Finding the half-life of an unknown substance

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    Half-life
In summary, the unknown radioisotope has a half-life of approximately 46 days, as determined by the formula N = N0 * (1/2)x/T1/2. After 75 days, approximately 324.656 particles will remain, and it will take approximately 460.521 days for the 1000 particles to decay to a single particle. These calculations were made using the formula N(x) = N0e^(-lambda x/T), where T is the half-life, t is the elapsed time, and N base0 is the initial number of particles.
  • #1
Sam
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1000 particles of an unknown radioisotope decays to 472.37 particles in 50 days.

(a) What is the half-life of this substance?


Any problems that I had done previously, the half-life was given.

Any help will be greatly appreciated.

Well, I know a few things, but don't know if they apply to this problem:

The decay constant = .693/T base 1/2
t 1/2 = half-life

Time to decay to a single particle = LN(Nbase0)/decay constant

N = number of particles remaining at some elapsed time
Nbase0 = number of particles we started with
e = a symbol that represents the irrational number 2.718281828...
lambda = decay constant
x = elapsed time

The formula to determing how much of the substance will be remaining at any particular time is: N = Nbase0 e^-lambda x
 
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  • #2
Hi Sam,
you give the correct formulae
N = N0e-[lamb]x
and
[lamb]=.693/T1/2.
You could combine these, solve for T1/2, and plug in N0, N, and x.

Or, more instructive, you could use the equation
N = N0 * (1/2)x/T1/2.

In both cases, the important step is applying the ln() to both sides of the equation.
 
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  • #3
Thank you.

I'll give it a try.
 
  • #4
Is this correct?

I used, based on the above:

N(t) = N base0^(-lambda x /T),
where T is the half-life
t = 50 days
N base0 = 1000

I rounded to the nearest whole number for days and came up with 46.

Is that correct?
 
  • #5


Originally posted by Sam
I used, based on the above:

N(t) = N base0^(-lambda x /T)
I think it should read
N(x) = N0eln(1/2)x/T, but that's probably just a typo, since
T is the half-life
t = 50 days
N base0 = 1000
is correct (except x=50 days), and your answer is also correct, although they probably expect you to come up with some more decimals...:wink:
 
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  • #6
More Help, Please...

I have two more parts to this problem:

(a) How much will be left after 75 days?
(b) How long will it take the 1000 particles to decay to a single particle?

For (a), I came up with: 324.656 particles
For (b), I came up with: 460.521 days

Will you please verify my answers?

Thank you!
 
  • #7
Both of your answers are correct.
 

1. How do you determine the half-life of an unknown substance?

The half-life of an unknown substance can be determined by conducting experiments in a controlled environment. This involves measuring the decay rate of the substance over a specific period of time and using mathematical equations to calculate the half-life.

2. Why is it important to find the half-life of an unknown substance?

Knowing the half-life of an unknown substance is important because it can provide valuable information about the stability and potential toxicity of the substance. It can also help in determining the appropriate storage and disposal methods for the substance.

3. What factors can affect the accuracy of determining the half-life of an unknown substance?

Several factors can affect the accuracy of determining the half-life of an unknown substance, including the measurement equipment used, the condition of the substance, and the presence of impurities or other substances that may interfere with the decay rate.

4. Can the half-life of an unknown substance change over time?

In most cases, the half-life of an unknown substance remains constant. However, certain factors such as temperature, pressure, and exposure to radiation can cause the half-life to change over time.

5. What is the relationship between the half-life and the rate of radioactive decay?

The half-life and the rate of radioactive decay are inversely related. This means that as the half-life of a substance increases, the rate of radioactive decay decreases, and vice versa. The mathematical equation for this relationship is R = ln(2)/t1/2, where R is the rate of decay and t1/2 is the half-life.

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