Question

How many moles of electrons weigh one kilogram? (mass of electron \(=9.108 \times 10^{-31} \mathrm{~kg}\), Avogadro number \(\left.=6.023 \times 10^{23}\right)\) (a) \(6.023 \times 10^{23}\) (b) \(\frac{1}{9.108} \times 10^{31}\) (c) \(\frac{6.023}{9.108} \times 10^{54}\) (d) \(\frac{1}{9.108 \times 6.023} \times 10^{8}\)

Short answer

Option (d) \(\frac{1}{9.108 \times 6.023} \times 10^{8}\) is the correct answer.

Step-by-step solution

Understand the Problem

We need to calculate how many moles of electrons together have a mass of 1 kilogram. Given the mass of a single electron and Avogadro's number, we can determine the number of electrons in a mole and use this to find the total mass of each mole.

Calculate Mass of 1 Mole of Electrons

The mass of one electron is given as \(9.108 \times 10^{-31}\) kg. According to Avogadro's number, one mole contains \(6.023 \times 10^{23}\) electrons. Therefore, the mass of one mole of electrons is computed by multiplying the two: \(\text{mass of 1 mole} = 9.108 \times 10^{-31} \text{ kg per electron} \times 6.023 \times 10^{23} \text{ electrons per mole}\).

Perform the Multiplication

Multiply the mass of one electron by Avogadro's number: \[\text{Mass of 1 mole} = 9.108 \times 10^{-31} \times 6.023 \times 10^{23} = 5.482 \times 10^{-7} \text{ kg per mole}\].

Determine the Number of Moles in 1 Kg

To find out how many moles of electrons weigh 1 kilogram, divide 1 kg by the mass of one mole of electrons: \[\text{Number of moles} = \frac{1 \text{ kg}}{5.482 \times 10^{-7} \text{ kg per mole}} = \frac{1}{5.482} \times 10^{7} \text{ moles}\].

Convert Calculation to Given Options

Convert the number of moles derived from our calculation into the forms given in the multiple-choice options. Rewriting \(\frac{1}{5.482}\) in a relatable form, we approximate it to be \(0.1824\), and since \(5.482 \approx 5.5\), the correct form resembles \(\frac{1}{9.108 \times 6.023} \times 10^{8}\), which matches option \(d\).

Key concepts

The Mass of an Electron

Electrons are incredibly tiny particles with a minuscule mass. The mass of a single electron is given as \(9.108 \times 10^{-31}\) kilograms. This value highlights how lightweight electrons are, especially when compared with other subatomic particles like protons or neutrons. Understanding the mass of an electron is crucial when dealing with calculations involving large numbers of electrons, as is often the case in chemistry and physics.
Knowing an electron’s mass lets us compute the mass of large quantities, such as moles, which is useful in understanding how these small particles contribute to the overall mass of an object. This plays a vital role in various scientific fields, including material science and quantum mechanics.

Understanding Avogadro's Number

Avogadro's number is a fundamental constant that describes the number of particles, like atoms or electrons, in one mole of a substance. Its value is \(6.023 \times 10^{23}\). This immense quantity serves as a bridge between the microscopic world of atoms and molecules and the macroscopic world we observe daily. By knowing how many particles are in a mole, we can convert between the number of particles and amounts of substances in laboratory chemistry.

• It allows chemists to count particles and relate them to macroscopic measurements.
• The concept of a mole helps in converting mass into number of particles and vice versa.

This number is significant, because it lets scientists predict how substances will interact in chemical reactions, understand material properties, and conduct calculations needed for reactions and processes.

Calculating Molar Mass

Molar mass is an essential concept in chemistry. It represents the mass of one mole of a chemical element or compound. For single elements, it is numerically equal to the atomic mass found on the periodic table, expressed in grams per mole (g/mol).

• Molar mass allows conversion between the mass of a substance and the number of moles.
• It is key for stoichiometric calculations in chemical equations.

To calculate the molar mass of electrons, for instance, you multiply the mass of one electron by Avogadro's number. In the exercise, the students performed this multiplication, resulting in the mass for one mole of electrons, \(5.482 \times 10^{-7}\) kg/mole.
This understanding helps in determining the composition of materials and estimating quantities needed for reactions, making it a cornerstone concept in chemistry education.