8.02 Radioactivity & Half-Life Practice Quiz

The half-life is defined as the time required for half of a given amount of radioactive substance to decay. It is a basic concept underlying exponential decay in radioactivity.

Alpha decay is the process in which an unstable nucleus emits an alpha particle, comprised of two protons and two neutrons. This emission reduces the atomic number by two and the mass number by four, making it distinct from other decay processes.

After one half-life, exactly half of the radioactive material remains regardless of the initial amount. This property is the foundation of the exponential decay process.

Alpha particles, due to their relatively large mass and charge, interact strongly with matter and are easily stopped by a sheet of paper or human skin. This makes them the least penetrating form of radiation compared to beta particles and gamma rays.

Radioactive decay occurs randomly with each nucleus having a fixed probability of decaying in a given time interval. This randomness leads to an overall exponential decay behavior when observing large numbers of atoms.

The formula N = N₀ * (1/2)^(t/T₝/₂) accurately captures the exponential decay of a radioactive substance based on its half-life. Every time the elapsed time reaches another half-life, the remaining quantity is halved.

The decay constant is related to the half-life by the equation λ = ln(2) / T₝/₂. This equation arises from equating N/N₀ to 1/2 when t equals the half-life in the exponential decay formula.

In one half-life, the amount decreases from 100 grams to 50 grams, and in a second half-life it reduces from 50 grams to 25 grams. Therefore, two half-lives have elapsed.

Radioactive decay is probabilistic, meaning that while the overall decay rate is predictable, the decay time of any single nucleus is random. This characteristic is a fundamental aspect of quantum mechanics.

When the daughter nuclide decays much faster than the parent, it does not accumulate significantly, meaning the decay rate of the entire chain is effectively determined by the parent nuclide's half-life. This simplifies the analysis of decay chains.

After one half-life (24 hours) 50% of the substance remains, and after two half-lives (48 hours) only 25% remains. This is a direct consequence of the halving process in exponential decay.

Exponential decay is based on the premise that each radioactive nucleus possesses a fixed probability of decaying per unit time. This assumption leads directly to the observed exponential decrease in the number of undecayed nuclei.

The decay of a radioactive substance follows an exponential trend, resulting in a curve that decreases rapidly at first and then gradually approaches zero. This distinguishes it from a linear or parabolic behavior.

One curie is defined as 3.7 � - 10^10 disintegrations per second. It is a metric used to quantify the radioactivity of a substance, indicating the number of atomic decays occurring each second.

The hazard of a radioactive isotope is largely determined by the type and energy of the radiation it emits, as these dictate how deeply it can penetrate and cause biological damage. While half-life also plays a role, radiation quality is key when evaluating potential risks.

Since 24 days equals three half-lives (8 days each), the quantity remaining is (1/2)^3, or 1/8 of the original amount. Multiplying the remaining 10 grams by 8 gives an initial amount of 80 grams.

Using the relation T₝/₂ = ln(2) / λ, substituting λ = 0.05 day❻¹ gives T₝/₂ = 0.693 / 0.05, which is roughly 13.9 days. This demonstrates the inverse relationship between the decay constant and the half-life.

Radiometric dating relies on the assumption that the starting amount of the daughter isotope is either negligible or well-known. This, together with the constant decay rate, allows for accurate age determination based on the remaining parent isotope.

Since isotope B has a longer half-life, it decays more slowly than isotope A. Over time, the quantity of isotope B remains higher, causing it to dominate the mixture.

Using the decay law N = N₀e^( - λt) with N/N₀ = 0.2 and λ = 0.1, we solve for t as t = - ln(0.2)/0.1, which yields roughly 16.1 years. This calculation illustrates how logarithms are used to back-calculate elapsed time from radioactive decay data.