The isotope californium-252 has a half life of 162 days. How many days would it take for 61.7% of the original material to decay?
I am using the equation t= (ln(nt/no))/ k
and I am first solving for K using: k=(.693/t1/2)
the suppose answer can range from (10.7-10.9)
but Im not even close to that answer. What am I doing wrong??
amount left=amountoriginal* e^(-.693*timedecaying/halflife)
Memorize that as
A=O*e^(-.693t/thalf)
in your problem, thalflife=162
1-.617= .393=1*e^(-.693t/162)
taking natural log of each side
ln(.383)=-.693 t/162 or
t= -162*ln(.383)/.693
putting that into the calculator (or your google search window)
-162*ln(.383)/.693 =
224 days
Now think. If 61 percent has decayed, then over half has decayed, so it must be well over the half life of 162 days, so 224 makes sense. Your 10 days is wrong.
#1 is that I think you are using the wrong equation. Although you didn't define your symbols, I think the equation should be
t = ln(No/Nt)/k where No is what we start with (100) and Nt is what we end up with (100-61.7 - 38.3) and k is 0.004278 using your formula (which is correct). Actually, I prefer not to do the algebra and use the standard ln(No/Nt) = kt. By the way, your formula would be correct if you stuck in a negative sign. Having said all of that, #2 reason is that 10.7 days CAN'T possibly be the correct answer. Think about it. It will take 162 days just to decay to 50% of the original amount. And you have more to go after that so the correct answer MUST be more than 162 days but less than twice that to get to 25%. So the correct answer is somewhere in the 225 day range.
I ended up mixing two questions which is my fault. It should have read:
The isotope californium-252 has a half life of 2.64 years. How many years would it take for 94.1% of the original material to decay?
I got the percentage and days mixed up. It should have been years not days.
To find the number of days it would take for 61.7% of the original material to decay, you are on the right track by using the equation t = (ln(nt/no)) / k, where t is the time in days, nt is the final amount of material, no is the initial amount of material, and k is the decay constant.
First, let's calculate the value of k using k = 0.693 / t1/2, where t1/2 is the half-life of the isotope californium-252, which is 162 days. Plugging in the value, we get:
k = 0.693 / 162 ā 0.00428
Now, we can use the calculated value of k to find the time required for 61.7% of the original material to decay:
t = (ln(0.617) / k)
Using a calculator or software that can calculate natural logarithms, we find:
t ā (ln(0.617) / 0.00428) ā 144.62 days
Here's what you might be doing wrong:
1. Make sure you are using the natural logarithm function (ln) and not the common logarithm (log).
2. Check your arithmetic calculations to ensure accuracy.
3. Verify that you are using the correct values for the half-life and the given percentage of decay.
4. Check if any rounding errors or approximations while using the calculator have affected the result.
By following these steps and using the correct values, you should be able to arrive at a more accurate answer within the given range of 10.7 to 10.9 days.