Suppose a single electron orbits about a nucleus containing two protons (+2e), as would be the case for a helium atom from which one of the two naturally occurring electrons is removed. The radius of the orbit is 2.65 10-11 m. Determine the magnitude of the electron's centripetal acceleration.
V^2/R is the centripetal acceleration. You know the value of R. You can get the velocity V by setting the centripetal force equal to the Coulomb force:
m V^2/R = k Q^2/R^2
In this case, Q = 2e
k is the Coulomb constant.
m is the electron mass and e is its charge.
You will need to look up these numbers.-
To determine the magnitude of the electron's centripetal acceleration, we can first find the velocity of the electron using the formula for the centripetal force:
F = mv²/r
Where F is the electrostatic force between the electron and the nucleus, m is the mass of the electron, v is the velocity of the electron, and r is the radius of the orbit.
The electrostatic force between the electron and the nucleus can be calculated using Coulomb's law:
F = k * (q₁*q₂) / r²
Where k is the electrostatic constant, q₁ is the charge of the electron (-e), q₂ is the charge of the nucleus (+2e), and r is the distance between the electron and the nucleus.
The mass of the electron (m) is a known constant. The charge of the electron (q₁) is -1.6 x 10^-19 C, and the charge of the nucleus (q₂) is +3.2 x 10^-19 C.
Given:
Radius of the orbit (r): 2.65 x 10^-11 m
Mass of the electron (m): 9.11 x 10^-31 kg
Charge of the electron (q₁): -1.6 x 10^-19 C
Charge of the nucleus (q₂): +3.2 x 10^-19 C
Electrostatic constant (k): 8.99 x 10^9 N m²/C²
Step 1: Calculate the electrostatic force (F) between the electron and the nucleus.
F = k * (q₁*q₂) / r²
F = 8.99 x 10^9 N m²/C² * (-1.6 x 10^-19 C) * (+3.2 x 10^-19 C) / (2.65 x 10^-11 m)²
Step 2: Calculate the velocity (v) of the electron.
F = mv²/r
v = √(Fr/m)
Substitute the calculated value of F from Step 1:
v = √(Fr/m) = √(F * 9.11 x 10^-31 kg / 2.65 x 10^-11 m)
Step 3: Calculate the centripetal acceleration (a) of the electron.
a = v²/r = (v * v) / r
Substitute the calculated value of v:
a = (v * v) / r = (√(F * 9.11 x 10^-31 kg / 2.65 x 10^-11 m))² / (2.65 x 10^-11 m)
Now you can substitute the given values and calculate the magnitude of the centripetal acceleration.