Radioactive Decay and Integrated Rate Laws

Radioactive Decay: Rate Laws and Calculations

Concept: Rate of Radioactive Decay

Radioactive decay is a process in which unstable atomic nuclei lose energy by emitting radiation. In general chemistry, radioactive decay is treated as a chemical kinetic process that follows a specific rate law.

• Radioactive decay is a first-order process, meaning the rate depends linearly on the amount of radioactive substance present.

• The rate law for radioactive decay is analogous to other first-order reactions in chemical kinetics.

Radioactive Integrated Rate Law

The integrated rate law for a first-order radioactive decay process relates the concentration (or amount) of a radioactive substance at any time to its initial concentration and the decay constant.

• General equation:

$\ln[N_t] = -kt + \ln[N_0]$

• $N_t$ = Amount (or concentration) of radioactive substance at time $t$

• $N_0$ = Initial amount (or concentration) of radioactive substance

• $k$ = Decay constant (units: $\text{time}^{-1}$)

• $t$ = Time elapsed

This equation can be rearranged to solve for any variable, depending on the information given.

Graphical Representation

• A plot of $\ln[N_t]$ versus $t$ yields a straight line with slope $-k$ and intercept $\ln[N_0]$.

• This linear relationship is characteristic of first-order kinetics.

Example Calculation

Example: The radioactive element astatine-210 has a decay constant of 0.084 min-1. How many minutes would it take for its concentration to go from 9.3 × 107 disintegrations/sec to 2.7 × 107?

• Use the integrated rate law:

$\ln[N_t] = -kt + \ln[N_0]$

• Plug in values and solve for $t$:

$t = \frac{\ln(N_t) - \ln(N_0)}{-k}$

Answer: 2.54731 min

Practice Problems

• Practice 1: For the radioactive decay of lead-202, the decay constant is 1.32 × 10-4 yr-1. How long (in hours) will it take to decrease to 52% of its initial amount?

• Practice 2: During World War I, radium-226 was used in the manufacturing of luminous paint. If it takes 1.2 × 103 days for its degradation to be 2.4%, compute its decay constant.

• Practice 3: If the decay constant for polonium-209 is 8.80 × 10-3 yr-1, what fraction of it remains after 1.1 × 102 years?

Key Terms and Definitions

• Decay constant ($k$): A proportionality constant that characterizes the rate of radioactive decay for a given isotope.

• First-order kinetics: A reaction rate that depends linearly on the concentration of one reactant.

• Half-life ($t_{1/2}$): The time required for half of the radioactive substance to decay. For first-order reactions:

$t_{1/2} = \frac{0.693}{k}$

Summary Table: Radioactive Decay Equations

Equation	Purpose
$N_t = N_0 e^{-kt}$	Find amount remaining after time $t$
$\ln[N_t] = -kt + \ln[N_0]$	Linear form for plotting and calculations
$t_{1/2} = \frac{0.693}{k}$	Calculate half-life from decay constant

Applications

• Radioactive decay calculations are essential in nuclear chemistry, dating archaeological samples, medical diagnostics, and nuclear energy.

• Understanding first-order kinetics allows prediction of how quickly a radioactive substance will decrease over time.

Additional info: Practice problems reinforce the use of the integrated rate law and decay constant in real-world scenarios, such as historical uses of radioactive materials and calculation of remaining fractions over time.