Problem 24 - Chemistry: Principles and Reactions

Question

Bromine-82 has a half-life of 36 hours. A sample containing Br-82 was found to have an activity of \(1.2 \times 10^{5}\) disintegrations \(/ \mathrm{min}\). How many grams of Br-82 were present in the sample? Assume that there were no other radioactive nuclides in the sample.

Step-by-Step Solution

Answer

Answer: To find the mass of Br-82 in a sample, follow these steps: 1. Find the decay constant: λ = 0.693 ÷ 36 2. Convert the activity to decays per second: 1.2 x 10^5 disintegrations/min × (1 min ÷ 60 s) 3. Calculate the number of Br-82 atoms: N = Activity (s^-1) ÷ λ 4. Calculate the mass of Br-82 in grams: Br-82 mass = N × (81.92 g/mol ÷ 6.022 x 10^23) By following these steps, the mass of Br-82 in the sample can be determined.

Step 1: Find the decay constant

To find the decay constant(λ), we can use the formula for half-life(T) which is; $$T = \frac{0.693}{\lambda}$$ We are given the half-life T = 36 hours. Now, we can calculate the decay constant(λ) as follows: $$\lambda = \frac{0.693}{T} = \frac{0.693}{36}$$

Step 2: Convert activity to decays per second

We are given the activity in disintegrations per minute (1.2 x 10^5 disintegrations/min). To make our calculations consistent with the time units used for half-life (hours), we must convert this activity to decays per second. We can achieve this by multiplying the activity with the conversion factor of 60 seconds. $$Activity (s^{-1}) = Activity (min^{-1}) \times \frac{1 \thinspace min}{60 \thinspace s} = 1.2 \times 10^{5} \times \frac{1}{60} \thinspace disintegrations/second$$

Step 3: Calculate the number of Br-82 atoms

Now, we will use the decay constant(λ) and activity to find the number of Br-82 atoms(N) in the sample. We can use the formula: $$Activity (s^{-1}) = \lambda \times N$$ Rearrange the equation to find the number of atoms (N): $$N = \frac{Activity (s^{-1})}{\lambda}$$ Compute N using the values from Step 1 and Step 2.

Step 4: Calculate the mass of Br-82 in grams

We can now use the number of Br-82 atoms(N) to calculate the mass of Br-82 present in the sample. First, we need to find the molar mass of Br-82. The molar mass of Br-82 is 81.92 g/mol. Using Avogadro’s number (6.022 x 10^23), we can calculate the mass of one Br-82 atom by using: $$1 \thinspace atom \thinspace mass = \frac{molar \thinspace mass}{Avogadro's \thinspace number} = \frac{81.92}{6.022 \times 10^{23}} \thinspace g/atom$$ Now, we can calculate the mass of Br-82 in grams by multiplying the number of Br-82 atoms(N) with 1 atom mass: $$Br-82 \thinspace mass = N \times 1 \thinspace atom \thinspace mass \thinspace (g)$$

Key Concepts

Half-Life of Isotopes

The half-life of an isotope is the time taken for half of the radioactive nuclei in a sample to decay. It's an intrinsic property of radioactive substances and is symbolically denoted by the letter 'T'. During each half-life period, the amount of the radioactive isotope decreases by 50%. Knowing the half-life is essential in calculating how long it will take for a given isotope to reduce to a certain amount.

For instance, in the exercise with Bromine-82 ('Br-82'), we're told that its half-life is 36 hours. That means in every 36-hour period, the number of active Br-82 atoms will halve due to radioactive decay. Importantly, the half-life can help us determine the decay constant and, from there, the initial quantity of the radioactive material.

Decay Constant

The decay constant, represented by the Greek letter λ, is a probability factor that describes the likelihood of a single atom decaying per unit time. In simpler terms, it's a measure of how quickly radioactive atoms are expected to decay. The decay constant and the half-life of an isotope are inversely related; as shown in the exercise solution, you can find one if you know the other using the formula: \[ \lambda = \frac{0.693}{T} \].

This relation is critical since the decay constant directly feeds into our calculations for the original activity or number of atoms present in a sample. In nuclear chemistry, understanding the decay constant is vital for making accurate predictions about a substance's behavior over time, which has practical applications in fields like medicine, archaeology, and environmental science.

Nuclear Chemistry

Nuclear chemistry is the branch of chemistry that deals with nuclear reactions and their implications. It encompasses various phenomena, including fission, fusion, and radioactive decay. Among these, radioactive decay is particularly important for understanding how unstable isotopes, like Br-82, transform into stable ones over time by emitting radiation in the form of particles or photons.

This type of chemistry is not only about comprehending the processes that go on within a nucleus but also involves calculations like those performed in our exercise. By using principles of nuclear chemistry, we can predict quantities, dates, and even the potential energy released in nuclear reactions. The subject is integral to energy generation, medical diagnostic and treatment techniques, and it is also key to ensuring safety from radiation hazards.