Problem 22 - Chemistry: Principles and Reactions

Question

Consider an isotope of yttrium, Y-90. This isotope is incorporated into cancer-seeking antibodies so that the cancer can be irradiated by the yttrium and destroyed. How many neutrons are in (a) twenty-five atoms of yttrium? (b) \(0.250\) mol of yttrium? (c) one nanogram \(\left(10^{-9} \mathrm{~g}\right)\) of yttrium?

Step-by-Step Solution

Answer

Answer: (a) There are 1,275 neutrons in twenty-five atoms of Y-90. (b) There are approximately \(7.676\times10^{24}\) neutrons in \(0.250\) mol of Y-90. (c) There are approximately \(3.406\times10^{14}\) neutrons in one nanogram of Y-90.

Step 1: Determine the number of neutrons in one Y-90 atom

Yttrium has an atomic number of 39, which means it has 39 protons. In the Y-90 isotope, we know that the mass number equals the sum of protons and neutrons. Therefore: Number of neutrons = Mass number - Number of protons = 90 - 39 = 51 Now that we know there are 51 neutrons in one Y-90 atom, we can calculate the number of neutrons in the given amounts.

Step 2: Calculate neutrons in twenty-five atoms of Y-90

Since there are 51 neutrons in one atom of Y-90, in 25 atoms there will be: Number of neutrons = 25 Y-90 atoms * 51 neutrons/atom = 1275 neutrons

Step 3: Calculate neutrons in \(0.250\) mol of Y-90

We know that one mole contains Avogadro's number of atoms, which is approximately \(6.022 \times 10^{23}\) atoms. To find the number of Y-90 atoms in 0.250 mol, we can use the formula: Number of atoms = moles * Avogadro's number = \(0.250 \ mol \times6.022 \times10^{23}\ atoms/mol \approx1.505 \times10^{23}\ atoms\) Now, we can calculate the number of neutrons in \(0.250\) mol of Y-90: Number of neutrons = \(1.505 \times10^{23}\ atoms\times51\ neutrons/atom ≈7.676\times10^{24}\ neutrons\)

Step 4: Calculate neutrons in one nanogram of Y-90

First, we need to find the number of moles in one nanogram (\(10^{-9}\) g) of Y-90. The molar mass of Y-90 is 90 g/mol, so we can use the formula: Moles = mass / molar mass = \(\frac{10^{-9}\ \mathrm{g}}{90\ \mathrm{g/mol}}=1.111\times10^{-11}\ \mathrm{mol}\) Next, we need to calculate the number of Y-90 atoms in this amount: Number of atoms = moles * Avogadro's number = \(1.111\times10^{-11}\ \mathrm{mol}\times6.022\times10^{23}\ \mathrm{atoms/mol}\approx6.678\times10^{12}\ \mathrm{atoms}\) Finally, we can calculate the number of neutrons in one nanogram of Y-90: Number of neutrons = \(6.678\times10^{12}\ \mathrm{atoms}\times51\ \mathrm{neutrons/atom}\approx3.406\times10^{14}\ \mathrm{neutrons}\)

Key Concepts

Isotopic Composition

Isotopic composition refers to the make-up of an element's isotopes within a sample. Isotopes are versions of an element with the same number of protons but different numbers of neutrons. Therefore, although they're chemically similar, they have different mass numbers.

Understanding isotopic composition is crucial for various scientific fields, including chemistry, physics, geology, environmental science, and medicine. For instance, in the medical field, specific isotopes, like the Y-90 used in cancer treatment mentioned in our exercise, are chosen for their unique properties – in this case, the emission of radiation that can target cancer cells.

• Isotopes are denoted by their element symbol followed by their mass number (e.g., Y-90).
• The mass number is the sum of protons and neutrons in the nucleus.
• The atomic number, which is specific to each element, denotes the number of protons in an isotope.
• Isotopic abundance refers to the relative amount of each isotope in a natural sample.

Understanding isotopic composition helps in calculating the number of neutrons and determining the molar mass of the isotope – key steps in our exercise's solution process.

Neutron Calculation

Neutron calculation in an atom is foundational to understanding isotopic composition. Since the number of protons in an isotope does not change, the neutron number is what differentiates isotopes of an element. To calculate the number of neutrons in an atom, you subtract the atomic number (number of protons) from the mass number:

\[\begin{equation} \text{Number of neutrons} = \text{Mass number} - \text{Number of protons} \tag{1}\end{equation}\]

In practice, if you want to find the number of neutrons in a larger sample, you multiply the number of neutrons in a single atom by the number of atoms in that sample. This was applied in our exercise when calculating the neutron content of various amounts of Y-90 isotope. The calculation is essential not just in theoretical chemistry, but also in applications such as material science, nuclear physics, and radiopharmaceuticals, as the neutron number affects the isotope's stability and nuclear behavior.

• To find the number of neutrons in more complex samples, use the formula: \[\begin{equation} \text{Neutrons in sample} = \text{Atoms in sample} \times \text{Neutrons per atom} \tag{2}\end{equation}\]
• This calculation requires familiarity with Avogadro's number for mole-to-atom conversions when dealing with molar quantities of a substance.

Molar Mass

Molar mass is a fundamental concept in chemistry defined as the mass of one mole of a substance. It is typically expressed in grams per mole (g/mol). The molar mass directly relates to the isotopic composition of an element because it is dependent on the weighted average mass of the element's isotopes in a given sample.

For instance, the molar mass of the Y-90 isotope used in the original exercise is 90 g/mol. This is straightforward since Y-90 is a mono-isotopic element, meaning it consists of one isotope in nature, and thus, its molar mass is the same as its mass number.

• The molar mass is used to convert between grams and moles of a substance using the formula: \[\begin{equation} \text{Moles} = \frac{\text{Mass}}{\text{Molar mass}} \tag{3}\end{equation}\]
• Having the molar mass allows you to calculate the number of atoms in a given mass of a sample, essential for stoichiometric calculations in chemical reactions.
• In real-world scenarios, the average molar mass of elements with multiple isotopes is used, which incorporates the isotopic abundance and the molar mass of each individual isotope.

Understanding molar mass is vital for precision in laboratory procedures, such as preparing solutions with accurate concentrations or in medical applications, as seen with dosages in the treatment of diseases.