12.4 The Bohr Model

Question 1

What was Bohr’s first postulate?

Answer 1

Bohr objected to the idea of an electron orbiting a nucleus in a circular orbit. He argued that such an electron experiences centripetal acceleration and an accelerated charge radiates away energy. So such an orb would be stable the electron would spiral into the nucleus. However Bohr argue that id the following condition is satisfied mvr = n(h/2pi) where n is a positive integer the orbits would be stable. The quantity mvr is the angular momentum of the orbiting electron.

Question 2

What is the total energy of one of the electron orbiting the nucleus in one of the allowed stable orbits?

Answer 2

E = 1/2mv^2 - (ke^2)/r where r is the orbital radius. The force on the electron is the coulomb force between the electron and the proton in the hydrogen nucleus F = (ke^2)/r
Hence (Ke^2)/r = m(v^2)/r and so
(ke^2)/r = mv^2 which simplifies the expression for total energy to E= -1/2(ke^2)/r

Question 3

What was a consequence of the Bohr postulate?

Answer 3

v=n(h/rm2pi). Substituting this not
(ke^2)/r = mv^2 gives
)ke^2)/r = m(n(h/rm2pi)^2) —> r = n^2(h^2/4pi^2ke^2m). Substituting this radius into the formula for the total energy given gives: E = - (2pi^2k^2e^4m)/n^2h^2. Putting numerical values of the constant we find the energy in eV is given by E = -13.6. This gives the discrete energy level structure.

Question 4

How is Bohr’s second postulate represented numerically?

Answer 4

Since the energy of a photon is given by E=hf=hc/lambda it follows that in a transition to a state with n = n1 to a state with n = n2 we must have 1/lambda = 2pi^2k^2e^4m/h^3c times (1/n2 - 1/n2).
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Question 5

What do Bohr’s two postulates thus explain?

Answer 5

The atomic spectrum of hydrogen.