Question

For the ground state, the electron in the $ H - $ atom has an angular momentum $ = \rlap{-} h $ , according to the simple Bohr model. Angular momentum is a vector and hence will be infinitely many orbits with the vector pointing in all possible directions. In actuality, this is not true,
(A) Because Bohr model gives incorrect values of angular momentum
(B) Because only one of these would have a minimum energy
(C) Angular momentum must be in the direction of spin of electron
(D) Because electrons go around only in horizontal orbits

Answer 1

Hint: The Bohr’s model quantized the magnitude of the angular momentum of an electron revolving around the nucleus. But it does not give any idea of the direction of the angular momentum vector. Using this fact we can answer the above question.

Complete Step-by-Step solution:
We know that the information regarding the angular momentum of an electron of a single electron species is given by the second postulate of the Bohr’s model.
The Bohr’s second postulate, as we know, states that the electron revolves around the nucleus only in those orbits for which the angular momentum is an integral multiple of $ h/2\pi $ . In other words, the angular momentum of an electron revolving around the nucleus is quantized. So it is given by
 $ L = \dfrac{{nh}}{{2\pi }} $
Here $ n $ is the principal quantum number.
We know that a hydrogen atom contains only one electron. So the Bohr’s model is valid for it.
Now, for the ground state, the value of the principal quantum number is equal to $ 1 $ . Therefore substituting $ n = 1 $ in the above equation, we get the angular momentum of the electron in the ground state as
 $ L = \dfrac{h}{{2\pi }} = \rlap{-} h $
But we know that the angular momentum is a vector quantity. So it is true that according to the Bohr’s model, infinitely many angular momentum vectors of magnitude $ \rlap{-} h $ are possible, which point in every possible direction.
Since the Bohr’s model is unable to fix the direction of the angular momentum vector of the electron, this means that it gives incorrect values of the angular momentum.

Note:
You might think as to what would be the exact value of the angular momentum if the Bohr’s value is incorrect. The answer is that the exact expression for the angular momentum of an electron depends not on the principle quantum number, but on the azimuthal quantum number $ \left( l \right) $ . The azimuthal quantum number depicts the shape of the orbital, or the path of the electron. Hence, it fixes the locus of the electron and thus, the direction of the angular momentum becomes unique.