Question

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

Short answer

54.0 g of Zn-64 is produced.

Step-by-step solution

Understanding the Decay

The radioactive decay involves \[{ }^{64} \mathrm{Cu} \rightarrow{ }^{64} \mathrm{Zn}+{ }_{-1}^{0} \beta\]This tells us that copper-64 (\({ }^{64} \mathrm{Cu}\)) decays to produce zinc-64 (\({ }^{64} \mathrm{Zn}\)) and a beta particle.

Calculate the Number of Halflives

The half-life \( t_{1/2} \) of \({ }^{64} \mathrm{Cu}\) is \(12.8\) hours. We want to find out how many half-lives have passed in \(18.4\) hours. Calculate:\[\text{Number of half-lives} = \frac{18.4}{12.8} = 1.4375\]This means \(1.4375\) half-lives have elapsed.

Calculate the Remaining Copper-64

Use the formula for exponential decay:\[ N = N_0 \times \left(\frac{1}{2}\right)^{t/t_{1/2}} \]where \(N_0 = 84.0\,\mathrm{g}\), \(t = 18.4\,\mathrm{h}\) \[N = 84.0 \times \left(\frac{1}{2}\right)^{1.4375} \approx 30.0\,\mathrm{g}\]Thus, \(30.0\,\mathrm{g}\) of \({ }^{64} \mathrm{Cu}\) remains.

Calculate the Produced Zinc-64

Since the amount of copper-64 has decreased by decay to form zinc-64, the amount of zinc-64 produced is the difference between the initial and remaining amounts of copper-64.Calculate:\[84.0 \, \text{g initial} - 30.0 \, \text{g remaining} = 54.0 \, \text{g of Zn-64 produced}\]Thus, \(54.0\,\mathrm{g}\) of zinc-64 is produced.

Key concepts

Half-Life Calculation

Half-life is a very important concept when it comes to understanding radioactive decay. It's the time it takes for half of the radioactive nuclei in a sample to decay. So, just imagine if you start with 100 pieces of radioactive matter. After one half-life, only 50 pieces will remain. After another half-life, half of those 50 pieces will decay, leaving you with 25 pieces.

In the context of our exercise, we're looking at copper-64 ( ) which has a half-life of 12.8 hours. This means every 12.8 hours, half of the copper-64 will decay into zinc-64 ( ). To figure out how much has decayed over a certain period, you calculate how many half-lives fit into that time frame. In this instance, 18.4 hours have passed. Therefore, we divide 18.4 by the half-life (12.8) to see how many half-lives have occurred, which is approximately 1.4375 half-lives.

Understanding this basic division helps to track the decay process over time and is fundamental to solving decay-related problems.

Exponential Decay

Exponential decay is a process that describes how the quantity of a radioactive material decreases over time. This type of decay is characterized by the fact that the rate of decay decreases as the quantity decreases. In mathematical terms, it follows an exponential formula. For our copper-64 example, the formula looks like this:\[ N = N_0 \times \left(\frac{1}{2}\right)^{t/t_{1/2}} \]- **\( N_0 \)** is the initial quantity of the substance, which in our case is 84.0 grams of copper-64.- **\( t \)** is the total time elapsed (18.4 hours here).- **\( t_{1/2} \)** is the half-life of the substance (12.8 hours for copper-64).This formula gives us the remaining amount of copper-64 after a given time. Using our numbers, we find approximately 30.0 grams remain after 18.4 hours.

This principle of exponential decay is not only pivotal in calculating the remaining material but also in various disciplines like chemistry, physics, and environmental science.

Beta Particle Emission

Beta particle emission is an essential part of radioactive decay involving changes in the nucleus of an atom. In beta decay, a neutron converts into a proton and a beta particle, which is an electron or positron, is emitted. This process alters the atomic number of the element.

In the case of copper-64 ( ) decaying into zinc-64 ( ), a beta particle is emitted. This means the copper atom loses a neutron that becomes a proton. As a result, the copper-64 becomes zinc-64 because the atomic number increases by one, as proton count determines the element.

Understanding beta decay provides insight into the transmutation of elements and their nuclear reactions. It is also crucial in medical applications, nuclear energy, and studying the universe.