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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 force
Afework Reply
The different examples for collision
Afework
What is polarization and there are type
Muhammed Reply
Polarization is the process of transforming unpolarized light into polarized light. types of polarization 1. linear polarization. 2. circular polarization. 3. elliptical polarization.
Eze
Describe what you would see when looking at a body whose temperature is increased from 1000 K to 1,000,000 K
Aishwarya Reply
how is tan ninety minus an angle equals to cot an angle?
Niicommey Reply
please I don't understand all about this things going on here
Jeremiah Reply
What is torque?
Matthew Reply
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study.
Teka
Torque refers to the rotational force. i.e Torque = Force × radius.
Arun
Torque is the rotational equivalent of force . Specifically, it is a force exerted at a distance from an object's axis of rotation. In the same way that a force applied to an object will cause it to move linearly, a torque applied to an object will cause it to rotate around a pivot point.
Teka
Torque is the rotational equivalence of force . So, a net torque will cause an object to rotate with an angular acceleration. Because all rotational motions have an axis of rotation, a torque must be defined about a rotational axis. A torque is a force applied to a point on an object about the axis
Teka
When a missle is shot from one spaceship towards another, it leaves the first at 0.950c and approaches the other at 0.750c. what is the relative velocity of the two shipd
Marifel Reply
how to convert:m^3/s^2 all divided by kg to cm^3/s^2
Thibaza Reply
Is there any proof of existence of luminiferious aether ?
Zero Reply
mass conversion of 58.73kg =mg
Proactive Reply
is Space time fabric real
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What's the relationship between the work function and the cut off frequency in the diagram above?
frankline Reply
due to the upthrust weight of the object varise with force in which the body fall into the water pendincular with the reflection of light with it
Gift
n=I/r
Gift
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falanga
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falanga
what is meant by fluctuated
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If n=cv then how v=cn? and if n=c/v then how v=cn?
Natanim
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Mbah Reply
what is electrolysis
Mbah
Electrolysis is the chemical decomposition of electrolyte either in molten state or solution to conduct electricity
Ayomide
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Ayesha Reply
can someone help explain why v2/c2 is =1/2 Using The Lorentz Transformation For Time Spacecraft S′ is on its way to Alpha Centauri when Spacecraft S passes it at relative speed c /2. The captain of S′ sends a radio signal that lasts 1.2 s according to that ship’s clock. Use the Lorentz transformati
Jennifer

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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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