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

Suppose that a radioactive sample contains equal numbers of two radioactive nuclides $\mathrm{A}$ and $\mathrm{B}$ at $t=0 .$ A has a half-life of $3.0 \mathrm{~h}$, while $\mathrm{B}$ has a half-life of $12.0 \mathrm{~h}$. Eind 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) $t=24.0 \mathrm{~h}$.

Suppose that a radioactive sample contains equal numbers of two radioactive nuclides $\mathrm{A}$ and $\mathrm{B}$ at $t=0 .$ A has
a half-life of $3.0 \mathrm{~h}$, while $\mathrm{B}$ has a half-life of $12.0 \mathrm{~h}$. Eind 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

Activity (Decay Rate)

Activity, or decay rate, is the number of decay events per unit time in a radioactive sample and is directly proportional to the number of remaining radioactive nuclei. It is given by the expression Activity = ?N, where N is the number of undecayed nuclei. This concept is essential for comparing how quickly different radioactive substances decay over time.

Radioactive Decay Law

The radioactive decay law describes the process by which unstable nuclei lose energy by emitting radiation. It is commonly expressed by an exponential decay formula, N(t) = N?e^(??t), which shows that the number of radioactive nuclei decreases exponentially over time. This law is central to understanding how samples of radioactive material diminish as each nuclide decays at its characteristic rate.

Half-life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is a specific property of each radioactive nuclide and provides a convenient way of characterizing the decay process. A shorter half-life indicates a faster decay rate, while a longer half-life indicates a slower decay rate.

Decay Constant

The decay constant (?) is a parameter that quantifies the probability per unit time that a nucleus will decay. It is directly related to the half-life by the relation ? = ln(2) / (half-life). This constant is used in the exponential decay law to determine how rapidly the number of undecayed nuclei decreases over time.

Ratio of Activities

When comparing the decay rates of two different radioactive nuclides, the ratio of activities can be determined by considering both the initial number of nuclei present and their respective decay constants. Even if two nuclides start with equal numbers, their activities will change differently over time if their half-lives (and thus decay constants) differ. Calculating this ratio at different times involves applying the exponential decay law to both and forming a ratio R_A / R_B.

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