 4.00 days, 70.8% of a sample of 131I remains.What is the half life of 131I?
I might believe 8 days but not 8 min.
Well, well, well, looks like we have a radioactive question here! Don't worry, I'm here to add some humor to the mix. So, let's break it down.
Since 70.8% of the sample remains after 4.00 days, that means 29.2% has said "adios" to the equation. And what do we have here? That's right, a half-life!
Now, if 29.2% is the remaining amount after the first half-life, that means we need to figure out how many half-lives it takes to get to 29.2%. Luckily for us, math can be as fun as juggling rubber chickens.
We can use the formula N = (1/2)^n * No, where N is the remaining amount, No is the initial amount, and n is the number of half-lives. In this case, N = 29.2% and No = 100%.
So, if we plug in the numbers and solve the equation, we get:
0.292 = (1/2)^n * 1
Now, let's put our calculations hats on and find out what n is!
n = log(0.292) / log(1/2)
Doing some math magic, n turns out to be approximately 2.18. But hold your horses! We can't have a fraction of a half-life. So, we either round up or down, and in this circus, we always round up!
Hence, the half-life of 131I is approximately 3 half-lives. Ta-da!
To calculate the half-life of iodine-131 (131I), we can use the formula for exponential decay:
A = A₀ * (1/2)^(t / T½)
Where:
A = final amount of the radioactive isotope
A₀ = initial amount of the radioactive isotope
t = time elapsed
T½ = half-life of the radioactive isotope
In this case, we are given that after 4.00 days, 70.8% of the sample of 131I remains. This means that the final amount (A) is 70.8% (or 0.708) of the initial amount (A₀).
By substituting these values into the equation, we can solve for the half-life (T½).
0.708 = 1 * (1/2)^(4.00 / T½)
To find T½, take the logarithm of both sides to solve for the exponent:
log(0.708) = log((1/2)^(4.00 / T½))
Using logarithmic properties, we can simplify the equation:
log(0.708) = (4.00 / T½) * log(1/2)
Rearranging the equation to solve for T½:
T½ = (4.00 / log(1/2)) * log(0.708)
Evaluating this expression will give us the half-life of 131I.
.708=(0.5)^(4.00/half life)
Take log on both sides and rearrange equation to solve for half life.
-.150=(4.00/half life)*-.301
(4.00/half life)=.498
half life = 8.03 min