A quantity of \(1.9 \times 10^{-4} \mathrm{~g}\) of the metal having density \(19 \mathrm{~g} / \mathrm{ml}\) is dispersed in 11 of water to give a sol having spherical metal particles of radius \(10 \mathrm{~nm} .\) The approximate number of metal sol particles per \(\mathrm{cm}^{3}\) of the sol is (a) \(2.35 \times 10^{9}\) (b) \(4 \times 10^{10}\) (c) \(1.9 \times 10^{9}\) (d) \(2.8 \times 10^{9}\)

One of the fundamental skills in physical chemistry is performing volume calculations. To understand these concepts, let's use the context of the problem at hand, which involves calculating the volume of a spherical particle. Volume is a measure of the space occupied by a substance, and in chemistry, this is often linked to the quantitative analysis of substances involved in a reaction or a solution.

In the given exercise, we need to calculate the volume of a single spherical metal particle using the formula for the volume of a sphere: \[ V = \frac{4}{3} pi r^3 \]where \( r \) is the radius. However, one common mistake is neglecting the importance of unit conversion. For instance, the exercise provides the radius in nanometers, but standard volume calculations in chemistry require the measurements to be in centimeters. Thus, converting 10 nm to centimeters becomes a critical step before applying the formula.

When dealing with colloidal chemistry or solutions such as metal sols, computations involving spherical particle dimensions become very relevant. The particle size can affect everything from the solution's properties to its behavior. For computational purposes, it's vital to understand how to relate a particle's volume to its physical attributes.

To continue with our example problem, after calculating the volume of a single particle, one will often need to relate this to the total number of particles, which can be described using a density parameter. Understanding these relationships requires not just formulaic knowledge but also a conceptual understanding of the dimensions of particles and how they fit into the larger system of the solution they are a part of.

A strategic approach to solving problems in chemistry often consists of a series of logical steps. Step-by-step processes provide a structured pathway and reduce the complexity of problems, especially when dealing with multilayered calculations or when multiple concepts are intertwined.

The steps laid out in the example above demonstrate an effective approach: starting with basic volume calculations, then moving on to determining the total volume of metal based on its mass and density, and finally deriving the number of particles. The final computation involves distributing these particles uniformly across the volume of the solution, highlighting the importance of a systematic method in tackling chemistry problems. To optimize this strategy, always verify units, keep an eye out for conversion factors, and remember to consider the physical implications your calculation.

The mole concept is a bridge between the macroscopic world we observe and the microscopic world of atoms and molecules. Avogadro's Law states that equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules. In a broader sense, knowing the number of entities relates to the concept of moles and serves as the basis for understanding chemical relationships.

In the exercise provided, while Avogadro's Law is not directly applied, the concept of determining the number of entities (in this case, metal sol particles) from a given mass is closely related. Here, we're focusing on solid particles, but the underlying method of using Avogadro's number, which defines the mole, could be used to relate atoms or molecules to physical quantities of a substance. The ability to traverse between these scales is a testament to the elegance and utility of the mole concept in chemistry.