The activity of a radioisotope is 3000 counts per minute at one time and 2736 counts per minute 48 hours later. What is the half-life of the radioisotope??
a) 831hr
b)521hr
c)361 hr
d)1.44hr
ln(No/N) = kt
No = 3000
N = 2736
k = ?
t = 48 hrs.
Solve for k.
Then k = 0.693/t1/2
Substitute k from above and solve to t1/2
Dr. Bob,
Where are you getting the 0.693 from?
log base 3 of 1 is .693
typo
log base e (as in ln) of 2 is .693
Bob Pursley is right. More explicitly, it comes from this.
If we start with 100 counts and drop to 50 counts (which is 1 half life or t1/2), then
ln(No/N) = kt
ln(100/50) = kt1/2
ln 2 = kt1/2
0.693 = kt1/2
and k = 0.693/t1/2
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To determine the half-life of a radioisotope, we can use the following formula:
Activity(t) = Activity₀ * (1/2)^(t / half-life)
Where:
- Activity(t) is the activity at time t
- Activity₀ is the initial activity
- t is the time elapsed
- half-life is the time it takes for the radioisotope's activity to halve
In this case, we have two data points:
- At time t = 0, the activity is 3000 counts per minute.
- 48 hours later, at t = 48 hours, the activity is 2736 counts per minute.
Using these points, we can set up the following equations:
3000 = Activity₀ * (1/2)^(0 / half-life) --> Equation 1
2736 = Activity₀ * (1/2)^(48 / half-life) --> Equation 2
Simplifying Equation 1:
3000 = Activity₀
Substituting this into Equation 2:
2736 = 3000 * (1/2)^(48 / half-life)
Next, let's simplify further:
2736 / 3000 = (1/2)^(48 / half-life)
After simplifying, we get:
0.912 = (1/2)^(48 / half-life)
To isolate the exponent, we can take the logarithm of both sides:
log(0.912) = log((1/2)^(48 / half-life))
Using logarithm properties, we can bring down the exponent:
log(0.912) = (48 / half-life) * log(1/2)
Dividing both sides by log(1/2), we get:
log(0.912) / log(1/2) = 48 / half-life
Simplify further:
log(0.912) / log(1/2) = 48 / half-life
We can now solve for the half-life by rearranging the equation:
half-life = 48 / (log(0.912) / log(1/2))
Calculating this value:
half-life ≈ 521 hours
Therefore, the answer is: b) 521 hours.