The Radius of Electron Orbit within a Hydrogen Atom

A hydrogen atom contains a single electron that moves in a circular orbit about a single proton. Assume the proton is stationary, and the electron has a speed of 7.5 x 10^5 m/s. We need to find the radius between the stationary proton and the electron orbit within the hydrogen atom.

Calculation:

Given:

Electron speed (v) = 7.5 x 10^5 m/s

Charge of electron (q1) = -1.6 x 10^-19 C

Electric constant (e0) = 8.85 x 10^-12 F/m

Mass of electron (me) = 9.1 x 10^-31 kg

Using the equations for centripetal acceleration and electric force, we can find the radius (r) of the electron orbit:

Centripetal acceleration (a) = v^2 / r

Electric force (F) = q1 * q2 / (4 * π * e0 * r^2)

Dividing the electric force by the mass of the electron, we get:

a = q1 * q2 / (4 * π * e0 * me * r^2)

Substitute the values and simplify the equation:

r= q1 * q2 / (4 * π * e0 * me * v^2)

r= 1.6 x 10^-19 * 1.6 x 10^-19 / (4 * π * 8.85 x 10^-12 * 9.1 x 10^-31 * (7.5 x 10^5)^2)

After calculation, we get:

Radius (r) = 4.5 x 10^-10 m = 450 pm

Explanation:

The electron is held in orbit by an electric force, acting as the centripetal force. By equating the equations for the centripetal acceleration and electric force, we can find the radius of the electron orbit within a hydrogen atom.

A hydrogen atom contains a single electron that moves in a circular orbit about a single proton. What is the radius between the stationary proton and the electron orbit within the hydrogen atom?

450 picometers (pm)