Problem 81

Question

How many atoms are present in a \(1.00 \mathrm{m}\) length of 20-gauge copper wire? A 20-gauge wire has a diameter of 0.03196 in., and the density of copper is \(8.92 \mathrm{g} / \mathrm{cm}^{3}\)

Step-by-Step Solution

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Answer

Finally, calculate the number of atoms by substituting the values in the formula obtained in the previous step.

1Step 1: Conversion of Diameter to Radius and Length to Volume

Start by converting the diameter of wire into radius as the formula to find volume of cylinder (which the wire essentially is) takes radius as a parameter. Given diameter is 0.03196 in., converting to cm by multiplying it with 2.54 gives approximately 0.0812 cm. Hence, the radius becomes approximately \(0.0812 / 2 = 0.0406\) cm. Also, convert 1.00 m of length to cm by multiplying it by 100 to get 100.0 cm. Now, the volume \(V\) of wire can be calculated using the formula for volume of a cylinder \(V= \pi r^{2} h\), where \(r\) is radius and \(h\) is height. Substituting the values, we get \(V = \pi * (0.0406)^{2} * 100.0\) cm³.

2Step 2: Calculate Mass Using Density

Knowing that the density \(d\) of an object equals its mass \(m\) divided by its volume \(V\), rearrange the formula to solve for the mass: \(m = d * V\). Substitute the given density of copper (8.92 g/cm³) and the calculated volume from the previous step to find the mass of copper in the wire.

3Step 3: Determine the Number of Atoms

First, find the number of moles (\(n\)) in the copper wire by using the formula \(n = m / M\), where \(M\) is the molar mass of copper, which is approximately 63.546 g/mol. After finding the number of moles, use Avogadro's number \(6.022 * 10^{23}\) to determine the number of atoms present in the wire. The formula is \(N = n * N_{A}\), where \(N_{A}\) is Avogadro's number and \(n\) is the number of moles calculated earlier.

Key Concepts

Cylinder Volume CalculationDensity and Mass RelationshipMole ConceptAvogadro's Number

Cylinder Volume Calculation

The first step in understanding how many atoms are in a wire is to calculate the volume of the wire itself. Because the wire is cylindrical in shape, we can use the formula for the volume of a cylinder:

\[ V = \pi r^2 h \]

where:

\( V \) is the volume
\( r \) is the radius of the base of the cylinder
\( h \) is the height (or length of the wire in this case)

To find the radius, you take half of the diameter. The problem provides the diameter as \( 0.03196 \; \text{in.} \). Convert this into centimeters by multiplying by 2.54, and then divide by 2. Afterward, convert the length of the wire from meters to centimeters since density is often given in \( \text{g/cm}^3 \). Put these values into the formula to get the volume.

Density and Mass Relationship

Understanding how density relates to mass and volume is crucial. Density is defined as mass per unit volume and is expressed as \( d = \frac{m}{V} \).

\( d \) is the density
\( m \) is the mass
\( V \) is the volume

When given the density and volume of copper, you can rearrange the formula to solve for mass: \( m = d \times V \). By calculating the volume of the copper wire and knowing its density \( 8.92 \; \text{g/cm}^3 \), you can determine the mass in grams. This step is essential because the mass will help determine the number of moles of copper, which is our pathway to finding out the number of atoms.

Mole Concept

The mole concept is a bridge between the mass of a material and the number of particles contained in it. A mole is defined as the amount of substance containing the same number of entities as atoms in 12 grams of carbon-12.
The molar mass of copper is approximately \( 63.546 \; \text{g/mol} \). To find the number of moles in a given mass of copper, use the formula:

\[ n = \frac{m}{M} \]

where:

\( n \) is the number of moles
\( m \) is the mass of the substance (calculated from density and volume)
\( M \) is the molar mass of the substance

This enables you to convert the mass of copper into moles, giving a counted quantity that can easily be used to find the number of atoms.

Avogadro's Number

Avogadro's number acts as a scaling factor to move from the macroscopic scale to the atomic scale. It is numerically equal to \( 6.022 \times 10^{23} \) particles/mole. This constant tells us how many atoms, ions, or molecules there are in one mole of substance.
After determining the number of moles using the mole concept, applying Avogadro's number allows calculating the total number of atoms in that mole quantity: