Question

Naturally occurring cobalt consists of only one isotope, \(^{59} \mathrm{Co}\) , whose relative atomic mass is \(58.9332 .\) A synthetic radioactive isotope of cobalt, \(^{60}\) Co, relative atomic mass \(59.9338,\) is used in radiation therapy for cancer. A \(1.5886-\mathrm{g}\) sample of cobalt has an apparent "atomic mass" of \(58.9901 .\) Find the mass of \(^{60} \mathrm{Co}\) in this sample.

Short answer

The mass of \(^{60}Co\) in the sample can be found by solving the equation and subtracting \(x\) from the total sample mass.

Step-by-step solution

Define Variables

Let the mass of the naturally occurring isotope, \(^{59}Co\), in the sample be \(x\) grams. Therefore, the mass of the synthetic radioactive isotope, \(^{60}Co\), will be \(1.5886 - x\) grams since the total mass of the sample is 1.5886 grams.

Set up the Equation Based on Average Atomic Mass

Using the formula for average atomic mass: (Mass of Isotope 1 \(\times\) its relative atomic mass) + (Mass of Isotope 2 \(\times\) its relative atomic mass) = Total mass of the sample \(\times\) the apparent 'atomic mass'. So the equation is \(x \cdot 58.9332 + (1.5886 - x) \cdot 59.9338 = 1.5886 \cdot 58.9901\).

Solve for x

Simplify the equation to solve for \(x\): \((58.9332\cdot x + 59.9338(1.5886 - x) = 93.7263566\). Expanding and combining like terms yields a linear equation you can solve for \(x\).

Calculate the Mass of \(^{60}Co\)

With the value of \(x\), calculate the mass of \(^{60}Co\) in the sample by subtracting \(x\) from the total mass of the sample: \(1.5886 - x\).

Summary

The calculated mass of \(^{60}Co\) gives us the amount of the radioactive isotope present in the 1.5886-gram sample.

Key concepts

Average Atomic Mass

Understanding the average atomic mass of an element is crucial in chemistry because it considers the relative abundance of each isotope of that element. The isotopes of an element have identical chemical properties but differ in mass due to the varying number of neutrons in their nuclei.

The average atomic mass is calculated by considering the mass and the natural abundance of each isotope. Mathematically, the formula for calculating the average atomic mass (\(M_{\text{avg}}\)) is a weighted average:
\[ M_{\text{avg}} = (f_1 \times m_1) + (f_2 \times m_2) + \.\.\. + (f_n \times m_n) \]
where \(f_i\) represents the relative abundance and \(m_i\) is the atomic mass of isotope \(i\). For simplicity, you can think of \(f_i\) as the percentage of each isotope present. When only two isotopes are involved, such as in the exercise example with cobalt, the calculation simplifies to a basic algebraic problem. By mastering this calculation, students can gain insights into not only the properties of isotopes but also the practical aspects of their uses in fields like medicine and environmental science.

Isotopes in Chemistry

Isotopes play a significant role in many areas of chemistry and beyond, such as in medical diagnostics, treatment, and radiocarbon dating for archaeology. An isotope is a variant of a particular chemical element that has the same number of protons but a different number of neutrons in its nucleus, thus resulting in different mass numbers.

Isotopes are typically represented by the notation \(^{A}_{Z}X\), where \(A\) is the mass number, \(Z\) is the atomic number, and \(X\) is the chemical symbol of the element. For example, the naturally occurring isotope of cobalt mentioned in the exercise, \(^{59}_{27}Co\), has 27 protons and 32 neutrons. Its synthetic counterpart, radioactive \(^{60}_{27}Co\), has an additional neutron.

Studying isotopes helps us to understand the stability of atoms and the ways in which they undergo radioactive decay, which has pivotal applications in energy generation, medical treatment, and understanding the history of objects based on their isotope compositions.

Radioactive Isotopes

Radioactive isotopes, also known as radioisotopes, are atoms that have excess nuclear energy, making them unstable. This instability leads to the emission of radiation as the atom decays to a more stable form. This decay process can occur over short or long timescales, characterized by a property known as the half-life—the time it takes for half of the atoms in a sample of the radioisotope to decay.

Applications of Radioactive Isotopes

Radioisotopes have wide-reaching applications due to their ability to decay and emit radiation. In medicine, for example, the \(^{60}Co\) isotope is used in cancer radiation therapy, as highlighted in the exercise. This particular isotope emits gamma rays, which are effective at killing cancer cells.

Other uses include dating archaeological samples using carbon-14, treating food to extend its shelf life, and as a tracer in biochemical research. Safety precautions are paramount when handling radioactive materials because their radiation can be harmful.

Understanding the behavior of radioisotopes, their applications, and safety measures is indispensable for students to appreciate the complexities and benefits that these unique atoms bring to various scientific fields.