24.2 The Kinetics of Radioactive Decay
Radioisotopic Dating
A method for determining the age of an object based on the rate of decay of a particular radioactive nuclide relative to a stable nuclide; uses unstable radioistopes to determine the ages of objects. Based on measuring the amounts of Carbon-14 and Carbon-12 in materials of biological origin. Accuracy of the method falls off after about six half-lives of Carbon-14 (t1/2 = 5730 years), so it is used to date objects up to about 36,000 years old.
Half-Life of Radioactive Decay
Decay rates are commonly expressed in terms of the fraction of nuclei that decay over a given time interval--this is known as the half life (t1/2)
How do we determine the half-life of a nuclear reaction from its rate constant?
Determine from rate constant--find number of nuclei remaining, Nt, after a given time t: ln(Nt/N0) = -kt OR Nt = N0e^-kt AND ln(N0/Nt) = kt THEREFORE: ln(N0/0.5N0) = kt1/2 This half-life (first-order) is NOT dependent on the number of nuclei and is inversely related to the decay constant large k-->short half-life small k--->long half-life
Commonalities Between Chemical and Nuclear Systems
- Both chemical and nuclear systems tend toward maximum stability. - In chemical systems, concentrations change in a predictable direction to give a stable equilibrium ratio; while in nuclear systems, the type and number of nucleons in an unstable nucleus change in a predictable direction to give a stable N/Z ratio. - The tendency of a system (either chemical or nuclear) to become more stable tells nothing about how long that process will take.
How can we determine the rate of radioactive decay?
- Since radioactive emissions interact with surrounding atoms, we can determine rate by measuring radioactivity by observing the effects of these interactions over time. - These effects can be electrically amplified billions of times, and the decay of a single nucleus can be detected. - Ionization Counters and Scintillation Counters are devices used to detect decay
Ionization Counter
- detects radioactive emissions as they ionize a gas. The ionization produces free electrons and gaseous cations, which are attracted to electrodes that conduct a current to a recording device. - EX: Geiger-Müller Counter
Scintillation Counter
- detects radioactive emissions by their ability to excite atoms and cause them to emit light - light-emitting substance in the counter, called a phosphor, is coated onto part of a photomultiplier tube--a device that increases the original electrical signal. - Incoming radioactive particles strike the phosphor, which emits photons. -Each photon strikes a cathode, which releases an electron through the photoelectric effect. These electrons hit other portions of the tube, increasing the numbers of electrons, and the resulting current is recorded.
Decay Rate and Decay Constant
- specific activity--the decay rate per gram - For a large collection of radioactive nuclei, the number decaying per unit time is proportional to the number present: - Decay Rate (A) ∝ N or A=kN where k is the decay constant and is characteristic of each type of nuclide. - The larger the value of k, the higher the decay rate. A = -dN/dt = kN Radioactive Decay is a first order process (only one reactant) with respect to the number of nuclei, rather than their concentration.
How does a Geiger-Müller Counter work?
- type of ionization counter consisting of a tube filled with an argon-methane mixture. - Tube housing acts as cathode, and a thin wire in the center of the tube acts as anode. - Emissions from samples enter the tube through a window and strike argon atoms. - These collisions produce Ar+ ions that migrate toward the cathode and free electrons that are accelerated toward the anode. - In an "avalanche effect," the already freed electrons collide with other Ar atoms and free more electrons. - Thusly, the current originally created is amplified and appears as a meter reading or audible click. - An initial release of 1 electron can release 10^10 electrons in a microsecond, making the Geiger-Müller very sensitive.
Liquid Scintillation Counters
- use an organic mixture, "cocktail," that contains phosphor and a solvent. - The cocktail dissolves the radioactive sample and emits pulses of light when excited by emission. - The number of pulses is proportional to the concentration of the radioactive substance (remember Curie's discovery!) - These counters are often used to measure β- emissions from dissolved biological samples, particularly those containing compounds of Hydrogen-3 and Carbon-14.
Summary
-Ionization and Scintillation counters measure the number of emissions from a radioactive sample. -The decay rate (activity) of a sample is proportional to the number of radioactive nuclei. Nuclear decay is a first order process, so the half-life of the process is a constant. -Radioisotopic methods, such as Carbon-14 dating, determine the age of an object by measuring the ratio of specific isotopes in it.
Units of Radioactivity
1) SI unit: becquerel (Bq), defined as one disintegration per second (d/s) 2) Curie (Ci), much larger and more common unit; 3.70*10^10 d/s. millicurie (mCi) and microcurie (μ) are often used
Rate of Radioactive Decay
Radioactive nuclei decay at a characteristic rate, regardless of the chemical substance in which they occur - Decay Rate (or activity (A)) is the change in number of Nuclei divided by the change in time--the magnitude of this value is considered the rate.
In what terms can the half-life of a nuclide be expressed?
The half-life can be express in terms of number of nuclei, mass of radioactive substance, and activity: Number of Nuclei: number of nuclei remaining is halved after each half-life Mass: the amount of mass remaining is halved after each half-life Activity: activity depends on the number of nuclei, so the activity is halved after each succeeding half-life
How does radiocarbon dating work?
t = (1/k)ln(A0/At)
