Calculating Half-Life

• The Decay Law: The fundamental equation governing this process is $N(t) = N_0 e^{-\lambda t}$, where $N(t)$ is the quantity remaining at time $t$, $N_0$ is the initial quantity, and $e$ is Euler's number. This formula shows that the rate of change is proportional to the current amount.

• Deriving the Half-Life Formula: By setting $N(t) = \frac{1}{2}N_0$ and $t = t_{1/2}$, the equation simplifies to $\frac{1}{2} = e^{-\lambda t_{1/2}}$. Taking the natural logarithm of both sides leads to the critical relationship: $t_{1/2} = \frac{\ln(2)}{\lambda}$.

• Constant Probability: The principle of half-life relies on the fact that every nucleus in a sample has the same probability of decaying in a given time interval. This means that whether you have one gram or one ton of a substance, the time it takes for half of it to disappear remains identical.

Method 1: The Power-of-Two Approach

• Step 1: Determine the number of half-lives ($n$) that have passed by dividing the total time ($t$) by the half-life ($t_{1/2}$): $n = \frac{t}{t_{1/2}}$.
• Step 2: Calculate the remaining fraction by raising $0.5$ to the power of $n$. The final amount is $N = N_0 \times (0.5)^n$.
• Application: This method is most efficient when the total time is an integer multiple of the half-life.

Method 2: The Logarithmic Approach

• Step 1: When solving for time or the half-life itself, use the formula $n = \frac{\log(N/N_0)}{\log(0.5)}$.
• Step 2: Rearrange the formula to isolate the unknown variable. For example, to find the half-life given the time and the remaining amount: $t_{1/2} = \frac{t \cdot \log(0.5)}{\log(N/N_0)}$.
• Application: Use this method for non-integer time intervals or when the final mass is not a simple fraction of the initial mass.

• Half-Life vs. Mean Life: While half-life is the time for 50% decay, the mean life ($\tau$) is the average lifetime of a nucleus, calculated as $\tau = \frac{1}{\lambda}$. The mean life is always longer than the half-life by a factor of approximately $1.44$.

• Physical vs. Biological Half-Life: In medical contexts, the physical half-life is the isotope's natural decay rate, while the biological half-life is the time it takes for a living organism to eliminate half of the substance through metabolic processes.

• Unit Consistency: Always ensure that the units for total time ($t$) and half-life ($t_{1/2}$) match before performing calculations. If the half-life is in days and the elapsed time is in hours, convert one to match the other to avoid massive errors.

• The Fraction Shortcut: Memorize common powers of 2 to quickly estimate remaining amounts. After 1 half-life, 50% remains; after 2, 25%; after 3, 12.5%; after 4, 6.25%; and after 5, 3.125%.

• Sanity Checks: If your calculated final mass is larger than the initial mass, or if the time elapsed is positive but the mass increased, you likely swapped the $N$ and $N_0$ variables or used a positive exponent instead of a negative one.

• Logarithm Base: You can use either natural logs ($\ln$) or common logs ($\log_{10}$) as long as you are consistent within the equation, as the ratio of logs remains the same.

• Linear Decay Error: A common mistake is assuming that if half of a sample decays in 10 years, the other half will decay in the next 10 years. In reality, only half of the remaining sample decays in the second interval, leaving 25% of the original.

• Confusing Amount with Activity: Students often confuse the number of atoms ($N$) with the activity ($A$), which is the rate of decay. While both decrease by half every half-life, they are distinct physical quantities related by $A = \lambda N$.

• Rounding Intermediate Steps: Rounding the decay constant (e.g., using $0.69$ instead of $0.6931...$) early in a multi-step calculation can lead to significant errors in the final answer, especially for long time periods.