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Chapter 13: Problem 100

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 \(1.00 \mathrm{mg} ?\)

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.
04

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 \[A = 4 \pi r^{2}\].
05

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 \[a = \sqrt[3]{v}\].
06

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.

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