The disintegration of a radioisotope, such as I-131 to Xe-131, is another example of a first-order reaction.

See your post above.

A vial contains radioactive iodine-131 with an activity of 3.7 mCi per milliliter. If the thyroid test requires 2.9 mCi in an "atomic cocktail," how many milliliters are used to prepare the iodine-131 solution?

To solve this problem, we need to use the formula for calculating the amount of substance remaining after a certain time in a first-order reaction:

N(t) = N(0) * e^(-kt)

Where:
N(t) = amount of substance remaining after time t
N(0) = initial amount of substance
k = rate constant of the reaction
t = time

The half-life of the reaction is given as 8.0 days, which means that after 8.0 days, the amount of I-131 will be reduced to half of its initial value.

Let's calculate the rate constant, k, using the equation:

k = ln(2) / t_1/2

Plugging in the given value of the half-life (t_1/2 = 8.0 days), we can calculate the rate constant:

k = ln(2) / 8.0

k ≈ 0.0866 (rounded to four decimal places)

Now we have the rate constant, we can use the formula to calculate the amount of I-131 remaining after a specific number of days.

Example: If we want to know how much I-131 remains after 10 days:

N(t) = N(0) * e^(-kt)

N(10) = 1.40 g * e^(-0.0866 * 10)

N(10) ≈ 1.40 g * e^(-0.866)

N(10) ≈ 1.40 g * 0.4189

N(10) ≈ 0.585 g

Therefore, after 10 days, approximately 0.585 grams of I-131 will remain.

You can repeat this calculation for any specific number of days to find the amount of I-131 remaining. Just replace t with the desired number of days in the equation.