Problem 150 - Chemistry An Atoms First Approach

Question

It takes \(476 \mathrm{kJ}\) to remove 1 mole of electrons from the atoms at the surface of a solid metal. How much energy (in kJ) does it take to remove a single electron from an atom at the surface of this solid metal?

Step-by-Step Solution

Answer

The energy required to remove a single electron from an atom at the surface of this solid metal is \(7.910 \times 10^{-22}\, kJ\).

Step 1: Identify known values and conversion factor

\ We know the following: 1. Energy to remove 1 mole of electrons: \(476 kJ\) 2. Conversion factor: Avogadro's number \(N_A = 6.022 \times 10^{23} \, \mathrm{mol}^{-1}\) We need to find the energy required to remove a single electron.

Step 2: Use the conversion factor to find the energy required for a single electron

\ To calculate the energy required to remove one electron, we will divide the given energy for 1 mole of electrons (476 kJ) by the Avogadro's number. Energy per electron = \(\frac{Energy \, for \, 1 \, mol}{Avogadro's \, number}\) Energy per electron = \(\frac{476\, kJ}{6.022 \times 10^{23}\, \mathrm{electron\cdot mol^{-1}}}\)

Step 3: Calculate the energy required for a single electron

\ Now, perform the calculation: Energy per electron = \(\frac{476 \times 10^3\, J}{6.022 \times 10^{23}\, electrons}\) Energy per electron = \(7.910 \times 10^{-19}\, J\)

Step 4: Convert Joules to Kilojoules

\ To express the result in kJ, we need to convert Joules to Kilojoules by dividing the value by \(10^3\). Energy per electron = \(\frac{7.910 \times 10^{-19}\, J}{10^3\frac{J}{kJ}}\) Energy per electron = \(7.910 \times 10^{-22}\, kJ\)

Step 5: Write the final answer

\ The energy required to remove a single electron from an atom at the surface of this solid metal is \(7.910 \times 10^{-22}\, kJ\).

Key Concepts

Mole Concept

The mole is a fundamental concept in chemistry that provides a bridge between the atomic world and the measurable, macroscopic world we interact with. It is a unit used to quantify the amount of a substance. One mole of any substance contains approximately the same number of entities, whether atoms, molecules, ions, or electrons.

• This amount is defined by Avogadro's number, which we will explore in more detail later.
• In this exercise, the mole helps us understand how much energy is needed to remove electrons from a mole of atoms in a metallic substance.
• The energy given for a mole of electrons—476 kJ—is a unit that makes it easier to compare and calculate chemical reactions, just like we use kilograms to measure sugar or flour.

By calculating the energy needed for a mole of electrons, we gain insights into the processes occurring at the atomic level, while using amounts we can work with in the laboratory or industry.

Avogadro's Number

Avogadro's number, denoted as \(N_A\), is a critical constant in chemistry, defined as approximately \(6.022 \times 10^{23}\). It represents the number of atoms, molecules, or particles in one mole of a substance.

• This number is essential for translating measurements and calculations from the molecular to the macroscopic scale, where we can observe and measure them.
• In the provided exercise, Avogadro's number was used to determine the energy required to remove a single electron from its surface atom.
• The calculation involved dividing the total energy for a mole of electrons by \(N_A\) to isolate the energy for one electron. This gives us incredible precision in understanding interactions at the smallest scales.

Thus, Avogadro's number is a cornerstone of understanding chemical quantities, giving us the power to relate countless particles to quantities we can manipulate and measure.

Unit Conversion

Unit conversion is a fundamental skill in chemistry, allowing us to express measurements in different units. In this situation, understanding how to convert the energy from Joules to Kilojoules is crucial to the problem.

• Energy in physics and chemistry is often measured in Joules (J), which can also be expressed in Kilojoules (kJ) for larger quantities.
• To perform the conversion between Joules and Kilojoules, remember that 1 kJ equals 1000 J (or \(10^3\) J).
• After calculating the energy in Joules to release one electron, dividing by 1000 allows us to easily express this in Kilojoules, which is more suitable for larger energy comparisons in chemistry.

Therefore, efficiently converting units helps standardize results, ensuring consistent communication and understanding across various scientific contexts.