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By the end of this section, you will be able to:
  • Describe the decay of a radioactive substance in terms of its decay constant and half-life
  • Use the radioactive decay law to estimate the age of a substance
  • Explain the natural processes that allow the dating of living tissue using 14 C

In 1896, Antoine Becquerel discovered that a uranium-rich rock emits invisible rays that can darken a photographic plate in an enclosed container. Scientists offer three arguments for the nuclear origin of these rays. First, the effects of the radiation do not vary with chemical state; that is, whether the emitting material is in the form of an element or compound. Second, the radiation does not vary with changes in temperature or pressure—both factors that in sufficient degree can affect electrons in an atom. Third, the very large energy of the invisible rays (up to hundreds of eV) is not consistent with atomic electron transitions (only a few eV). Today, this radiation is explained by the conversion of mass into energy deep within the nucleus of an atom. The spontaneous emission of radiation from nuclei is called nuclear radioactivity    ( [link] ).

A yellow triangle with a black outline, enclosing a fan shaped structure is shown. The “fan” is black and has three blades.
The international ionizing radiation symbol is universally recognized as the warning symbol for nuclear radiation.

Radioactive decay law

When an individual nucleus transforms into another with the emission of radiation, the nucleus is said to decay    . Radioactive decay occurs for all nuclei with Z > 82 , and also for some unstable isotopes with Z < 83 . The decay rate is proportional to the number of original (undecayed) nuclei N in a substance. The number of nuclei lost to decay, d N in time interval dt , is written

d N d t = λ N

where λ is called the decay constant    . (The minus sign indicates the number of original nuclei decreases over time.) In other words, the more nuclei available to decay, the more that do decay (in time dt ). This equation can be rewritten as

d N N = λ d t .

Integrating both sides of the equation, and defining N 0 to be the number of nuclei at t = 0 , we obtain

N 0 N d N N = 0 t λ d t .

This gives us

ln N N 0 = λ t .

Taking the left and right sides of the equation as a power of e , we have the radioactive decay law    .

Radioactive decay law

The total number N of radioactive nuclei remaining after time t is

N = N 0 e λ t

where λ is the decay constant for the particular nucleus.

The total number of nuclei drops very rapidly at first, and then more slowly ( [link] ).

A graph of N versus t is shown. It is labeled N equal to N subscript 0 e to the power minus lambda t. The value of N is maximum, N subscript 0, at t =0 and it reduces with time till it reaches 0. At t = T subscript half, N = N subscript 0 by 2 and at t = 2T subscript half, N = N subscript 0 by 4.
A plot of the radioactive decay law demonstrates that the number of nuclei remaining in a decay sample drops dramatically during the first moments of decay.

The half-life     ( T 1 / 2 ) of a radioactive substance is defined as the time for half of the original nuclei to decay (or the time at which half of the original nuclei remain). The half-lives of unstable isotopes are shown in the chart of nuclides in [link] . The number of radioactive nuclei remaining after an integer ( n ) number of half-lives is therefore

N = N 0 2 n

If the decay constant ( λ ) is large, the half-life is small, and vice versa. To determine the relationship between these quantities, note that when t = T 1 / 2 , then N = N 0 / 2 . Thus, [link] can be rewritten as

Questions & Answers

what is bohrs model for hydrogen atom
Swagatika Reply
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
what is the value of speed of light
Propessor Reply
1.79×10_¹⁹ km per hour
Swagatika
what r dwarf planet
Sivalakshmi Reply
what is energy
Isiguzo Reply
কাজের একক কী
Jasim
কাজের একক কী
Jasim
friction ka direction Kaise pata karte hai
Rahul Reply
friction is always in the opposite of the direction of moving object
Punia
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
Varia Reply
b) symmetric of time only
Swagatika
fundamental note of a vibrating string
fasoyin Reply
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
what are matter waves? Give some examples
mallam Reply
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
Mathematical expression of principle of relativity
Nasir Reply
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
obia Reply
What is the importance of de-broglie's wavelength?
Mukulika Reply
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant
how those weight effect a stable motion at equilibrium
Nonso Reply
how do I differentiate this equation- A sinwt with respect to t
Evans Reply
just use the chain rule : let u =wt , the dy/dt = dy/du × du/dt : wA × cos(wt)
Jerry
I see my message got garbled , anyway use the chain rule with u= wt , etc...
Jerry
de broglie wave equation
LoNE Reply
vy beautiful equation
chandrasekhar
what is electro statics
fitsum Reply
when you consider systems consisting of fixed charges
Sherly

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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