What is the half-life of radon-222?

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In summary, the conversation discusses how long it will take for a sample of radon-222 to decay to 10% of its original amount. After 3.817 days, it will have decayed to 1/4 of its original amount.
  • #1
noboost4you
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After 3 days a sample of radon-222 decayed to 58% of its original amount:
a) what is the half-life of radon-222?
b) how long would it take the sample to decay to 10% of its original amount?

check my answers please:

a) m(t)=mo*e^kt
t = 3 and m(t)=.58mo
0.58mo = mo*e^3k
ln(.58) = 3k
k = -0.1816

1/2 = e^(-0.1816t)
ln(1/2)= -.1816t
t = 3.817 half life is approx 3.817 days

b) e^(-0.1816t) = .10
-0.1816t = ln(.10)
t = 12.68 it will take approx 12.68 days to reach 10%

the answer to (b) throws me. if half life is 3.817 days, 0% would be 7.634 days. where does this 12.68 days come from? i must be doing something wrong, unless (a) is wrong too.

thanks in advance
 
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  • #2
You have a fundamential misunderstanding of [tex]\lambda[/tex]. Half-life is the time required for 1/2 of a sample to decay thus forming another product. If 1/2 of a sample decays in x days, the other half still remains... That other half would then undergo a decay for x days leaving 1/2 of the sample (1/4 of the original sample). This process continues on an on and on dividing a sample by 1/2 each time, so simply assuming 2 half lifes equals 100% of a sample is flawed.

Look here, I plugged the numbers you got into excel and spred the data out beginning from t=0[tex]\lambda[/tex] to t=15[tex]\lambda[/tex]. If you notice each step is approximetly 1/2 of the prevuous (this was due to in-precise rounding). It took about 11 1/2 lives after time zero for the sample to theoretically decay to an insignificant amount which is substantially greater than 7.6 days (more along the lines of 42 days):

t0=100
t1= 50.15359446
t2= 25.15383037
t3= 12.61555008
t4= 6.327151823
t5= 3.173294066
t6= 1.591521037
t7= 0.798205007
t8= 0.400328502
t9= 0.200779133
t10=0.100697952
t11=0.050503643
t12=0.025329392
t13=0.012703601
t14=0.006371312
t15=0.003195442
 
  • #3
Saying that the half-life of any substance is "T" is the same as saying that [tex]X(t)= X(0)\({\frac{1}{2}}\)^{\frac{t}{T}}[/tex]

where t is the time measured in whatever units T is in.

Since it takes 3 days to decay to 58% of the original amount, we have [tex]X(3)= X(0)\({\frac{1}{2}}\)^{\frac{3}{T}}= 0.58X(0)[/tex]
so that [tex]\({\frac{1}{2}}\)^{\frac{3}{T}}= 0.58[/tex]
Then [tex]\frac{3}{T}ln(\frac{1}{2})= ln .58[/tex]
[tex]\frac{3}{T}= 0.786 [/tex]
[tex]\frac{T}{3}= \frac{1}{0.786}= 1.27 [/tex]

and, finally T= 3(1.27)= 3.82 days. ('cuz 0.58 is pretty close to 0.5!)

Knowing now that [tex]X(3)= X(0)\({\frac{1}{2}}\)^{\frac{3}{3.82}}= 0.58X(0)[/tex]
We can answer the second question by solving
[tex]\(\frac{1}{2}\)^{\frac{t}{3.82}= 0.10 [/tex]
[tex]\frac{t}{3.82}ln(0.5)= ln(.10) [/tex]
[tex] t= 3.82(\frac{ln(.10)}{ln(0.5)}) [/tex]
t= 12.68 days.

"the answer to (b) throws me. if half life is 3.817 days, 0% would be 7.634 days. "

No, that would be a linear function. The whole reason this has a "half-life" is that it is an exponential function. If it decreases to half in 3.817 days, it will decrease to half of that (that is to 1/4 of the original amount) in another 3.817 days. It decreases by the same proportion not the same amount.
(Decreasing by the same amount is linear.)
 
Last edited by a moderator:

1. What is the half-life of radon-222?

The half-life of radon-222 is approximately 3.8 days. This means that after 3.8 days, half of the initial amount of radon-222 will decay into other elements.

2. Why is it important to know the half-life of radon-222?

Radon-222 is a radioactive gas that is known to cause lung cancer. By understanding its half-life, we can better understand its decay rate and potential health risks.

3. How is the half-life of radon-222 measured?

The half-life of radon-222 is typically measured through radioactive decay experiments in a controlled laboratory setting. This involves tracking the decay of radon-222 over a period of time and analyzing the data.

4. Can the half-life of radon-222 vary?

The half-life of radon-222 is a constant value and does not vary. It is a fundamental characteristic of this radioactive element.

5. What factors can affect the half-life of radon-222?

The half-life of radon-222 is not affected by external factors such as temperature or pressure. However, it can be influenced by the presence of other elements or compounds that can alter its decay rate.

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