Decay Equations

• The Fundamental Rate Law: The rate of change of the number of nuclei is directly proportional to the number of nuclei present, expressed as $\frac{dN}{dt} = -\lambda N$.

• Exponential Nature: Because the decay rate depends on the current population, the resulting solution is an exponential decay function, meaning the sample decreases by a constant percentage rather than a constant amount.

• Statistical Probability: The decay constant $\lambda$ represents the statistical likelihood of decay; it is independent of external factors like temperature, pressure, or chemical bonding.

Calculating Remaining Nuclei

• Use the Exponential Decay Equation: $N = N_0 e^{-\lambda t}$, where $N_0$ is the initial quantity and $e$ is the base of the natural logarithm.

Determining Activity

• Activity follows the same exponential trend: $A = A_0 e^{-\lambda t}$. Alternatively, calculate instantaneous activity using $A = \lambda N$.

Solving for Time or Decay Constant

• Rearrange the decay equation using natural logarithms: $\ln\left(\frac{N}{N_0}\right) = -\lambda t$. This linearizes the relationship, making it easier to solve for unknown variables.

• It is vital to distinguish between the physical quantity of the sample and its rate of decay.

• Decay Constant vs. Half-Life: $\lambda$ is a rate (probability per time), while $T_{1/2}$ is a duration (time for 50% decay). They are inversely related: $\lambda = \frac{\ln 2}{T_{1/2}}$

• Unit Consistency: Always ensure that the units for the decay constant $\lambda$ (e.g., $s^{-1}$) match the units for time $t$ (e.g., $s$) before calculating the exponent.

• The 'n' Half-Lives Shortcut: For simple problems involving integer multiples of half-lives, use the formula $N = N_0 \left(\frac{1}{2}\right)^n$, where $n = \frac{t}{T_{1/2}}$.

• Sanity Check: After calculating $N$, verify that $N < N_0$. If your answer is larger than the starting amount, you likely omitted the negative sign in the exponent or used the wrong growth/decay formula.

• Logarithmic Graphs: Remember that a plot of $\ln(N)$ vs. $t$ results in a straight line with a gradient of $-\lambda$ and a y-intercept of $\ln(N_0)$.

• Mass vs. Nuclei: Students often use mass directly in the formula $A = \lambda N$. You must first convert mass to moles and then to the number of atoms using Avogadro's constant.

• Decayed vs. Remaining: Read questions carefully to determine if they ask for the amount remaining ($N$) or the amount that has decayed ($N_0 - N$).

• Background Radiation: In practical scenarios, the measured activity includes background radiation. You must subtract the background count rate before applying decay equations to the source data.