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Wize University Physics Textbook (Master) > Modern Physics

Radioactivity

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Radioactive Decays



Radioactive nuclei decay until they become stable. The decay happens with probability rr, also called the decay rate.

The activity AA is defined as the number of decay processes in a sample per second, given by:
 A=Nr \boxed{ \ A=Nr \ }
  • AA is the activity of the sample
  • rr is the decay rate
  • NN is the total number of radioactive particles that have not decayed yet
The unit for activity is Becquerel: 1Bq=1 decay/second1 Bq = 1 \ decay/second



The number of radioactive nuclei in a sample decreases based of the following formula:
 N(t)=N0 e−tτ=N0 e−rt \boxed{ \ N(t)=N_0 \ e^{-\frac{t}{\tau}}=N_0 \ e^{-rt} \ }

  • N0N_0 is the initial number of radioactive particles in the sample
  • τ\tau is the lifetime, the time it takes for the number of radioactive particles drop by a factor of 1/e1/e



Wize Concept
The lifetime and the decay rate are inverses of each other.



Exam Tip
The final and initial quantities NN and N0N_0 may represent any of the following: number of atoms, mass, activity etc.

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Half-Life



The rate at which the decay happens is related to the half-life of the nuclei.


The half-life thalft_{half} of a substance is the time it takes for half of the quantity to decay. That is, after one half -life the final quantity NN is half of the original quantity N0N_0.



The following equation describes this relationship:

 N=N0(12)tthalf \boxed{\ N=N_0\bigg(\dfrac{1}{2}\bigg)^{\frac{t}{t_{half}}} \ }

  • NN is the final amount
  • N0N_0 is the initial amount
  • thalft_{half} is the half-life







The half-life is related to the decay rate rr and the lifetime τ\tau by the following:

 thalf=τ ln⁡2=ln⁡2r \boxed { \ t_{half}=\tau \ \ln2=\frac{\ln2}{r} \ }

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Example: Cobalt-60 is a radioactive isotope that emits gamma rays and is used to treat cancer. It has a half-life of 5.27 years. The cobalt-60 used for the cancer treatment must be replaced regularly to continue to be effective.







Wize Concept
The longer the half-life, the more stable the element is.


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Example: Half-Life


A radioactive substance has a half-life of 5050 years. At the current time t=75t=75 years, 200200 g of the substance remains. How long will it take from now for the substance to be reduced down to 2525 g?

Challenge: solve without a calculator!


We know that thalf=50t_\text{half}=50 , and when t=75t=75 the amount left is N=200N=200.

Let's put these into our formula to solve for the initial value:

N=N0(12)tthalfN=N_0\bigg(\dfrac{1}{2}\bigg)^{\frac{t}{t_{half}}}

200=N0(12)7550200=N_0\bigg(\dfrac{1}{2}\bigg)^{\frac{75}{50}}

200=N0(2)−32200=N_0(2)^{-\frac{3}{2}}

N0=200⋅232N_0=200\cdot2^{\frac{3}{2}}


NOTE: There's no need to compute this value. We'll just put it in our equation, which becomes:

N=200⋅232(12)t50N=200\cdot2^{\frac{3}{2}}\bigg(\dfrac{1}{2}\bigg)^{\frac{t}{50}}


Solve for time by using the final value N=25N=25:

25=200⋅232(12)t5025=200\cdot2^{\frac{3}{2}}\bigg(\dfrac{1}{2}\bigg)^{\frac{t}{50}}

18=232(12)t50\dfrac{1}{8}=2^{\frac{3}{2}}\bigg(\dfrac{1}{2}\bigg)^{\frac{t}{50}}


Write both sides using the same base of 22:

2−3=232⋅2−t502^{-3}=2^{\frac{3}{2}}\cdot2^{-\frac{t}{50}}

2−3=2(32−t50)2^{-3}=2^{\left(\frac{3}{2}-\frac{t}{50}\right)}

Equate exponents to get:

−3=32−t50\displaystyle-3=\frac{3}{2}-\frac{t}{50}

Multiply everything by 5050 to eliminate the fractions:
−150=75−t-150=75-t
t=225t=225


Finally, the time from now (t=75t=75) to then (t=225t=225) is the difference:

Δt=225−75=150\Delta t=225-75=150 (yrs)


Practice: Half-Life


A radioactive element has an unknown half-life. The original sample weighs 3535 g. Two years later, the sample weighs 77 g. What is the half-life of the element?