A copper wire 81.9 cm long and 1.10 mm in diameter has a mass of 6.43 g.
(a) Find the number of electrons in the wire. (Copper has an atomic number of 29; that is, there are 29 protons in the Copper atom. Copper's atomic mass is 63.5.)
how many electrons:
(b) There is one free electron per atom in copper. Find the number of free electrons in the wire.
how many electrons :
from the mass, calculate the number of moles of Cu.
From the moles, count the number of atoms (moles*avag number)
number of electrons= that * 29
To find the number of electrons in a copper wire, we need to use some key information:
1. The length of the wire: 81.9 cm
2. The diameter of the wire: 1.10 mm
3. The mass of the wire: 6.43 g
4. The atomic number of copper: 29
5. The atomic mass of copper: 63.5
Let's start with part (a) and find the number of electrons in the wire.
(a) Find the number of electrons in the wire:
To find the number of electrons, we need to determine the number of copper atoms in the wire. We will use the formula:
Number of atoms = (Mass of wire / Atomic mass of copper) * Avogadro's number
where Avogadro's number is approximately 6.022 x 10^23.
Step 1: Convert the diameter to meters:
1.10 mm = 1.10 x 10^-3 m
Step 2: Calculate the cross-sectional area of the wire:
The cross-sectional area of a wire can be calculated using the formula:
Area = (π * (diameter / 2)^2)
Substituting the values:
Area = π * ((1.10 x 10^-3) / 2)^2
Step 3: Calculate the volume of the wire:
The volume of the wire can be calculated using the formula:
Volume = Area * Length
Substituting the values:
Volume = (π * ((1.10 x 10^-3) / 2)^2) * 81.9 x 10^-2
Step 4: Calculate the mass per unit volume:
The mass per unit volume of the wire can be calculated using the formula:
Mass per unit volume = Mass of wire / Volume of wire
Substituting the values:
Mass per unit volume = 6.43 g / [(π * ((1.10 x 10^-3) / 2)^2) * 81.9 x 10^-2]
Step 5: Calculate the number of copper atoms:
Number of atoms = (Mass per unit volume / Atomic mass of copper) * Avogadro's number
Substituting the values:
Number of atoms = [(6.43 g / [(π * ((1.10 x 10^-3) / 2)^2) * 81.9 x 10^-2]) / 63.5] * 6.022 x 10^23
Step 6: Calculate the number of electrons:
Since copper atoms have 29 protons, they each have 29 electrons. Therefore, the number of electrons in the wire is equal to the number of atoms multiplied by 29.
Number of electrons = Number of atoms * 29
Now, you can calculate the number of electrons using the above equation.
(b) There is one free electron per atom in copper. Find the number of free electrons in the wire:
Since there is one free electron per atom in copper, the number of free electrons in the wire is equal to the number of atoms.
Number of free electrons = Number of atoms
Using the calculated number of atoms from part (a), you can determine the number of free electrons in the wire.
I hope this explanation helps you understand how to solve these problems!