For my homework the question is...
What is the half life of polonium 214 if a 1 gram sample decays to 31.25mg after 820 seconds? Im not sure how you arrive at the answer, my friend says the answer is 5 but were not sure how to do it..thanks for any help u may have
ln(No/N)= kt
No = 1 gram
N = 0.03215 g
t = 820 seconds.
solve for k. THEN
k=0.693/t1/2
.03125g=1g * e^(-.692*820sec/thl)
where thl is halflife time.
thl*ln .03125 = -.692*820
thl= -.692*820/ln.03125
Check you friend:
1g
1/2g
1/4 g
1/8g
1/16g
1/32g=.0312sec
So you friend is right, it takes 5 half lives, or 820/5 seconds. Check that with the answer above.
Your friend is correct that it goes through 5 half-lives BUT the question is what is the half-life, not how many half-lives it goes through.
My equation or the one Bob Pursley gave you will give you t1/2
which is the half-life.
t1/2 = -.692*820/ln.03125
t1/2 = 164 seconds
To find the half-life of polonium 214, you can use the formula:
ln(No/N) = kt
Where:
No is the initial amount of the substance (1 gram)
N is the final amount of the substance (0.03125 grams)
t is the time it takes for the decay (820 seconds)
k is the decay constant
First, rearrange the formula to solve for k:
k = ln(No/N) / t
Substituting the given values:
k = ln(1 / 0.03125) / 820
Calculate the value of k using a calculator:
k = ln(32) / 820
This gives you the value of k. Next, you can use the formula for half-life:
k = 0.693 / t(1/2)
Rearranging the formula to solve for t(1/2):
t(1/2) = 0.693 / k
Substituting the value of k:
t(1/2) = 0.693 / (ln(32) / 820)
Calculate the value of t(1/2) using a calculator:
t(1/2) = 0.693 / (-0.1195)
This gives you the value of t(1/2), which is approximately -5.8 seconds.
The negative sign indicates that the half-life is decreasing, which is not possible. Therefore, there might be a calculation error or incorrect data given for the problem. Please double-check your values and calculations.
If you're sure the given data is correct, then the half-life of polonium 214 would be approximately 5 seconds, as your friend stated.
To find the half-life of polonium 214, you can use the equation:
ln(No/N) = kt
Where:
No = initial quantity of polonium 214 (in this case, 1 gram)
N = final quantity of polonium 214 (in this case, 31.25 mg or 0.03125 g)
t = time elapsed (in this case, 820 seconds)
k = decay constant
First, let's solve the equation for k:
ln(No/N) = kt
ln(1/0.03125) = k * 820
ln(32) = k * 820
Now we can solve for k:
k = ln(32)/820 ≈ -0.00972
Next, we can use the relationship between the decay constant and half-life:
k = 0.693/t1/2
Solving for t1/2:
t1/2 = 0.693/k
t1/2 = 0.693/(-0.00972) ≈ 71.3 seconds
So the half-life of polonium 214 is approximately 71.3 seconds.