Radioactive Decay

• The Exponential Decay Law states that the probability of a nucleus decaying per unit time is constant. This leads to the mathematical relationship $N(t) = N_0 e^{-\lambda t}$, where $N(t)$ is the number of nuclei remaining at time $t$, $N_0$ is the initial number, and $\lambda$ is the decay constant.

• Half-life ($T_{1/2}$) is the time required for half of the radioactive nuclei in a sample to decay. It is a constant property for a specific isotope and is related to the decay constant by the formula $T_{1/2} = \frac{\ln(2)}{\lambda} \approx \frac{0.693}{\lambda}$.

• Because the process is random, we can only predict the behavior of large populations of atoms using statistics. For a single atom, we can only state the probability that it will decay within a certain timeframe.

• Decay is unaffected by external physical conditions such as temperature, pressure, or chemical bonding. This makes radioactive isotopes reliable 'clocks' for geological and archaeological dating.

Balancing Nuclear Equations

• To represent decay, we use nuclear equations where the total Mass Number ($A$) and Atomic Number ($Z$) must be conserved (balanced) on both sides of the reaction arrow.
• In Alpha Decay, the parent nucleus emits a helium nucleus ($^4_2\text{He}$), resulting in a daughter with $A-4$ and $Z-2$.
• In Beta-minus Decay, a neutron converts into a proton and an electron ($^0_{-1}e$); the mass number $A$ remains constant while the atomic number $Z$ increases by 1.

Calculating Remaining Activity

• To find the amount of a substance remaining after $n$ half-lives, use the formula $N = N_0 \times (0.5)^n$, where $n = \frac{\text{total time}}{\text{half-life}}$.
• For example, after 3 half-lives, the remaining activity is $1/2 \times 1/2 \times 1/2 = 1/8$ (or $12.5\%$) of the original value.

• It is vital to distinguish between the three primary types of radiation based on their physical properties and interactions with matter.

• Activity vs. Count Rate: Activity is the total number of decays occurring in the source per second, while count rate is what a detector measures. Count rate is always lower than activity because radiation is emitted in all directions and some is absorbed before reaching the detector.

• Account for Background Radiation: In many problems, you must subtract the background count rate from the measured count rate before performing half-life calculations. Failure to do this will result in an incorrectly high calculated half-life.

• Check the Units: Ensure time units for half-life and total elapsed time match (e.g., both in years or both in seconds) before calculating the number of half-lives.

• Graph Interpretation: When given a decay graph, always identify the initial value ($y$-intercept) and find the time it takes for that value to drop exactly by half. Repeat this for a second interval (e.g., from $1/2$ to $1/4$) to verify the half-life is constant.

• Sanity Check: If a substance has a half-life of 10 minutes and 30 minutes have passed, the remaining amount must be exactly $1/8$ of the start. If your calculation gives a larger number, you likely multiplied instead of divided.