The radius of circular electron orbits in the Bohr model of the hydrogen atom are given by (5.29 ✕ 10−11 m)n2, where n is the electron's energy level (see figure below). The speed of the electron in each energy level is (c/137n), where c = 3 ✕ 108 m/s is the speed of light in vacuum.
(a) What is the centripetal acceleration of an electron in the ground state (n = 1)of the Bohr hydrogen atom?
magnitude( )m/s2
direction( )
(b) What are the magnitude and direction of the centripetal force acting on an electron in the ground state?
magnitude( )N
direction( )
(c) What are the magnitude and direction of the centripetal force acting on an electron in the
n = 2 excited state?
magnitude( )N
direction( )
To find the centripetal acceleration of an electron in the ground state (n = 1) of the Bohr hydrogen atom, we can use the formula for centripetal acceleration, which is given by:
ac = v^2 / r
Where ac is the centripetal acceleration, v is the velocity of the electron, and r is the radius of the orbit.
First, let's find the velocity of the electron. According to the given information, the velocity of the electron in each energy level is given by:
v = c / (137n)
Substituting n = 1, we have:
v = c / (137*1) = c / 137
Now, let's find the radius of the orbit. According to the given information, the radius of circular electron orbits in the Bohr model is given by:
r = (5.29 × 10^-11 m) * n^2
Substituting n = 1, we have:
r = (5.29 × 10^-11 m) * 1^2 = 5.29 × 10^-11 m
Now, substitute the values of v and r into the formula for centripetal acceleration:
ac = (c / 137)^2 / (5.29 × 10^-11 m)
Calculating this expression will give us the magnitude of the centripetal acceleration in m/s^2.
To determine the direction of the centripetal acceleration, we need to know the direction of the velocity of the electron. In the Bohr model, the electron moves in a circular orbit around the nucleus, so the direction of the centripetal acceleration is towards the center of the orbit.
(a) The magnitude of the centripetal acceleration of an electron in the ground state is calculated by substituting the given values into the formula and solving for ac.
(b) The magnitude of the centripetal force acting on an electron in the ground state can be calculated using Newton's second law:
F = ma
Where F is the centripetal force, m is the mass of the electron, and a is the centripetal acceleration. The direction of the centripetal force is also towards the center of the orbit.
(c) To find the magnitude and direction of the centripetal force acting on an electron in the n = 2 excited state, follow the same procedure as in part (b), but use the appropriate values of n for the radius and velocity of the electron.