A single electron orbits a lithium nucleus that contains three protons (+3e). The radius of the orbit is 1.76*10^-11m. Determine the kinetic energy of the electron.
For a rotating object the equation for acceleration is a = v^2 / r. The force is F = m*a = m*v^2 / r. The force in this problem is the coulomb force.
F = k*3q*q / r^2 = m*v^2 / r
Since the kinetic energy is just m*v^2 / 2, and that's all we need, let's multiply both sides by r / 2:
k*3q*q / (2*r) = m*v^2 / 2
Plug in numbers on the left and we'll get the kinetic energy:
8.99*10^9 N*m^2 / c^2 * 3*(1.60*10^-19 c)*(1.60*10^-19 c) / (2*1.76*10^-11 m) = 1.96*10^-17 Joules
1.96*10^-17 Joules = about 122 electron volts (eV), which is probably a better answer.
To determine the kinetic energy of the electron, we need to use the formula for kinetic energy:
KE = (1/2) * m * v^2
Where:
- KE is the kinetic energy
- m is the mass of the electron
- v is the velocity of the electron
Given that we are dealing with a single electron, the mass of the electron, m, is approximately 9.11 × 10^-31 kg.
To find the velocity, v, of the electron, we can use the concept of electrostatic attraction between the electron and the lithium nucleus. The centripetal force due to this attraction can be equated to the electrostatic force:
F_electrostatic = F_centripetal
Where:
- F_electrostatic is the electrostatic force between the electron and the lithium nucleus
- F_centripetal is the centripetal force acting on the electron
The electrostatic force can be calculated using Coulomb's law:
F_electrostatic = k * (q1 * q2) / r^2
Where:
- k is the electrostatic constant, approximately 9 × 10^9 N m^2 C^-2
- q1 and q2 are the charges of the interacting particles (e.g., the charge of the electron and the charge of the lithium nucleus)
- r is the distance between the particles
Since the lithium nucleus contains three protons, its charge is +3e, where e is the elementary charge (approximately 1.6 × 10^-19 C). Therefore, q1 is -e and q2 is +3e.
By equating the electrostatic force with the centripetal force, we get:
k * (q1 * q2) / r^2 = m * v^2 / r
Simplifying this equation, we get:
v^2 = (k * q1 * q2) / (m * r)
Finally, substituting the given values, we can calculate the kinetic energy:
KE = (1/2) * m * v^2
To determine the kinetic energy of the electron, we need to make use of the equation for kinetic energy.
The kinetic energy (KE) of an object can be calculated using the equation:
KE = (1/2) * m * v^2
where m is the mass of the object and v is its velocity.
In this case, since we are dealing with an electron, we know that the mass of an electron is approximately 9.11 * 10^-31 kg.
To find the velocity of the electron, we need to consider the centripetal force acting on it due to its circular motion around the nucleus. The centripetal force is provided by the electric force between the proton and the electron.
The electric force (F) can be calculated using Coulomb's law, which states that the force between two charged particles is proportional to the product of their charges and inversely proportional to the square of the distance between them.
F = k * (q1 * q2) / r^2
where k is the electrostatic constant, q1 and q2 are the charges of the particles, and r is the distance between them.
In this case, q1 (charge of the lithium nucleus) is +3e, where e is the elementary charge (+1.6 * 10^-19 C) and q2 (charge of the electron) is -e.
Substituting the values, we get:
F = k * (3e * -e) / r^2
Now, equating this force to the centripetal force:
F = (m * v^2) / r
We can now equate the two equations:
k * (3e * -e) / r^2 = (m * v^2) / r
Now, we can solve for the velocity (v) of the electron:
v^2 = (k * (3e * -e) * r) / (m * r^2)
v = sqrt((k * (3e * -e) * r) / (m * r^2))
Finally, substitute the values into the equation for kinetic energy:
KE = (1/2) * m * v^2
Now you can calculate the kinetic energy of the electron.