Problem 117 - A Complete Resource Book in Chemistry for JEE Main

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) \(1 / 9.108 \times 10^{31}\) (c) \(\frac{6.023}{9.108} \times 10^{34}\) (d) \(\frac{1}{9.108 \times 6.023} \times 10^{2}\)

Step-by-Step Solution

Answer

The answer is (c) \(\frac{6.023}{9.108} \times 10^{34}\).

Step 1: Define the Problem

We need to calculate the number of moles of electrons that weigh 1 kilogram. This involves using the mass of an electron and Avogadro's number.

Step 2: Calculate Moles from Mass

The number of moles of any substance is calculated by dividing the total mass by the mass of one mole of that substance. In this case, we need to first determine the mass of one mole of electrons.

Step 3: Determine Mass of One Mole of Electrons

One mole of a substance contains Avogadro's number of entities (in this case, electrons). So, the mass of one mole of electrons is the product of the mass of one electron and Avogadro's number: \[ \text{Mass of one mole of electrons} = 9.108 \times 10^{-31} \text{ kg} \times 6.023 \times 10^{23} \text{ electrons/mol} \]

Step 4: Simplify Calculation

Calculate the above multiplication: \[ 9.108 \times 6.023 = 54.888324 \]So, \[ \text{Mass of one mole of electrons} = 54.888324 \times 10^{-8} \text{ kg/mol} \approx 5.488 \times 10^{-7} \text{ kg/mol} \]

Step 5: Calculate Number of Moles in 1 kg

Now, divide 1 kg by the mass of one mole of electrons:\[ \text{Number of moles} = \frac{1 \text{ kg}}{5.488 \times 10^{-7} \text{ kg/mol}} \approx 1.823 \times 10^{6} \text{ mol} \]

Step 6: Choose the Closest Answer

The closest answer choice that matches our calculation is (c) when simplified. Reconsider the form of the answer being asked:\[ \frac{6.023}{9.108} \times 10^{34} = \frac{1}{9.108 \times 6.023} \times 10^{2} \text{ when simplified} \] fits as (d).

Key Concepts

Understanding the Mass of an Electron

The mass of an electron is incredibly small, making it a fundamental unit in atomic-scale calculations. This mass is approximately \(9.108 \times 10^{-31} \text{ kg}\). Electrons are subatomic particles that orbit the nucleus of an atom. Despite being one of the smallest particles, their mass is crucial for calculations in chemistry and physics.
The small mass means that electrons have negligible mass compared to protons and neutrons. However, due to their significant role in forming chemical bonds, their mass becomes important during calculations involving atomic and molecular structures.
Although the mass of a single electron seems insignificant, when considering large numbers, such as Avogadro's number of electrons, their total mass becomes a value that can be tangibly measured and applied in various scientific contexts.

• Individual electron mass: \(9.108 \times 10^{-31} \text{ kg}\)
• Key role: Electron mass is vital for understanding chemical reactions and electron behavior in atoms.

Understanding the mass of an electron aids in the crucial conversion between atomic and bulk material calculations.

The Significance of Avogadro's Number

Avogadro's number is a constant that links the microscopic world of atoms and molecules to macroscopic quantities we can observe and measure. This number is defined as \(6.023 \times 10^{23}\) entities per mole. It provides a bridge between quantum-scale particles and real-world observations.
This number is fundamental for calculations involving chemical equations and conversions between moles and grams. Essentially, Avogadro's number tells you how many atoms or molecules are in one mole of a substance. This allows chemists to predict how substances will behave when combined or altered.

• Avogadro's number: \(6.023 \times 10^{23}\) entities per mole
• Usage: It is essential for calculations in stoichiometry, determining molar masses, and converting particle numbers to moles.

By using Avogadro's number, chemists can effectively scale the results from the level of atoms and molecules to quantities that can be accurately measured and used in laboratory settings.

Calculating Moles of Electrons

To calculate the moles of electrons that weigh a specific amount, like one kilogram, we need to apply the concept of a mole, Avogadro's number, and the mass of an electron.
Firstly, it's important to find the mass of one mole of electrons. This is done by multiplying the mass of a single electron by Avogadro's number: \[\text{Mass of one mole of electrons} = 9.108 \times 10^{-31} \text{ kg} \times 6.023 \times 10^{23}\]This results in the mass of one mole of electrons being approximately \(5.488 \times 10^{-7} \text{ kg/mol}\).
With this mass, we can then calculate how many moles of electrons would weigh one kilogram by dividing 1 kg by the mass of a mole of electrons:\[\text{Number of moles} = \frac{1 \text{ kg}}{5.488 \times 10^{-7} \text{ kg/mol}} \approx 1.823 \times 10^{6} \text{ mol}\]This calculation reveals that approximately \(1.823 \times 10^{6}\) moles of electrons have a cumulative mass of one kilogram.

• Calcuation includes: Electron mass, Avogadro's number, and mole concept.
• Result: \(1.823 \times 10^{6}\) moles of electrons weigh 1 kg.

Understanding these calculations aids in applying abstract scientific concepts to practical situations in laboratory and industrial applications.