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/min. How many grams of Br- 82 were present in the sample? Assume that there were no other radioactive nuclides in the sample.

Short answer

Answer: To calculate the mass of Br-82 in the sample, first, determine the decay constant, \(\lambda\), using the half-life formula: $$\lambda = \frac{ln2}{t_{1/2}}$$ $$\lambda = \frac{ln2}{36}$$ Next, find the initial activity, \(A_0\) using the decay constant and final activity, A: $$A_0 = \frac{A}{e^{-\lambda t}}$$ $$A_0 = \frac{1.2 \times 10^{5}}{e^{-\frac{ln2}{36} \times 0}}$$ Calculate the number of atoms of Br-82 using the initial activity and decay constant: $$N = \frac{A_0}{\lambda}$$ $$N = \frac{1.2 \times 10^{5}}{\frac{ln2}{36}}$$ Finally, determine the mass of Br-82 using the number of atoms, Avogadro's number, and the atomic weight of Br-82: $$mass_{Br-82} = \frac{N \times (81.918 \ \text{g/mol})}{6.022\times10^{23}}$$ $$mass_{Br-82} = \frac{\frac{1.2 \times 10^{5}}{\frac{ln2}{36}} \times (81.918 \ \text{g/mol})}{6.022\times10^{23}}$$ Now, calculate the mass to obtain the answer.

Step-by-step solution

Find Decay Constant, \(\lambda\)

Remember that the half-life formula is, $$t_{1/2} = \frac{ln2}{\lambda}$$, where \(t_{1/2}\) is the half-life, and \(\lambda\) is the decay constant. We can rearrange the formula to solve for the decay constant. $$\lambda = \frac{ln2}{t_{1/2}}$$ We know that the half-life is \(t_{1/2} = 36\) hours. Plug the value into the formula. $$\lambda = \frac{ln2}{36}$$

Calculate Initial Activity, \(A_0\)

Now we have the decay constant, so we can use it in the radioactive decay formula to find the initial activity, \(A_0\). Let's rearrange the formula for \(A_0\): $$A_0 = \frac{A}{e^{-\lambda t}}$$ We have the final activity \(A = 1.2 \times 10^{5}\) disintegrations/min. Since the problem asks for the amount of Br-82 at the time the sample was found, the time, \(t, = 0\). Therefore, plug in the values and decay constant into the formula. $$A_0 = \frac{1.2 \times 10^{5}}{e^{-\frac{ln2}{36} \times 0}}$$

Calculate the number of atoms of Br-82

We know the initial activity and the decay constant. We can now calculate the number of atoms of Br-82 using the relation between activity and mass based on Avogadro's number. $$N = \frac{A_0}{\lambda}$$ Putting the value of initial activity and decay constant, we get $$N = \frac{1.2 \times 10^{5}}{\frac{ln2}{36}}$$

Calculate the mass of Br-82

Now we have the number of atoms of Br-82. We can calculate the mass of the sample using the atomic weight of Br-82 and Avogadro's number. First, we need to convert the atomic weight of Br-82 to grams per mole. The atomic weight of Br-82 is 81.918 amu. We will calculate the mass in terms of g $$mass_{Br-82} = \frac{N \times (81.918 \ \text{amu})}{6.022\times10^{23}} = \frac{N \times (81.918 \ \text{g/mol})}{6.022\times10^{23}} \ (\text{since} \ 1\ \text{amu} = 1 \ \text{g/mol})$$ Now, put the calculated value of N into the equation to get the mass. $$mass_{Br-82} = \frac{\frac{1.2 \times 10^{5}}{\frac{ln2}{36}} \times (81.918 \ \text{g/mol})}{6.022\times10^{23}}$$ Calculate the mass of Br-82 in grams.

Key concepts

Understanding Half-life in Radioactive Decay Calculations

Half-life is a vitally important concept in the context of radioactive decay, as it signifies the time required for half of the radioactive atoms within a sample to decay. A clear understanding of half-life is essential because it helps in calculating how long a radioactive substance will remain active, which, in turn, affects safety measures and the planning of scientific studies involving radioactive materials.

For example, in the case of Bromine-82 with a half-life of 36 hours, after this duration, only half of the original atoms will still be intact while the rest will have decayed. This exponential decay process continues with each successive half-life period. By knowing the half-life, we can determine the decay constant equation: display: div style:$$div class:div number:$$term:, which creates a link between the time elapsed and the amount of substance remaining. The calculation is fundamental when estimating the original quantity of the substance from the current activity level, as illustrated in the textbook exercise.

Decay Constant and Its Role in Decay Calculations

The decay constant, represented by the symbol equation: term:term: display: div style:, plays a critical role in the realm of radioactive decay. It is a probability factor that gives the fraction of a number of radioactive atoms expected to decay per unit time. To calculate the decay constant, we utilize the formula connecting it to the half-life:display: div style:$$equation: div number:$
By determining the decay constant, one can comprehend the rate at which a radioactive isotope will transform over time. For Bromine-82, finding the value of the decay constant allows us to proceed with further calculations, such as identifying the sample's initial activity or overall mass at any given time.

Determining the Activity of a Radioactive Sample

Activity is the term used to describe the number of disintegrations or decays occurring per unit time within a radioactive sample. It is often measured in disintegrations per minute (dpm) or becquerels (Bq), where 1 Bq equals to 1 disintegration per second. The activity of a sample reflects its radioactivity level, and it decreases over time as atoms decay.

To find the activity of a sample at any moment, use the decay law, which involves the decay constant and the number of undecayed nuclei. For instance, given Bromine-82's activity of equation: term: term: term: term: disintegrations/min at a certain point, we can trace back to its initial activity using the principles of radioactive decay. This process, as demonstrated in the exercise, involves understanding and applying the relationship between the initial and final activities, alongside the decay constant and elapsed time.

Using Avogadro's Number in Radioactivity

Avogadro's number is an indispensable concept used in chemistry and physics, representing the number of atoms, ions, or molecules in one mole of a substance. Its value is approximately equation: . Avogadro's number allows us to bridge the microscopic world of atoms with the macroscopic world we can measure, providing the ability to convert between the number of atoms and the amount of substance in moles.

When working with radioactive materials, like Bromine-82, Avogadro's number gives the link between the quantity of decay events (activity) and the mass of a sample. By dividing the number of atoms by Avogadro's number and multiplying by the atomic mass, we can calculate the mass of a radioactive sample. This concept is crucially utilized in solving exercises involving calculating the grams of a radioactive isotope present in a sample, as showcased in the problem provided.