Problem 41 - Chemistry

Question

The bromine- 82 nucleus has a half-life of \(1.0 \times 10^{3}\) min. If you wanted 1.0 g \(^{82}\mathrm{Br}\) and the delivery time was 3.0 days, what mass of NaBr should you order (assuming all of the Br in the NaBr was \(^{82} \mathrm{Br}\) )?

Step-by-Step Solution

Answer

To obtain 1.0 g of Bromine-82, you should order: \[\text{Mass of NaBr} = \frac{1.0 \, \text{g}}{e^{-\frac{\ln 2}{1.0 \times 10^3 \,\text{min}} \times 4320 \, \text{min}}} \times \frac{22.99 \, \text{g/mol}}{81.92 \, \text{g/mol}}\]

Step 1: Calculate the decay constant for Bromine-82

We are given that the half-life of Bromine-82 is \(1.0 \times 10^3\) minutes. We can use this information to calculate the decay constant 'λ' using the following formula: \[t_{1/2} = \frac{\ln 2}{\lambda}\] Rearranging for λ, we get: \[\lambda = \frac{\ln 2}{t_{1/2}}\] Plug in the values: \[\lambda = \frac{\ln 2}{1.0 \times 10^3 \, \text{min}}\]

Step 2: Find the total decay time

The delivery time is given as 3.0 days, we need to convert this into minutes. \[\text{Delivery time} = 3.0 \, \text{days} \times \frac{24 \, \text{hours}}{1 \, \text{day}} \times \frac{60 \, \text{min}}{1 \, \text{hour}} = 4320 \, \text{min}\]

Step 3: Calculate the initial mass of Bromine-82

Using the decay equation, we can find the initial mass of Bromine-82 by considering its final mass after 3 days. \[N_t = N_0 e^{-\lambda t}\] We want to find \(N_0\). Rearranging for \(N_0\), we get: \[N_0 = \frac{N_t}{e^{-\lambda t}}\] Now, plug in the values for \(N_t\), \(t\), and the decay constant λ that we calculated in Step 1: \[N_0 = \frac{1.0 \, \text{g}}{e^{-\frac{\ln 2}{1.0 \times 10^3 \, \text{min}} \times 4320 \, \text{min}}}\]

Step 4: Calculate the mass of NaBr needed

Now that we have the initial mass of \(^{82}\mathrm{Br}\), we can calculate the mass of NaBr required. We will use the molar masses of Na (22.99 g/mol) and \(^{82}\mathrm{Br}\) (81.92 g/mol). The mass ratio of Na and \(^{82}\mathrm{Br}\) in NaBr can be determined as follows: \[\text{Mass ratio} = \frac{\text{Molar mass of Na}}{\text{Molar mass of } ^{82}\mathrm{Br}} = \frac{22.99 \, \text{g/mol}}{81.92 \, \text{g/mol}}\] So, the mass of NaBr needed can be calculated by multiplying the initial mass of \(^{82}\mathrm{Br}\) with the mass ratio: \[\text{Mass of NaBr} = N_0 \times \text{Mass ratio} = \frac{1.0 \, \text{g}}{e^{-\frac{\ln 2}{1.0 \times 10^3 \,\text{min}} \times 4320 \, \text{min}}} \times \frac{22.99 \, \text{g/mol}}{81.92 \, \text{g/mol}}\] Finally, compute the mass of NaBr needed to obtain 1.0 g \(^{82}\mathrm{Br}\).

Key Concepts

Half-life

The concept of half-life is crucial in understanding radioactive decay. It refers to the amount of time it takes for half of a given sample of a radioactive isotope to decay. For instance, if you start with 100 grams of a radioactive substance, and its half-life is 1 year, only 50 grams will remain after 1 year. This process continues, so after another year, you'd have 25 grams remaining.

Half-life is determined by the equation:

• \( t_{1/2} = \frac{\ln 2}{\lambda} \)

where \( \ln 2 \) (about 0.693) is the natural logarithm of 2, and \( \lambda \) is the decay constant specific to each substance.

This characteristic time period helps scientists predict how quickly a radioactive isotope will become non-hazardous or how much of it will remain after a certain time. In the context of the exercise, Bromine-82 has a half-life of \(1.0 \times 10^3\) minutes, meaning every 1,000 minutes, half of it will decay.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are variants of chemical elements that have an unstable combination of protons and neutrons in the nucleus. This instability causes them to undergo radioactive decay, transforming into different elements or isotopes over time.

These isotopes emit radiation in the form of alpha, beta, or gamma rays as they decay, leading to a new, more stable nucleus. A common application of radioisotopes is in medical imaging and therapy, where they help diagnose and treat certain illnesses.

In our exercise, Bromine-82 is a radioactive isotope that decays over time. Since it has a distinct half-life, we use this property to calculate how much of it needs to be ordered to ensure a certain amount is available after a specific delivery time. This understanding is critical when working with short-lived radioisotopes that decay quickly.

Decay Constant

The decay constant \( \lambda \) is a fundamental parameter in the study of radioactive decay, representing the probability per unit time that a nucleus will decay. It provides a mathematical way to describe how quickly a radioactive isotope will decay over time.

The relationship between the decay constant and half-life is expressed by:

• \( \lambda = \frac{\ln 2}{t_{1/2}} \)

Where \( t_{1/2} \) is the half-life of the isotope.

A larger decay constant means a faster decay rate, and thus a shorter half-life. In the example of Bromine-82 from our exercise, knowing its decay constant allows us to calculate how much of the isotope is left after a certain period. This is essential for precise planning in practical applications such as medicine or research, where specific quantities of isotopes are needed for effective results.