Radioactive decay is a process that follows first-order kinetics. The half-life of 32P is 0.03920 years; how long (in minutes) would it take for the amount of 32P to decrease to 53.25% of its initial amount?

I don't understand at all what to do here

Use the following formula:

T1/2 = T*ln(2)/ln(Ao/At)

Where

T1/2=0.03920 years
T=?
Ao=100%
and
At=53.25%

Solve for T:

(T1/2)/[[ln(2)/ln(Ao/At)]= To

To=(0.03920)/[ln(2)/ln(100/53.25)]

Answer contains four significant figures.

The answer that is given to you will be in years.

Take To and do the following with it:

To*(365 days/1 year)*(24hrs/1 day)*(60mins/1 Hr)= answer in minutes.

Remember, no more than four significant figures.

To solve this problem, we need to use the concept of radioactive decay and the equation for first-order kinetics.

First, let's understand the given information. The half-life of 32P is stated as 0.03920 years. This means that after 0.03920 years, half of the original amount of 32P will remain.

Now, let's determine how much of the initial amount of 32P remains after a certain time t using the equation for first-order kinetics:

N(t) = N₀ * e^(-kt),

where:
N(t) is the amount of substance remaining at time t,
N₀ is the initial amount of substance,
k is the decay constant,
t is the time passed.

Since the half-life is given, we can use it to determine the decay constant (k) by rearranging the equation:

0.5 = e^(-kt).

Taking the natural logarithm (ln) of both sides:

ln(0.5) = -kt,

Thus:

k = -ln(0.5) / t₁/₂.

Now, we know the value of k. Let's use this information to find the time (in minutes) it takes for the amount of 32P to decrease to 53.25% of its initial amount. The remaining amount of 32P (N(t)) is 53.25% or 0.5325 times the initial amount (N₀).

0.5325 = N₀ * e^(-kt).

Rearranging this equation:

t = -ln(0.5325) / k.

Since the time unit in the equation for k was in years, we need to convert the half-life and the time to be in the same unit. Let's convert the half-life from years to minutes:

1 year = 365.25 days (considering leap years)
1 day = 24 hours
1 hour = 60 minutes

Now, we can plug in all the values into the equation and calculate the time it takes for the amount of 32P to decrease to 53.25% of its initial amount in minutes.