The radius of circular electron orbits in the Bohr model of the hydrogen atom ar

Question

The radius of circular electron orbits in the Bohr model of the hydrogen atom are given by (5.29 1011 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?

(b) What are the magnitude and direction of the centripetal force acting on an electron in the ground state?

(c) What are the magnitude and direction of the centripetal force acting on an electron in the

n = 2

excited state?

magnitude     m/s2 direction     ---Select--- inward outward clockwise counterclockwise :a n=l/ n=2

Explanation / Answer

a)

r = 5.29 1011 /n2

v = c/(137n)

at n = 1

r1 = 5.29 1011

v1 = (3 x 108)/(137) = 2.19 x 108 m/s

centripetal acceleration is given as

a1 = v12/r1 = (2.19 x 108 )2/(5.29 1011) = 9.1 x 1026 m/s2

direction : inward

b)

F1 = ma1 = (9.1 x 10-31) (9.1 x 1026) = 0.00083 N

direction : inward

c)

r = 5.29 1011 /n2

v = c/(137n)

at n = 2

r2 = 5.29 1011/4 = 1.32 1011

v2 = (3 x 108)/(137 x 2) = 1.095 x 108 m/s

centripetal acceleration is given as

a2 = v22/r2 = (1.095 x 108 )2/(1.32 1011) = 9.1 x 1026 m/s2

direction : inward

F2 = ma2 = (9.1 x 10-31) (9.1 x 1026) = 0.00083 N

direction : inward

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