If radon-222 has a half-life of 3.8 days, determine how long it takes for only 1/8 of a sample to remain.
Find k from the following.
k = 0.693/t1/2
Then substitute into the following
ln(No/N) = kt
Use a convenient number for No, then N will be 1/8 of that, solve for t.
To determine how long it takes for only 1/8 of a sample of radon-222 to remain, we can use the concept of half-life.
The half-life of radon-222 is given as 3.8 days. This means that in each 3.8-day interval, the amount of radon-222 in a sample will decrease by half.
To find the time it takes for only 1/8 (or 1/2^3) of the sample to remain, we need to determine how many half-lives it would take.
Let's use the equation for half-life decay:
N = N₀ * (1/2)^(t / T₁/₂)
Where:
N is the remaining amount of the sample,
N₀ is the initial amount of the sample,
t is the time passed, and
T₁/₂ is the half-life of the radionuclide.
Since we want only 1/8 (or 1/2^3) of the sample to remain, we can substitute N = N₀ / 8 into the equation:
N₀ / 8 = N₀ * (1/2)^(t / 3.8)
Now, we can proceed to solve for t:
1/8 = (1/2)^(t / 3.8)
To simplify the equation, we can take the logarithm of both sides:
log(1/8) = log((1/2)^(t / 3.8))
Using logarithm properties, we can bring down the exponent:
log(1/8) = (t / 3.8) * log(1/2)
Now, we can solve for t by isolating it:
t / 3.8 = log(1/8) / log(1/2)
Finally, we can solve for t by multiplying both sides by 3.8:
t = (log(1/8) / log(1/2)) * 3.8
Using a calculator, we can evaluate the right-hand side of the equation to find the value of t.