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R = R 0 e λt , size 12{R=R rSub { size 8{0} } e rSup { size 8{ - λt} } } {}

where R 0 size 12{R rSub { size 8{0} } } {} is the activity at t = 0 size 12{t=0} {} . This equation shows exponential decay of radioactive nuclei. For example, if a source originally has a 1.00-mCi activity, it declines to 0.500 mCi in one half-life, to 0.250 mCi in two half-lives, to 0.125 mCi in three half-lives, and so on. For times other than whole half-lives, the equation R = R 0 e λt size 12{R=R rSub { size 8{0} } e rSup { size 8{ - λt} } } {} must be used to find R size 12{R} {} .

Phet explorations: alpha decay

Watch alpha particles escape from a polonium nucleus, causing radioactive alpha decay. See how random decay times relate to the half life.

Alpha Decay

Section summary

  • Half-life t 1 / 2 size 12{t rSub { size 8{1/2} } } {} is the time in which there is a 50% chance that a nucleus will decay. The number of nuclei N size 12{N} {} as a function of time is
    N = N 0 e λt , size 12{N=N rSub { size 8{0} } e rSup { size 8{ - λt} } } {}
    where N 0 size 12{N rSub { size 8{0} } } {} is the number present at t = 0 size 12{t=0} {} , and λ size 12{λ} {} is the decay constant, related to the half-life by
    λ = 0 . 693 t 1 / 2 . size 12{λ= { {0 "." "693"} over {t rSub { size 8{1/2} } } } } {}
  • One of the applications of radioactive decay is radioactive dating, in which the age of a material is determined by the amount of radioactive decay that occurs. The rate of decay is called the activity R size 12{R} {} :
    R = Δ N Δ t . size 12{R= { {ΔN} over {Δt} } } {}
  • The SI unit for R size 12{R} {} is the becquerel (Bq), defined by
    1 Bq = 1 decay/s. size 12{1" Bq"="1 decay/s"} {}
  • R size 12{R} {} is also expressed in terms of curies (Ci), where
    1 Ci = 3 . 70 × 10 10 Bq. size 12{1" Ci"=3 "." "70" times "10" rSup { size 8{"10"} } " Bq"} {}
  • The activity R size 12{R} {} of a source is related to N size 12{N} {} and t 1 / 2 size 12{t rSub { size 8{1/2} } } {} by
    R = 0 . 693 N t 1 / 2 . size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {}
  • Since N size 12{N} {} has an exponential behavior as in the equation N = N 0 e λt size 12{N=N rSub { size 8{0} } e rSup { size 8{ - λt} } } {} , the activity also has an exponential behavior, given by
    R = R 0 e λt , size 12{R=R rSub { size 8{0} } e rSup { size 8{ - λt} } } {}
    where R 0 size 12{R rSub { size 8{0} } } {} is the activity at t = 0 size 12{t=0} {} .

Conceptual questions

In a 3 × 10 9 size 12{3 times "10" rSup { size 8{9} } } {} -year-old rock that originally contained some 238 U , which has a half-life of 4.5 × 10 9 years, we expect to find some 238 U remaining in it. Why are 226 Ra , 222 Rn , and 210 Po also found in such a rock, even though they have much shorter half-lives (1600 years, 3.8 days, and 138 days, respectively)?

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Does the number of radioactive nuclei in a sample decrease to exactly half its original value in one half-life? Explain in terms of the statistical nature of radioactive decay.

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Radioactivity depends on the nucleus and not the atom or its chemical state. Why, then, is one kilogram of uranium more radioactive than one kilogram of uranium hexafluoride?

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Explain how a bound system can have less mass than its components. Why is this not observed classically, say for a building made of bricks?

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Spontaneous radioactive decay occurs only when the decay products have less mass than the parent, and it tends to produce a daughter that is more stable than the parent. Explain how this is related to the fact that more tightly bound nuclei are more stable. (Consider the binding energy per nucleon.)

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To obtain the most precise value of BE from the equation BE= ZM 1 H + Nm n c 2 m A X c 2 size 12{"BE=" left lbrace left [ ital "ZM" left ("" lSup { size 8{1} } H right )+ ital "Nm" rSub { size 8{n} } right ]-m left ("" lSup { size 8{A} } X right ) right rbrace c rSup { size 8{2} } } {} , we should take into account the binding energy of the electrons in the neutral atoms. Will doing this produce a larger or smaller value for BE? Why is this effect usually negligible?

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How does the finite range of the nuclear force relate to the fact that BE / A size 12{ {"BE"} slash {A} } {} is greatest for A size 12{A} {} near 60?

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Problems&Exercises

Data from the appendices and the periodic table may be needed for these problems.

An old campfire is uncovered during an archaeological dig. Its charcoal is found to contain less than 1/1000 the normal amount of 14 C size 12{"" lSup { size 8{"14"} } C} {} . Estimate the minimum age of the charcoal, noting that 2 10 = 1024 size 12{2 rSup { size 8{"10"} } ="1024"} {} .

57,300 y

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Questions & Answers

sir dose it apply to the human system
Olubukola Reply
prove that the centrimental force Fc= M1V² _________ r
Kaka Reply
prove that centripetal force Fc = MV² ______ r
Kaka
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mitul Reply
griffts bridge derivative
Ganesh Reply
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please explain; when a glass rod is rubbed with silk, it becomes positive and the silk becomes negative- yet both attracts dust. does dust have third types of charge that is attracted to both positive and negative
Timothy Reply
what is a conductor
Timothy
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Timothy
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Timothy
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dust particles contains both positive and negative charge particles
Mbutene
corona charge can verify
Stephen
when pressure increases the temperature remain what?
Ibrahim Reply
what is frequency
Mbionyi Reply
define precision briefly
Sujitha Reply
CT scanners do not detect details smaller than about 0.5 mm. Is this limitation due to the wavelength of x rays? Explain.
MITHRA Reply
hope this helps
what's critical angle
Mahmud Reply
The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
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Chidalu
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Temiloluwa
the ratio of the density of a substance to the density of a standard, usually water for a liquid or solid, and air for a gas.
Chidalu
What is momentum
aliyu Reply
mass ×velocity
Chidalu
it is the product of mass ×velocity of an object
Chidalu
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Sean Reply
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Wat is the relationship between Instataneous velocity
Oyinlusi Reply
Instantaneous velocity is defined as the rate of change of position for a time interval which is almost equal to zero
Astronomy
The potential in a region between x= 0 and x = 6.00 m lis V= a+ bx, where a = 10.0 V and b = -7.00 V/m. Determine (a) the potential atx=0, 3.00 m, and 6.00 m and (b) the magnitude and direction of the electric ficld at x =0, 3.00 m, and 6.00 m.
what is energy
Victor Reply
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GIDEON
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Bitrus
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how to prove that Newton's law of universal gravitation F = GmM ______ R²
Kaka Reply
Practice Key Terms 8

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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