the half time of a radioisotope is found to be 4.55 minutes. if the decay follow first order kinetics. what percentage of isotope will remain after 2.00 hours?

To find the percentage of the isotope that remains after 2.00 hours, we can use the first-order decay equation:

N(t) = N0 * e^(-kt)

Where:
N(t) = Final amount of isotope remaining at time t
N0 = Initial amount of isotope
k = Decay constant
t = Time

First, calculate the decay constant (k) using the half-life information:

t(1/2) = ln(2) / k

Given that the half-life (t(1/2)) is 4.55 minutes, we can rearrange the equation to solve for k:

k = ln(2) / t(1/2)

k = ln(2) / 4.55 minutes

Next, convert 2.00 hours to minutes:

2.00 hours * 60 minutes/hour = 120 minutes

Now, substitute the values into the first-order decay equation:

N(t) = N0 * e^(-kt)

N(120 minutes) = N0 * e^(-k * 120 minutes)

Since we are interested in the percentage remaining, we need to divide N(t) by N0:

Percentage remaining = (N(t) / N0) * 100

Substituting the values:

Percentage remaining = (N(120 minutes) / N0) * 100

Now, we can solve for the percentage of isotope that remains after 2.00 hours.

To calculate the percentage of isotope that will remain after a given time, we need to use the concept of radioactive decay and the half-life of the radioisotope.

First, let's convert the half-life from minutes to hours. Since there are 60 minutes in an hour, we divide the half-life by 60:

Half-life = 4.55 minutes ÷ 60 = 0.0758 hours (approximately)

Next, we need to determine how many half-lives have passed during the given time of 2.00 hours. We calculate this by dividing the given time by the half-life:

Number of half-lives = 2.00 hours ÷ 0.0758 hours ≈ 26.35 half-lives (approximately)

Now, we will use the formula for radioactive decay of a substance following first-order kinetics:

N(t) = N₀ * (1/2)^(t / t₁/₂)

Where:
N(t) = Final amount of the radioisotope after time t.
N₀ = Initial amount of the radioisotope.
t = Time elapsed.
t₁/₂ = Half-life of the radioisotope.

In this case, we are interested in finding the percentage remaining, so we can calculate it using the formula:

Percentage remaining = (N(t) / N₀) * 100

Now, let's calculate the percentage remaining after 2.00 hours:

Percentage remaining = (1/2)^(26.35) * 100 ≈ 0.181% (approximately)

Therefore, approximately 0.181% of the radioisotope will remain after 2.00 hours.

99.99%

100% * (1/2)^(120 / 4.55)