find the number of free electrons in a copper conductor having a diameter of 0.064 in. and a length of 1000ft.
**copper = 1.64*10^24 free electrons per cu in
get the volume of the copper
**convert ft to inch 12*1000 = 12000
(π(0.064/2)^2)(12000) =38.6038
38.6038(1.64*10^24)= 6.33*10^25
Why did the electron go to the gym?
To get a copper-toned conductor body, of course!
But let's get serious for a second. To find the number of free electrons in a copper conductor, we can use the formula:
\(N = \frac{A \times L}{V}\)
Where:
N = Number of free electrons
A = Cross-sectional area of the conductor
L = Length of the conductor
V = Volume of a single free electron
Now, to calculate N, we need to determine the values of A and V first.
The cross-sectional area of a copper wire can be found using the formula for the area of a circle:
\(A = \pi \times r^2\)
Given that the diameter of the conductor is 0.064 inches, the radius (r) would be half of that, which is 0.032 inches (rounding to 3 decimal places).
After converting the radius to feet (since the length is given in feet), we have:
\(r_{ft} = \frac{0.032}{12}\) ft
Now, we can calculate the cross-sectional area:
\(A = \pi \times (r_{ft})^2\)
Let's assume the value of π to be approximately 3.1416.
Finally, to find the volume of a single free electron (V), we can use the atomic volume of copper. It turns out that each copper atom has a volume of approximately 7.11 x 10^-24 cm^3.
Now that we have all the values, we can substitute them into the formula for N and calculate the number of free electrons in the copper conductor.
However, keep in mind that the number will be an estimate, as it's impossible to count the exact number of free electrons in a given conductor.
To find the number of free electrons in a copper conductor, we need to use the formula:
n = (N * A * L) / V
where:
- n is the number of free electrons
- N is Avogadro's number (6.02214076 × 10^23 electrons/mol)
- A is the cross-sectional area of the copper conductor
- L is the length of the copper conductor
- V is the volume of the copper conductor
First, let's calculate the cross-sectional area (A) of the copper conductor:
Given diameter = 0.064 in.
Radius (r) = diameter / 2 = 0.064 / 2 = 0.032 in.
To convert inches to meters, we use the conversion factor 1 in. = 0.0254 m.
Radius (r) in meters = 0.032 * 0.0254 = 0.0008128 m
The cross-sectional area (A) of the copper conductor is given by:
A = π * r^2
A = 3.14159 * (0.0008128)^2 = 2.0751 × 10^-6 m^2
Now, let's calculate the volume (V) of the copper conductor:
Given length = 1000 ft.
To convert feet to meters, we use the conversion factor 1 ft = 0.3048 m.
Length (L) in meters = 1000 * 0.3048 = 304.8 m
The volume (V) of the copper conductor is given by:
V = A * L
V = 2.0751 × 10^-6 * 304.8 = 6.3247 × 10^-4 m^3
Finally, we can calculate the number of free electrons (n) in the copper conductor:
n = (N * A * L) / V
n = (6.02214076 × 10^23 * 2.0751 × 10^-6 * 304.8) / (6.3247 × 10^-4)
n ≈ 1.98 × 10^28 free electrons
Therefore, the copper conductor has approximately 1.98 × 10^28 free electrons.
To find the number of free electrons in a copper conductor, we need to use the formula for the volume of a cylinder. Here are the steps to calculate the number of free electrons:
1. Calculate the volume of the copper conductor:
- Given that the diameter of the conductor is 0.064 inches, we can find the radius by dividing the diameter by 2: radius = 0.064 in / 2 = 0.032 in.
- Convert the radius to feet by dividing it by 12: radius = 0.032 in / 12 = 0.00267 ft.
- Given that the length of the conductor is 1000 ft, the volume of the cylinder can be calculated as: volume = π * radius^2 * length.
- Plugging in the values: volume = π * (0.00267 ft)^2 * 1000 ft = (3.14) * (7.1289e-6) ft^3 = 2.242e-5 ft^3.
2. Calculate the number of atoms in a mole of copper:
- The atomic mass of copper (Cu) is 63.546 grams per mole (g/mol).
- Avogadro's number states that there are 6.022 × 10^23 atoms per mole.
- So, the number of atoms in a mole of copper is 6.022 × 10^23 atoms/mol.
3. Calculate the number of moles of copper in the conductor:
- To find the number of moles of copper in the conductor, we need to know the density of copper (ρ).
- The density of copper is approximately 8.96 grams per cubic centimeter (g/cm^3) or 8.96e3 kilograms per cubic meter (kg/m^3).
- To convert the volume of the conductor from ft^3 to m^3, multiply by the conversion factor: 2.242e-5 ft^3 * (0.3048 m / 1 ft)^3 = 6.348e-7 m^3.
- Now, we can calculate the mass of copper in the conductor: mass = density * volume.
- Plugging in the values: mass = 8.96e3 kg/m^3 * 6.348e-7 m^3 = 5.691e-3 kg.
- Using the molar mass of copper, we can calculate the number of moles: moles = mass / molar mass.
- Plugging in the values: moles = 5.691e-3 kg / 63.546 g/mol = 8.95e-5 mol.
4. Calculate the number of free electrons:
- Copper has one free electron per atom. Therefore, the number of free electrons is equal to the number of atoms in the conductor (since each atom contributes one free electron).
- The number of free electrons is calculated as: free electrons = number of atoms.
- Since we have already calculated the number of moles of copper in the conductor (8.95e-5 mol), we can use Avogadro's number to calculate the number of atoms:
free electrons = number of atoms = 8.95e-5 mol * 6.022 × 10^23 atoms/mol.
- Plugging in the values: free electrons = 8.95e-5 mol * 6.022 × 10^23 atoms/mol = 5.39e17 free electrons.
Therefore, in the copper conductor with a diameter of 0.064 in. and a length of 1000 ft, there are approximately 5.39 x 10^17 free electrons.