20.3 Radioactive Decay | Chemistry: Atoms First 2e

Learning Objectives

Identify decay modes and their changes to mass number and atomic number
Calculate remaining isotope amount, activity, half-life, or elapsed time
Explain radioactive decay series and radiometric dating

Core Concepts & Principles

Radioactive decay is the spontaneous change of an unstable parent nuclide into a daughter nuclide, which may be stable or may decay again. Decay often brings a nucleus closer to the band of stability.

Decay Modes

Mode	What happens	Change in $A$	Change in $Z$
Alpha (α) decay	Emits a helium nucleus, $^4_2\text{He}$	−4	−2
Beta-minus (β⁻) decay	A neutron becomes a proton and emits an electron, $^0_{-1}\text{e}$	0	+1
Positron (β⁺) emission	A proton becomes a neutron and emits a positron, $^0_{+1}\text{e}$	0	−1
Electron capture	A proton captures an inner-shell electron and becomes a neutron	0	−1
Gamma (γ) emission	An excited nucleus loses energy as a photon, $^0_0\gamma$	0	0

Neutron-rich nuclei tend to undergo beta-minus decay; proton-rich nuclei may undergo positron emission or electron capture. Very heavy nuclei commonly undergo alpha decay. Gamma emission often follows another decay that leaves the daughter excited. Electron capture can be followed by X-rays as electrons refill the vacancy. Beta decay and electron capture also involve neutrinos or antineutrinos; introductory nuclear equations usually omit these because they have $A=0$ and charge 0.

In a radioactive decay series, one daughter becomes the next parent. For example, uranium-238 passes through several alpha and beta decays before reaching stable lead-206.

First-Order Decay and Half-Life

Decay follows first-order kinetics: activity is $\lambda N$ , where $N$ counts the remaining nuclei of the radioactive isotope and $\lambda$ is its decay constant. The half-life, $t_{1/2}$ , is the time for half those nuclei to decay. Each half-life halves their amount; it does not remove half the entire physical sample, because daughter atoms remain.

Half-lives range from fractions of a second to billions of years and are characteristic of each isotope under ordinary conditions. After $k$ half-lives, the remaining fraction is $(1/2)^k$ . Amount, concentration, and activity ratios follow the same decay relation for the same isotope with consistent measurement conditions.

Radiometric Dating

Radiometric dating uses known decay rates and parent/daughter amounts to estimate age. Choose an isotope with a half-life appropriate to the timescale, and account for any daughter material present initially and later movement or contamination of isotopes. Uranium-238 ( $t_{1/2}\approx4.5\times10^9$ years) can date old rocks through its decay to lead-206.

Radiocarbon dating applies to once-living material. Cosmic-ray neutrons produce carbon-14 from atmospheric nitrogen:

^{14}_7\text{N} + ^1_0\text{n} \rightarrow ^{14}_6\text{C} + ^1_1\text{H}

Carbon-14 enters CO₂ and living organisms through photosynthesis and food chains. Living organisms continually exchange carbon; after death, exchange stops and C-14 decays ( $t_{1/2}=5730$ years), while stable C-12 remains. Compare the sample's C-14/C-12 ratio or activity per gram of carbon with a suitable reference. Atmospheric changes require calibration. The useful range is roughly up to 50,000 years; very old samples contain too little C-14 for reliable measurement. Other isotopes suit older geological samples.

The Same Fraction Decays in Each Equal Time Interval

Decay removes a constant fraction, not a constant number of nuclei. A smaller sample has lower activity but the same half-life.

Problem-Solving Routines & Methods

Half-Life and Decay Constant

t_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}

Variables & Constants

t_{1/2}

=half-life;

\lambda

=decay constant in inverse time units

Integrated Decay Law

N_t = N_0 e^{-\lambda t}, \qquad \ln\left(\frac{N_t}{N_0}\right)=-\lambda t

Variables & Constants

N_0

=initial nuclei count;

N_t

=remaining count;

t

=elapsed time

Finding Age or Elapsed Time

t = \frac{\ln(N_0/N_t)}{\lambda}

The same formula works with initial and current activities of the same isotope.

Solving Decay and Dating Problems

1
Identify the initial and remaining isotope amounts or activities, and form a ratio using matching units.
2
Find the decay constant from the half-life. Use the same time units for half-life and elapsed time.
3
Use repeated halving for a whole number of half-lives; otherwise use the exponential or logarithmic decay equation.
4
Check that a decaying amount decreases and that the calculated elapsed time is positive.

Practice & Concept Checks

Concept Check

An isotope has a half-life of 2.0 hours. What fraction remains after 6.0 hours?

Concept Check

Once-living material has one-quarter of its reference C-14 activity. Estimate its age before calibration.

Concept Check

How do beta-minus decay and positron emission change atomic number?

Key Terms & Vocabulary

alpha (α) decayRadioactive Decay

loss of an alpha particle during radioactive decay

beta (β) decayRadioactive Decay

breakdown of a neutron into a proton, which remains in the nucleus, and an electron, which is emitted as a beta particle

daughter nuclideRadioactive Decay

nuclide produced by the radioactive decay of another nuclide; may be stable or may decay further

electron captureRadioactive Decay

combination of a core electron with a proton to yield a neutron within the nucleus

gamma (γ) emissionRadioactive Decay

decay of an excited-state nuclide accompanied by emission of a gamma ray

half-life (t₁/₂)Radioactive Decay

time required for half of the atoms in a radioactive sample to decay

parent nuclideRadioactive Decay

unstable nuclide that changes spontaneously into another (daughter) nuclide

positron emissionRadioactive Decay

(also, β⁺ decay) conversion of a proton into a neutron, which remains in the nucleus, and a positron, which is emitted

radioactive decayRadioactive Decay

spontaneous decay of an unstable nuclide into another nuclide

radioactive decay seriesRadioactive Decay

chains of successive disintegrations (radioactive decays) that ultimately lead to a stable end-product

radiocarbon datingRadiometric Dating

highly accurate means of dating objects 30,000–50,000 years old that were derived from once-living matter; achieved by calculating the ratio of

{}_{\,6}^{14}\text{C}\text{:}\,{}_{\,6}^{12}\text{C}

in the object vs. the ratio of

{}_{\,6}^{14}\text{C}\text{:}\,{}_{\,6}^{12}\text{C}

in the present-day atmosphere

radiometric datingRadiometric Dating

use of radioisotopes and their properties to date the formation of objects such as archeological artifacts, formerly living organisms, or geological formations