Suppose that a radioactive sample contains equal numbers of...

Suppose that a radioactive sample contains equal numbers of two radioactive nuclides $A$ and $B$ at $t=0$. A has a half-life of $3.0 \mathrm{h}$, whereas $\mathrm{B}$ has a half-life of $12.0 \mathrm{h} .$ Find the ratio of the decay rates or activities $R_{\mathrm{A}} / R_{\mathrm{B}}$ at (a) $t=0,$ (b) $t=12.0 \mathrm{h},$ and (c) $t=24.0 \mathrm{h}$.

Key Concepts

Ratio of Decay Rates

When comparing the decay rates or activities of different radioactive substances within the same sample, especially when they start with the same number of nuclei, the ratio of their activities can be determined by the ratio of their decay constants and the exponential factors from the decay law. This concept is crucial for understanding how substances with different half-lives behave over time relative to one another.

Activity

The activity of a radioactive sample is the rate at which the nuclei decay, typically measured in disintegrations per second (becquerels) or decays per minute. Activity is directly proportional to the number of radioactive nuclei present, meaning that as the number of undecayed nuclei decreases, so does the activity.

Exponential Decay Law

The exponential decay law describes the decrease in the number of undecayed nuclei or the activity of a radioactive sample over time. It is expressed by the equation N(t) = N?e^(??t), where N? is the initial number of nuclei and ? is the decay constant. This law is fundamental in predicting the behavior of radioactive decay over time.

Half-life

The half-life of a radioactive substance is the time required for half of the nuclei in a sample to decay. It is a characteristic property of each radioactive isotope and is inversely related to its decay constant. Half-life provides a convenient way to compare the rates of decay of different isotopes.

Radioactive Decay

Radioactive decay is the spontaneous process by which unstable atomic nuclei lose energy by emitting radiation. This process is probabilistic in nature, meaning that it is impossible to predict exactly when a particular nucleus will decay, but the overall behavior of a large number of nuclei follows well-defined statistical laws.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time that a nucleus will undergo radioactive decay. It is mathematically related to the half-life through the expression ? = ln(2)/t?/?. A larger decay constant implies a faster rate of decay.

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