## Monday, March 10, 2008

### Two Questions (MCQ) on Bohr Atom Model

You can find the earlier questions (with solution) on Bohr atom model posted on this site by clicking on the label ‘Bohr model’ or ‘hydrogen atom’ below this post. You can get them also by using the search option at the top of this page. Today I give you two questions which are meant for gauging the depth of your understanding of Bohr’s theory.

(1) When the electron in a hydrogen atom of mass M undergoes transition from an orbit of higher quantum number n2 to an orbit of lower quantum number n1, the recoil velocity acquired by the atom is (Rydberg’s constant = R, Planck’s constant = h)

(a) (R/hM) (1/n12 1/n22)

(b) (hR/M) (n2 n1)

(c) 1/hRM (1/n12 1/n22)

(d) h/RM) (1/n12 1/n22)

(e) (hR/M) (1/n12 1/n22)

The wave number of the photon emitted because of the electron transition is

ν' = 1/λ = R(1/n12 1/n22) where λ is the wave length of the photon and R is Rydberg’s constant.

The momentum of the photon is p = h/λ = hR(1/n12 1/n22) where h is Planck’s constant.

When the photon is emitted with this momentum, the atom recoils (like a gun firing a bullet) with an equal and opposite momentum. Therefore, the recoil velocity of the atom is given by

v = p/M = (hR/M)(1/n12 1/n22).

(2) If the radius of the innermost electron orbit in a hydrogen atom is R1, the de Broglie wave length of the electron in the second excited state is

(a) πR1

(b) 3πR1

(c) 4πR1

(d) 6πR1

(e) 9πR1

The wave length of the electron in the nth orbit is given by

λ = 2πRn/n where Rn is the radius of the nth orbit.4

[This follows because the angular momentum of the electron in the nth orbit is

mvRn = nh/2π.

Therefore, de Broglie wave length, λ = h/mv = 2πRn/n ]

The second excited state has quantum number n = 3 (Third orbit). The radius of the 3rd orbit in terms of the radius R1 of the first orbit is given by (remembering Rn = n2 R1)

R3 = 9R1

Therefore, λ = 2πRn/n = 2π×9R1/3 = 6πR1

[It will be convenient to remember that the de Broglie wave length of the electron in the nth orbit is n times the the wave length in the innermost orbit].

You will find some useful posts on Atomic Physics and Quantum effects at apphysicsresources