A radioactive substance has a half-life of 3 minutes. After 9 minutes, the count rate was observed to be 200. What was the count rate at zero time?
You can do it manually by increasing it by two for 3 times since you've gone through 3 half lives. Or you can follow the formula.
k = 0.693/t1/2
Then ln(No/N) = kt
No = ?
N = 200
k from above
t = 9 min
Solve for No. Post your work if you get stuck.
To determine the count rate at zero time, we need to calculate how many half-lives have passed between zero time and 9 minutes.
Let's denote the count rate at zero time as C₀.
We know that the count rate at any given time is related to the initial count rate by the equation:
C(t) = C₀ * (1/2)^(t / half-life),
where C(t) is the count rate at time t, and t / half-life is the number of half-lives that have passed.
In this case, we are given that the half-life is 3 minutes, and the count rate at 9 minutes is 200. So we can write:
200 = C₀ * (1/2)^(9 / 3).
To solve for C₀, we can rearrange the equation:
C₀ = 200 / (1/2)^(9 / 3).
Now let's calculate the value of C₀:
C₀ = 200 / (1/2)^(3) = 200 / (1/2)^3 = 200 / (1/8) = 200 * 8 = 1600.
Therefore, the count rate at zero time was 1600.
To determine the count rate at zero time, we can use the concept of half-life. The half-life is the time it takes for half of the radioactive substance to decay.
First, let's determine the number of half-lives that have passed during the 9 minutes.
9 minutes / 3 minutes per half-life = 3 half-lives
Since three half-lives have passed, the radioactive substance has undergone three cycles of decay. At the end of each half-life, the count rate is halved.
Therefore, after three half-lives (9 minutes), the count rate has decreased to 1/2 * 1/2 * 1/2 = 1/8 (or 0.125) of the original count rate.
If the count rate at 9 minutes is given as 200, we can determine the count rate at zero time by multiplying 200 by 8:
Count rate at zero time = 200 * 8 = 1600.
So, the count rate at zero time was 1600.