Question

How many moles of electrons weigh one kilogram? (mass of electron \(=9.108 \times 10^{-31} \mathrm{~kg}\), Avogadro number \(=6.023 \times 10^{23}\) ) a. \(6.023 \times 10^{23}\) b. \(1 / 9.108 \times 10^{31}\) c. \(\frac{6.023}{9.08} \times 10^{54}\) d. \(\frac{1}{9.108 \times 6.023} \times 10^{9}\)

Short answer

Option c: \(\frac{6.023}{9.08} \times 10^{54}\) is correct.

Step-by-step solution

Calculate Number of Electrons in 1 kg

First, calculate how many electrons make up one kilogram. You know the mass of one electron is \(9.108 \times 10^{-31} \mathrm{~kg}\). Therefore, the number of electrons \(N\) in 1 kg is given by: \[ N = \frac{1}{9.108 \times 10^{-31}} \approx 1.098 \times 10^{30} \text{ electrons} \]

Convert Electrons to Moles

Next, convert the number of electrons to moles using Avogadro's number. Avogadro's number is \(6.023 \times 10^{23}\) electrons per mole, so the number of moles \(n\) is given by: \[ n = \frac{1.098 \times 10^{30}}{6.023 \times 10^{23}} \approx 1.823 \times 10^{6} \text{ moles} \]

Format the Answer

Finally, match this result with the given options in the problem statement. The number of moles \(1.823 \times 10^{6}\) is found in option \(c\), after evaluating \[ \frac{6.023}{9.08} \times 10^{54} = 1.823 \times 10^{6} \]. Thus, option \(c\) corresponds to the correct calculation.

Key concepts

Mass of an Electron

Electrons, though incredibly small particles, have a mass. The mass of a single electron is approximately \(9.108 \times 10^{-31} \mathrm{~kg}\). This value is tremendously small and comprehending its scale can be challenging, but it's a fundamental property of electrons. Understanding this mass is crucial when performing calculations that involve enormous numbers of electrons, like determining the number of electrons in one kilogram of material. Calculating this involves dividing 1 kg by the mass of a single electron. This allows us to find out how many electrons make up one kilogram. Though the electron is just a tiny fraction of a kilogram, when you consider the stupendous number of electrons it takes to weigh one kilogram, the calculations become fascinating. This process underpins our understanding of mass at a microscopic level.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry and physics. Defined as \(6.023 \times 10^{23}\), it represents the number of atoms, ions, or molecules in one mole of a substance. Think of it as a bridge connecting the tiny world of atoms to the macroscopic world we experience daily. Avogadro's number lets us take exceedingly large quantities of atomic-scale particles and equate them to more manageable terms, like moles. This is especially useful when converting the number of electrons into moles, as every mole contains exactly \(6.023 \times 10^{23}\) of any elemental unit, whether they be electrons or entire atoms. This conversion is an essential tool in chemistry and physics, allowing scientists to measure and predict reactions and phenomena that occur at atomic scales.

Conversion from Electrons to Moles

To convert electrons to moles, Avogadro's number is essential. Given a large number of electrons, you can find out how many moles they represent by dividing this number by Avogadro's number. This step is vital when dealing with extensive quantities of electrons, such as when determining how many moles are equivalent to a kilogram of electrons.In our exercise, once we determined that there are \(1.098 \times 10^{30}\) electrons in a kilogram, the next step was to convert this to moles. By dividing \(1.098 \times 10^{30}\) electrons by \(6.023 \times 10^{23}\) electrons/mole, we find the number of moles to be approximately \(1.823 \times 10^{6}\) moles.This conversion is critical for tasks requiring a shift from terms of individual particles to terms of moles, simplifying the understanding and communication of scientific ideas at an atomic scale.