Question

Calcium- 47 , a beta emitter, has a half-life of \(4.5\) days. a. Write the balanced nuclear equation for the beta decay of calcium-47. b. How many milligrams of a \(16-\mathrm{mg}\) sample of calcium- 47 would remain after 18 days? c. How many days have passed if \(4.8 \mathrm{mg}\) of calcium- 47 decayed to \(1.2 \mathrm{mg}\) of calcium- \(47 ?\)

Short answer

a. \[^{47}_{20}Ca \rightarrow ^{47}_{21}Sc + \beta^- + \bar{u}_e\] b. 1 mg c. 9 days

Step-by-step solution

Write the balanced nuclear equation for beta decay

In beta decay, a neutron in the nucleus is converted into a proton while emitting a beta particle (\beta^-), which is an electron, and an antineutrino. Calcium-47 can be written as \[^{47}_{20}Ca \rightarrow ^{47}_{21}Sc + \beta^- + \bar{u}_e\]

Determine the remaining sample after 18 days

First, calculate the number of half-lives that have passed. Given the half-life (\text{t}_{1/2}) is 4.5 days, the number of half-lives in 18 days is \[\frac{18 \text{ days}}{4.5 \text{ days/half-life}} = 4 \text{ half-lives}\]Using the half-life formula for remaining quantity: \[N = N_0 \times \frac{1}{2}^{n}\] where \( N_0 = 16 \text{ mg} \) and \( n = 4 \), we get \[N = 16 \text{ mg} \times \frac{1}{2}^{4} = 16 \text{ mg} \times \frac{1}{16} = 1 \text{ mg}\]Thus, 1 mg of calcium-47 would remain after 18 days.

Calculate time passed given initial and final amounts

Use the half-life formula rearranged for time elapsed. Start with the equation for exponential decay: \[N = N_0 \times \frac{1}{2}^{\frac{t}{t_{1/2}}} \]For \( N_0 = 4.8 \text{ mg} \) and \( N = 1.2 \text{ mg} \), solve for \( t \): \[1.2 = 4.8 \times \frac{1}{2}^{\frac{t}{4.5}}\]Divide both sides by 4.8: \[\frac{1.2}{4.8} = \frac{1}{2}^{\frac{t}{4.5}}\]This simplifies to: \[\frac{1}{4} = \frac{1}{2}^{\frac{t}{4.5}}\]Since \( \frac{1}{4} = \frac{1}{2}^{2} \), equate the exponents: \[2 = \frac{t}{4.5}\]Solving for \( t \): \[ t = 2 \times 4.5 = 9 \text{ days}\]Thus, 9 days have passed.

Key concepts

beta decay

In beta decay, a neutron inside an atom's nucleus transforms into a proton. During this process, it emits a beta particle (\beta^-), which is essentially an electron, and an antineutrino. This transformation increases the atomic number of the atom by one because the number of protons increases while the mass number remains the same. For example, in the case of calcium-47, the decay process can be represented by the nuclear equation: \(^{47}_{20}Ca \rightarrow ^{47}_{21}Sc + \beta^- + \bar{u}_e\).Here, calcium-47 changes into scandium-47. This is a fundamental concept as it shows how elements can transform over time through radioactive decay.

nuclear equations

Nuclear equations describe the changes that occur in the nucleus of an atom during radioactive decay. These equations are vital for understanding the processes inside the nucleus. The key components of a nuclear equation include:

• The original atom (parent isotope)
• The new atom (daughter isotope)
• The emitted particles (like the beta particle in beta decay)

For example, in the beta decay of calcium-47, the nuclear equation is balanced when the sum of atomic numbers and mass numbers on both sides of the equation are equal: \(^{47}_{20}Ca \rightarrow ^{47}_{21}Sc + \beta^- + \bar{u}_e\). This indicates a proper conservation of mass and charge.

exponential decay

Exponential decay describes how the amount of a radioactive substance decreases over time. The process is characterized by a constant percentage reduction in the quantity of the radioactive isotope per unit time. The core formula used to calculate the remaining quantity (\text{N}) after a certain time (\text{t}) is: \(N = N_0 \times \frac{1}{2}^{\frac{t}{t_{1/2}}}\), where \(N_0\) is the initial quantity and \(t_{1/2}\) is the half-life.For example, given \(N_0 = 16 \text{ mg}\), \(t_{1/2} = 4.5 \text{ days}\), and \(t = 18 \text{ days}\), the remaining quantity is calculated as \(N = 16 \text{ mg} \times \frac{1}{2}^{4} = 1 \text{ mg}\). This shows that using the exponential decay formula, we can predict the amount of radioactive isotope left after a certain period.

radioactive isotopes

Radioactive isotopes, or radioisotopes, are variants of chemical elements that have unstable nuclei. These isotopes undergo radioactive decay, resulting in the emission of radiation and the transformation into more stable isotopes. Some common characteristics of radioactive isotopes include:

• They can be naturally occurring or artificially produced.
• They have specific half-lives, which determine how quickly they decay.
• They are used in various fields, like medicine for imaging and treatment, archaeology for dating artifacts, and energy production in nuclear power plants.

Understanding radioactive isotopes, like calcium-47 with its 4.5-day half-life, allows scientists to utilize these materials for many beneficial applications, while also handling them safely to minimize radiation exposure.