Question

A radioactive isotope of copper decays as follows: $$ { }^{64} \mathrm{Cu} \longrightarrow{ }^{64} \mathrm{Zn}+{ }_{-1}^{0} \beta \quad t_{\frac{1}{2}}=12.8 \mathrm{~h} $$ Starting with \(84.0 \mathrm{~g}\) of \({ }^{64} \mathrm{Cu},\) calculate the quantity of \(^{64}\) Zn produced after 18.4 h.

Short answer

To find the amount of zinc-64 produced, apply the half-life formula to calculate how much copper-64 remained after 18.4 hours. Then, subtract this quantity from the initial amount of copper-64. This will give the amount of zinc-64 produced in 18.4 hours.

Step-by-step solution

Understand the half-life formula

The formula for the half-life of a radioactive substance is \(N=N_0 \times \frac{1}{2}^{\frac{t}{t_{1/2}}}\), where \(N\) is the final quantity of substance, \(N_0\) is the initial quantity of the substance, \(t\) is the time, and \(t_{1/2}\) is the half-life of the substance.

Calculate the remaining copper-64

Substitute the given values into the half-life formula \(N_0 = 84.0\, g\), \(t = 18.4\, h\), and \(t_{1/2} = 12.8\, h\). Calculate the final quantity \(N\) of copper-64 left after 18.4 h: \(N = 84.0\, g \times \frac{1}{2}^{\frac{18.4\, h}{12.8\, h}}\).

Calculate the amount of zinc-64 formed

The amount of zinc-64 formed will be the initial quantity of copper-64 minus the final quantity of copper-64 after 18.4 h. This is because in this decay, each atom of copper-64 decays into one atom of zinc-64. So, if the amount of copper-64 decreases by a certain amount, the amount of zinc-64 will increase by the same amount.

Calculate for result

Subtract the final quantity of copper-64 from the initial quantity to find the amount of zinc-64 produced.

Key concepts

Half-Life Formula

The half-life of a radioactive isotope is a measure of the time it takes for half of the isotope nuclei present in a sample to decay. It's a constant characteristic for any given radioactive substance and incredibly important for various applications, including medical imaging and nuclear power generation.

Understanding the half-life formula is fundamental when dealing with radioactive decay. The formula is expressed as \(N=N_0 \times \frac{1}{2}^{\frac{t}{t_{1/2}}}\), where \(N\) represents the final quantity of the substance, \(N_0\) is the initial quantity, \(t\) is the elapsed time, and \(t_{1/2}\) is the substance's half-life. This mathematical relationship allows us to calculate how much of a radioactive isotope remains after a certain amount of time. By using logarithms, we can also rearrange the formula to solve for any of the variables, making it a versatile tool in nuclear chemistry.

Radioactive Isotope Decay

Radioactive isotope decay refers to the process by which an unstable atomic nucleus loses energy by emitting radiation. This emission can take the form of alpha particles, beta particles, gamma rays, or other particles. Over time, this leads to a change in the identity of the element, as seen in the decay of copper-64 to zinc-64.

Each radioactive isotope has its own decay pathway and rate, which is characteristically defined by its half-life. Understanding the nature of this decay is essential not just for predicting the quantity of material after a time period, as in homework problems or real-life scenarios, but also when considering safety and storage issues related to radioactive materials. This understanding is also crucial for applications like dating archaeological finds or treating medical conditions with radiotherapy.

Copper-64 to Zinc-64 Decay

When copper-64 (\(^{64}Cu\)) decays, it transforms into zinc-64 (\(^{64}Zn\)) by emitting a beta particle (\(\_{-1}^{0} \beta\)). For each atom of copper that decays, one atom of zinc is formed. This type of decay is common in nuclear medicine, where copper-64 might be used as a tracer due to its relatively short half-life and the emission of both positrons and beta particles, useful in diagnostic imaging.

By applying the concepts of half-life and radioactive decay, we can calculate not only the remaining mass of copper-64 after a certain period but also the amount of zinc-64 that has been produced. This calculation is essential for creating precise diagnostic tools, in dosimetry calculations for radiation therapy, and in other scientific and industrial applications where the precise knowledge of the rate of decay and the quantity of product formed is critical.