Problem 100 - General Chemistry: Principles and Modern Applications

Question

Suppose that \(1.00 \mathrm{mg}\) of gold is obtained in a colloidal dispersion in which the gold particles are spherical, with a radius of \(1.00 \times 10^{2} \mathrm{nm}\). (The density of gold is \(\left.19.3 \mathrm{g} / \mathrm{cm}^{3} .\right)\) (a) What is the total surface area of the particles? (b) What is the surface area of a single cube of gold of mass \(1.00 \mathrm{mg} ?\)

Step-by-Step Solution

Answer

The total surface area of all the colloidal gold particles is significantly larger than the surface area of a single cube of gold of the same mass. This is due to the increased number of particles, and therefore increased surface area, when the gold is split into smaller particles.

Step 1: Calculate the volume of one colloidal particle of gold

The volume (v) of a sphere can be calculated using the formula: \[v = \frac{4}{3} \pi r^{3}\] where r is the radius of the sphere. Given, the radius is \(1.00 \times 10^{2}\) nm = \(1.00 \times 10^{-7}\) cm.

Step 2: Calculate the mass of one colloidal particle of gold

The mass (m) of a substance can be calculated using its volume (v) and density (ρ), with the formula: \[m = \rho \times v\] where the diameter of gold is 19.3 g/cm³.

Step 3: Calculate the number of particles in 1 mg of gold

To calculate the number of colloidal particles in 1 mg of gold, divide the total mass by the mass of one particle. Since the unit of mass should be the same, convert 1mg to grams. Thus, 1.00 mg = 0.001 g.

Step 4: Calculate the total surface area of the particles

The total surface area (A_total) of all the colloidal particles can be calculated by multiplying the number of particles (n) by the surface area (A) of one particle. The surface area of a sphere is given by \[A = 4 \pi r^{2}\].

Step 5: Calculate the side length of the cube

First, find out the volume of 1.00 mg of gold using the mass (m) and density (ρ) of gold with the formula \[v = \frac{m}{\rho}\]. Now, the side length of the cube (a) can be obtained using the formula \[a = \sqrt[3]{v}\].

Step 6: Calculate the surface area of the gold cube

The surface area (A_cube) of a cube can be calculated using the formula \[A_cube = 6a^2\], where a is the side length of the cube.

Key Concepts

Surface Area Calculation

Understanding how to calculate surface area is crucial for students dealing with various geometric shapes. When it comes to spheres, the surface area can be determined using the formula:

• \( A = 4 \pi r^{2} \)

where \( r \) is the radius of the sphere. This formula tells us that the total surface area of a sphere is directly proportional to the square of its radius. For a gold colloidal particle with a radius of \(1.00 \times 10^{2}\) nm which is equivalent to \(1.00 \times 10^{-7}\) cm after conversion, this formula allows us to find the surface area for just one individual particle.
To find the total surface area of multiple particles, you simply multiply the surface area of one particle by the number of particles.
This application can also be extended to other regular shapes, like cubes, using a pertinent formula like \(A_{\text{cube}} = 6a^2\) where \(a\) is the edge length of the cube. For instance, after calculating the side length of a cube formed from a given mass of gold, this information can help find the total surface area of the cube.

Density of Gold

Density is a fundamental property of matter, representing how mass is spread over a given volume. Specifically, the density of gold is a defining attribute in these calculations since it's given as \(19.3 \text{ g/cm}^3\).
This means that for every cubic centimeter of gold you have, it weighs 19.3 grams. In colloidal dispersions, which involve very small particles, using density allows for an efficient transition between knowing the volume of particles and their mass.
In mathematical terms, density is applied via the formula:

• \( \rho = \frac{m}{v} \)

where \( \rho \) is density, \( m \) is mass, and \( v \) is volume. Utilizing this formula, you can determine the mass of a single gold particle, given its volume.
Conversely, you can use the inverse relationship to determine the volume from the mass.
This principle also plays a critical role in calculating how many such particles you could derive from a known mass of gold.

Volume of a Sphere

Understanding the volume of a sphere is fundamental when working with spherical particles like colloidal gold particles. The volume of a sphere can be calculated using the formula:

• \( v = \frac{4}{3} \pi r^{3} \)

where \( r \) refers to the sphere's radius. The formula gives you the amount of space occupied by a sphere.
For instance, if you know the radius of the gold particles in a solution, you can use this equation to determine the individual particle volume.
Such calculations become especially critical when determining how a mass converts into a volume and vice versa.
As you require the volume of a single particle to then multiply by its total number, realizing the importance of this simple but effective formula is crucial.
Applications of this concept are not restricted to spheres alone but extend to any context where you need to understand space filling in three dimensions.