Problem 43 - Chemical Principles

Question

Estimate the value of \(n\) for the uppermost filled level in a one-dimensional line of silver atoms of length \(1.0 \mathrm{~mm}\). Hint: Take the radius of an Ag atom as \(144 \mathrm{pm}\).

Step-by-Step Solution

Answer

The number of Ag atoms that can fit in a 1.0 mm line is approximately 6944.

Step 1: Determine the diameter of a silver atom

Calculate the diameter of an Ag atom by doubling its radius. The diameter (d) is given by the formula: \(d = 2 \times r\), where \(r = 144 \mathrm{pm}\).

Step 2: Convert the diameter into millimeters

Convert the diameter from picometers to millimeters using the conversion: \(1 \mathrm{pm} = 10^{-12} \mathrm{mm}\).

Step 3: Calculate the number of Ag atoms that fit in 1.0 mm

Divide the length of the silver line (1.0 mm) by the diameter of an Ag atom in millimeters to find the number of atoms that can be placed in a line. The formula is: \(n = \frac{line \ length}{diameter \ of \ an \ Ag \ atom}\).

Key Concepts

Atomic Radius

The atomic radius is a measure of the size of an atom, typically its mean or typical distance from the nucleus to the boundary of the surrounding cloud of electrons. It's a critical concept in chemistry because the atomic radius influences several properties of an atom, including its volume, reactivity, and the nature of its bonding with other atoms. Specifically, in this exercise, we consider the radius of a silver (Ag) atom. Knowing that the radius of an Ag atom is approximately 144 pm (picometers) is crucial to understanding how many such atoms can fit into a given space, in this case, a one-dimensional line of length 1.0 mm.

Analysing atomic radii helps students understand the relative sizes of atoms across the periodic table, which can be affected by factors like the number of electrons and the shell structure determined by quantum mechanics—a topic we will delve into further in this article.

Unit Conversion

Unit conversion is an essential skill in chemistry and other sciences for comparing measurements that have been expressed in different units. In our example, we needed to convert the diameter of a silver atom from picometers to millimeters to correctly estimate how many atoms fit within a 1.0 mm length. This is done by using the conversion factor that 1 picometer equals 1x10^-12 millimeters (\(1 \text{pm} = 10^{-12} \text{mm}\)).

Understanding and performing unit conversions correctly is vital because failing to do so can lead to significant errors in experiments and scientific calculations. It ensures that all quantities are expressed in compatible units, enabling accurate calculations and solutions.

Quantum Mechanics

Quantum mechanics is the foundation of modern chemistry; it explains the behavior of particles at the atomic and subatomic levels. It introduces the concept of quantum numbers, which describe properties such as energy levels, angular momentum, and orbital shape. These numbers determine how electrons are arranged in atoms—an arrangement that explains chemical bonding and the periodic table.

Although our current exercise does not deal directly with quantum numbers, it hints at a one-dimensional quantum system, which is an idealized concept often used in quantum mechanics for theoretical analysis. The exercise's solution involves classical measurements and geometry, informed by an underlying quantum framework that determines the atomic properties.

Chemistry Problem Solving

Chemistry problem solving is about applying knowledge of chemical principles and mathematical computations to solve quantitative and qualitative problems. In the case of our exercise, problem solving involves a step-by-step approach: we first determine the diameter of a silver atom, then convert units for consistent measurements, and finally calculate the quantity of atoms that can be arranged in a given length. It requires critical thinking, understanding of the principles involved (such as the physics of atomic radii), and mathematical skills (unit conversion).

Robust problem-solving skills enable students to tackle a wide range of scenarios in chemistry, from the laboratory to real-world applications, ensuring their grasp of the subject is thorough enough to apply to practical situations.