Half-life Knowledge-Based Study

Study notes

1. What is Half-Life?

   Radioactive decay is a spontaneous nuclear process in which an unstable parent nuclide releases radiation to transform into a more stable daughter nuclide. Because decay is a random quantum event for any individual nucleus, it is impossible to predict the exact moment a specific atom will disintegrate.

   However, in a sample containing a large number of nuclei, the rate of decay is statistically constant and predictable. The half-life (symbolized as $t_{1/2}$) is defined as the time required for half of the radioactive nuclei in a given sample to undergo decay.

   Key Properties of Half-Life

   • Independence from environmental factors: Unlike chemical reaction rates, nuclear half-lives are unaffected by temperature, pressure, concentration, or chemical state.
   • Isotope specificity: Every radioisotope possesses a characteristic half-life, ranging from tiny fractions of a second to billions of years. For example, carbon-14 has a half-life of 5,730 years, whereas uranium-238 has a half-life of 4.5 billion years.

   Calculating Elapsed Half-Lives

   When analyzing a radioactive sample over a total elapsed time $t$, the number of half-lives that have passed, represented by $n$, is calculated using the relationship:

   $n = \frac{t}{t_{1/2}}$

   Both the elapsed time and the half-life must be expressed in the same units of time before performing the calculation.

   Remaining Amount of Parent Nuclide

   Each half-life reduces the quantity of the parent nuclide by half. After $n$ elapsed half-lives, the remaining amount $N_t$ is determined from the initial amount $N_0$ according to:

   $N_t = N_0 \left(\frac{1}{2}\right)^n$

   This relationship applies directly to mass, the number of nuclei, or radioactivity count rates.

   Stepwise Decay Progression

   As consecutive half-lives pass, the remaining parent fraction decreases in a geometric progression:

   • 0 half-lives: 100% of the original sample remains (fraction = 1).
   • 1 half-life: 50% remains, and 50% has decayed into daughter products (fraction = 1/2).
   • 2 half-lives: 25% remains, and 75% has decayed (fraction = 1/4).
   • 3 half-lives: 12.5% remains, and 87.5% has decayed (fraction = 1/8).
   • 4 half-lives: 6.25% remains, and 93.75% has decayed (fraction = 1/16).

   Plotting the remaining quantity against time produces a characteristic downward exponential decay curve. While the curve approaches zero, it does so asymptotically.

   Step-by-Step Sample Problem

   A pure sample initially contains 80.0 g of iodine-131. The half-life of iodine-131 is 8.0 days. Determine the mass of iodine-131 remaining after 24.0 days.

   Step 1: Calculate the number of elapsed half-lives ($n$):

   $n = \frac{24.0}{8.0} = 3$

   Step 2: Determine the remaining fraction:

   $\left(\frac{1}{2}\right)^3 = \frac{1}{8}$

   Step 3: Calculate the final remaining mass ($N_t$):

   $N_t = 80.0 \times \frac{1}{8} = 10.0$ g

   After 24.0 days, 10.0 g of the parent iodine-131 remains, meaning 70.0 g of the original sample has decayed into stable daughter products.