Question

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

Short answer

Answer: There are approximately 9.03 x 10^24 total elementary particles (protons, neutrons, and electrons) in 10 grams of Si-28.

Step-by-step solution

Calculate the number of moles

To calculate the number of moles, we can use the formula: Number of moles = \(\frac{\text{mass of substance}}{\text{molar mass}}\) Given that the mass of Si-28 is 27.977 amu, and 1 amu = 1 g/mol, the mass of substance is 10 g. Therefore, the molar mass of Si-28 is 27.977 g/mol. Number of moles = \(\frac{10\,\text{g}}{27.977\,\text{g/mol}}\) = 0.357 moles (approximately) (a) There are approximately 0.357 moles of Si-28 in 10 grams of the substance.

Calculate the number of atoms

We can find the number of atoms by multiplying the number of moles by Avogadro's constant. Number of atoms = Number of moles * Avogadro's constant Avogadro's constant is 6.022 x 10^23 entities/mole Number of atoms = 0.357 moles * 6.022 x 10^23 atoms/mole = 2.15 x 10^23 atoms (approximately) (b) There are approximately 2.15 x 10^23 atoms in 10 grams of Si-28.

Calculate the total number of protons, neutrons, and electrons

Si-28 has 14 protons, 14 neutrons, and 14 electrons, thus, 42 elementary particles per atom. To calculate the total number of protons, neutrons, and electrons in 10 grams of Si-28, we will multiply the number of atoms by the number of elementary particles per atom: Total number of elementary particles = Number of atoms * Number of particles per atom Total number of elementary particles = 2.15 x 10^23 atoms * 42 particles/atom = 9.03 x 10^24 elementary particles (approximately) (c) There are approximately 9.03 x 10^24 total elementary particles (protons, neutrons, and electrons) in 10 grams of Si-28.

Key concepts

Molar Mass

When studying chemistry, understanding the concept of molar mass is essential. Molar mass is the weight of one mole of a substance, often expressed in grams per mole (g/mol). To visualize this, think of the molar mass as the molecular weight of a substance but scaled up from individual atoms or molecules to Avogadro's number of particles.

In calculations, the molar mass acts as a conversion factor, allowing us to switch between the mass of a substance and the number of moles. For isotopic composition calculations, like the one involving Si-28, the molar mass is especially important because isotopes of an element have slightly different weights. In this case, Si-28 has a molar mass of 27.977 g/mol. Using this figure and the mass of the sample, we can determine how many moles of Si-28 are in a given sample.

Avogadro's Constant

Avogadro's constant is a fundamental number in chemistry, representing the number of particles in one mole of a substance. The value is approximately 6.022 x 10^23 particles/mole. It’s not just any number; it's a bridge that connects the microscopic world of atoms and molecules to the macroscopic world we can measure in the laboratory.

Whenever you've got a mole of something, no matter what it is, you have Avogadro's number of it. In our Si-28 problem, this constant is pivotal in determining the number of atoms. By multiplying the number of moles by Avogadro's constant, we can convert moles to atoms. This step is crucial for understanding the scale of chemical reactions on an atomic level.

Moles to Atoms Conversion

The process of converting moles to atoms might seem magical, but it's actually simple math anchored by Avogadro's constant. The relationship to remember is that one mole of any substance contains Avogadro's number of atoms or molecules.

In practical terms, for our example of Si-28, after finding the number of moles, we multiply it by 6.022 x 10^23 atoms/mole to convert to the actual number of Si-28 atoms in the sample. This process reveals not just a number for the sake of a number, but the scale at which we are observing our substance—far beyond what we can see or touch.

Sub-Atomic Particles Count

To understand the full scope of an atom, you must delve into the count of sub-atomic particles—protons, neutrons, and electrons. Each element has a unique number of these particles, which determines its identity and properties. In our Si-28 problem, Silicon has 14 protons and 14 electrons, essential for chemical behaviors, and 14 neutrons, which provide mass and stability to the nucleus.

Knowing the count of each type of particle in one atom allows us to scale up to the total count in any sample. Multiply the number of each particle by the number of atoms (which we got from the moles to atoms conversion), and you have the total count of protons, neutrons, and electrons in the sample. This data is fundamental for understanding both chemical reactions and the structure of matter.