A quantity of oxygen gas had 16.32 g of the radioactive isotope oxygen-19 in it. When measured exactly 10 minutes later, the amount of oxygen-19 was 0.964 g. What is the half-life, in seconds, of oxygen-19?
16.32(1/2)^(t/k) = .964 , where t is in minutes, and k is the half-life in minutes.
(.5)^(10/k) = .0590686.. ( I stored it)
ln both sides
(10/k) ln .5= ln (.059068...)
10/k = 4.081464...
k = 2.4504 minutes or 147.006 seconds
To find the half-life of oxygen-19, we need to calculate the time it takes for half of the initial amount of oxygen-19 to decay.
We know that the initial quantity of oxygen-19 is 16.32 g, and after 10 minutes, it becomes 0.964 g.
To find the half-life, we can use the formula:
amount = initial amount * (1/2)^(time / half-life)
Rearranging the formula to solve for half-life, we have:
half-life = time / log(base 1/2) (amount / initial amount)
Substituting the given values into the formula:
half-life = 10 minutes / log(base 1/2) (0.964 g / 16.32 g)
Now we will convert the time to seconds since the half-life is required in seconds:
half-life = 10 minutes * 60 seconds / log(base 1/2) (0.964 g / 16.32 g)
To evaluate the logarithm, we need to convert the fraction into a decimal:
half-life = 10 minutes * 60 seconds / log(0.0589)
Using a calculator to find the logarithm:
half-life ≈ 1800 seconds / (-2.83027)
Evaluating the expression:
half-life ≈ -637.06 seconds
Since time cannot be negative, we disregard the negative sign:
half-life ≈ 637.06 seconds
Therefore, the half-life of oxygen-19 is approximately 637.06 seconds.