calculate the free electrons of a copper wire having a 0.064 inches diameter and 1000 ft. long?
Copper has one free conducting electron per atom, in the metallic state. The other 28 electrons remain bound.
For the number of atoms, compute the mass of the wire and divide it by the mass of one copper atom. You will need to look up the density of copper.
The mass of one copper atom is
(63.5 g/mole)/(6.02*10^23 atoms/mole)
Show your work for further assistance, if needed.
To calculate the number of free electrons in a copper wire, we need to make a few assumptions and use some basic formulae. Here are the steps:
Step 1: Find the cross-sectional area of the wire
The diameter of the copper wire is given as 0.064 inches. To find the radius, divide the diameter by 2:
radius = 0.064 inches / 2 = 0.032 inches
The cross-sectional area of the wire can be calculated using the formula for the area of a circle:
A = π * r^2
where π is approximately 3.14159 and r is the radius.
A = 3.14159 * (0.032 inches)^2 = 0.003209 sq inches
Step 2: Convert the cross-sectional area to square feet
Since the length of the wire is given in feet, we need to convert the cross-sectional area from square inches to square feet. Since there are 12 inches in a foot:
A (in sq ft) = A (in sq inches) / 144
A (in sq ft) = 0.003209 sq inches / 144 = 2.226e-05 sq ft
Step 3: Find the volume of the wire
The volume of the wire can be calculated by multiplying the cross-sectional area by the length of the wire:
V = A * length
V = 2.226e-05 sq ft * 1000 ft = 0.02226 cubic ft
Step 4: Calculate the number of free electrons
The number of free electrons can be estimated using Avogadro's number (6.02214076 × 10^23 free electrons per mole) and the density of copper (8.96 grams per cubic centimeter) to convert the volume of the wire to the number of atoms, and then multiplying by the number of free electrons per atom of copper (1) since there is one free electron per atom in copper.
Let's assume the wire is made of pure copper. The molar mass of copper is approximately 63.55 grams.
First, convert the wire volume from cubic feet to cubic centimeters:
V (in cubic cm) = V (in cubic ft) * 28316.8466
V (in cubic cm) = 0.02226 cubic ft * 28316.8466 = 629.759 cubic cm
Next, calculate the number of atoms using the density and molar mass of copper:
number of atoms = (density * V) / molar mass
number of atoms = (8.96 g/cm^3 * 629.759 cm^3) / 63.55 g/mol = 89.061 moles
Finally, multiply the number of atoms by Avogadro's number to find the number of free electrons:
number of free electrons = number of atoms * Avogadro's number
number of free electrons = 89.061 moles * 6.02214076 × 10^23 free electrons/mole = 5.358 × 10^25 free electrons
Therefore, the copper wire would contain approximately 5.358 × 10^25 free electrons.
To calculate the number of free electrons in a copper wire, we need to know the volume of the wire and the density of copper.
First, let us convert the diameter of the wire from inches to meters:
1 inch = 0.0254 meters
So, the diameter of the wire is: 0.064 inches * 0.0254 meters/inch = 0.0016256 meters
Next, we need to calculate the cross-sectional area of the wire. The cross-sectional area of a wire is given by the formula:
Area = π * (radius)^2
The radius of the wire is half of the diameter, so:
Radius = 0.0016256 meters / 2 = 0.0008128 meters
Now, we can calculate the cross-sectional area:
Area = π * (0.0008128 meters)^2 = 2.0765 x 10^-6 square meters
Next, let us convert the length of the wire from feet to meters:
1 foot = 0.3048 meters
So, the length of the wire is: 1000 feet * 0.3048 meters/foot = 304.8 meters
Now we have the volume of the wire:
Volume = Area * Length = (2.0765 x 10^-6 square meters) * 304.8 meters = 0.0006326 cubic meters
Next, we need to know the density of copper. The density of copper is approximately 8.96 grams per cubic centimeter or 8,960 kilograms per cubic meter.
Now we can calculate the mass of the copper wire:
Mass = Volume * Density = 0.0006326 cubic meters * 8960 kilograms/cubic meter = 5.665 kg
Finally, to find the number of free electrons, we multiply the mass of copper by Avogadro's constant and divide by the atomic mass of copper.
Avogadro's constant = 6.02214076 x 10^23 mol^-1
Atomic mass of copper = 63.55 grams/mole (or 0.06355 kg/mol)
Number of free electrons = (Mass of copper wire * Avogadro's constant) / Atomic mass of copper
Number of free electrons = (5.665 kg * 6.02214076 x 10^23 mol^-1) / 0.06355 kg/mol
After performing the calculation, we find that the number of free electrons in the copper wire is approximately 5.38 x 10^25 free electrons.
Note: This calculation assumes that all the copper atoms in the wire contribute free electrons.