Question

The isotope Si-28 has a mass of 27.977 amu. For ten grams of Si-28, calculate (a) the number of atoms. (b) the total number of protons, neutrons, and electrons.

Short answer

Answer: In ten grams of Si-28 isotope, there are approximately 2.15 x 10^23 atoms. The total number of protons, neutrons, and electrons in that sample is approximately 3.01 x 10^24 each.

Step-by-step solution

Convert grams to moles using Avogadro's number

To convert the mass of Si-28 given in grams to moles, we'll use the molar mass of Si-28 (27.977 amu) and Avogadro's number (6.022 x 10^23 atoms per mole). First, calculate the number of moles of Si-28 in the sample: Moles of Si-28 = (grams of Si-28) / (molar mass of Si-28) = (10 g) / (27.977 amu) Let's calculate the moles of Si-28: Moles of Si-28 ≈ 0.357 moles

Calculate the number of atoms in the sample

Now that we have the number of moles of Si-28, we can use Avogadro's number to find the number of atoms: Number of atoms = (moles of Si-28) x (Avogadro's number) = (0.357 moles) x (6.022 x 10^23 atoms/mole) Let's calculate the number of atoms: Number of atoms ≈ 2.15 x 10^23 atoms

Find the total number of protons, neutrons, and electrons in the sample

To find the total number of protons, neutrons, and electrons in the sample, we'll first find the number of each element in a single Si-28 atom and then multiply each by the total number of atoms from Step 2. Silicon has an atomic number of 14, which means it has 14 protons. Si-28 has a mass number of 28, and mass number = protons + neutrons, therefore, Si-28 has 14 neutrons. Since silicon is neutral, it has an equal number of protons and electrons, which means it also has 14 electrons in one atom. Protons = 14 Neutrons = 14 Electrons = 14 Next, multiply these numbers by the total number of atoms: Total protons = (14 protons/atom) x (2.15 x 10^23 atoms) ≈ 3.01 x 10^24 protons Total neutrons = (14 neutrons/atom) x (2.15 x 10^23 atoms) ≈ 3.01 x 10^24 neutrons Total electrons = (14 electrons/atom) x (2.15 x 10^23 atoms) ≈ 3.01 x 10^24 electrons Now we have the results: (a) The number of atoms in ten grams of Si-28: ≈ 2.15 x 10^23 atoms (b) The total number of protons, neutrons, and electrons: ≈ 3.01 x 10^24 each.

Key concepts

Avogadro's Number

Avogadro's number, named after scientist Amedeo Avogadro, is a fundamental constant in chemistry. It represents the number of atoms, ions, or molecules in one mole of a substance. The value is approximately \( 6.022 \times 10^{23} \) entities per mole.

This constant is crucial when performing isotopic composition calculations, such as determining the number of atoms in a sample. It bridges the gap between the macroscopic scale, where we measure substances in grams or liters, and the microscopic scale, where measurements involve individual atoms or molecules.

For example, if you have one mole of any element, you'll have \( 6.022 \times 10^{23} \) atoms of that element. To calculate the number of atoms in a given mass, you simply convert the mass to moles and multiply by Avogadro's number, as seen in the solution to the Si-28 problem.

Molar Mass

Molar mass is another essential concept in chemistry, representing the mass of one mole of a substance. It's typically expressed in grams per mole (g/mol) and is equivalent to the atomic or molecular weight expressed in atomic mass units (amu).

Every element has a unique molar mass, which is found on the periodic table and corresponds to its average atomic mass. In calculations, having the correct molar mass is pivotal for converting between grams and moles, as seen in our exercise.

The molar mass of Si-28 is 27.977 amu, and we use this to convert 10 grams of Si-28 into moles to further explore the number of atoms present. The precise molar mass ensures accuracy when proceeding with isotopic composition calculations. Remember, accuracy in determining the molar mass is key to the accuracy of subsequent calculations involving the mass of atoms or molecules.

Atomic Structure

Understanding the atomic structure of an element is essential when dealing with isotopes and their respective calculations. An atom consists of a nucleus made up of protons and neutrons, surrounded by electrons.

The atomic number indicates the number of protons in an atom's nucleus, defining the element itself. Meanwhile, the mass number is the sum of protons and neutrons, thus providing insight into the specific isotope of the element.

In the Si-28 problem, we learn that silicon has an atomic number of 14 (hence 14 protons) and a mass number of 28 (14 protons and 14 neutrons for the Si-28 isotope). Since atoms are electrically neutral, the number of electrons equals the number of protons. This understanding of atomic structure is then extrapolated to calculate the total number of each subatomic particle in a given sample based on the number of atoms, as shown in the exercise solution.