The size of Rutherford's atom is about 0.1 nm.
(a) Calculate the speed of an electron that moves around a proton, based on their electrostatic attraction, if they are separated by 0.1 nm. (answer in m/s)
(b) Calculate the corresponding de Broglie wavelength for Rutherford's electron. (answer in nm)
To calculate the speed of the electron moving around the proton, we can use the equation for electrostatic attraction between two charged particles:
F = k * (q1 * q2) / r^2
Where:
F is the electrostatic force between the particles,
k is the electrostatic constant (9 x 10^9 N m^2 / C^2),
q1 and q2 are the charges of the particles (in this case, q1 represents the charge of the electron and q2 represents the charge of the proton),
r is the separation distance between the particles.
In this case, as the electrostatic attraction is the force keeping the electron in orbit around the proton, this force is equal to the centripetal force:
F = m * (v^2) / r
Where:
m is the mass of the electron, and
v is the speed of the electron.
By setting these two equations equal to each other, we can solve for v.
m * (v^2) / r = k * (q1 * q2) / r^2
Rearranging the equation, we get:
v = sqrt((k * (q1 * q2)) / (m * r))
Now, let's plug in the given values:
k = 9 x 10^9 N m^2 / C^2,
q1 = charge of the electron = -1.6 x 10^-19 C (in Coulombs),
q2 = charge of the proton = 1.6 x 10^-19 C (in Coulombs),
m = mass of the electron = 9.1 x 10^-31 kg (in kilograms),
r = separation distance = 0.1 nm = 0.1 x 10^-9 m (in meters).
Calculating the speed:
v = sqrt((9 x 10^9 N m^2 / C^2 * (-1.6 x 10^-19 C * 1.6 x 10^-19 C)) / (9.1 x 10^-31 kg * 0.1 x 10^-9 m))
v ≈ 2.19 x 10^6 m/s
Therefore, the speed of the electron is approximately 2.19 x 10^6 m/s.
To calculate the corresponding de Broglie wavelength for Rutherford's electron, we can use the equation:
λ = h / p
Where:
λ is the de Broglie wavelength,
h is Planck's constant (6.63 x 10^-34 J s),
p is the momentum of the electron.
The momentum of the electron can be calculated as:
p = m * v
Plugging in the given values:
m = 9.1 x 10^-31 kg (mass of the electron),
v = 2.19 x 10^6 m/s (speed of the electron, as calculated above).
Calculating the momentum:
p = (9.1 x 10^-31 kg) * (2.19 x 10^6 m/s)
p ≈ 1.993 x 10^-24 kg m/s
Now we can calculate the de Broglie wavelength:
λ = (6.63 x 10^-34 J s) / (1.993 x 10^-24 kg m/s)
λ ≈ 3.32 x 10^-10 meters
Therefore, the corresponding de Broglie wavelength for Rutherford's electron is approximately 3.32 x 10^-10 meters or 0.332 nm.