Consider two different hydrogen atoms. The electron in each atom is in an excited state.

(a) Is it possible for the electrons to have different energies but the same orbital angular momentum L, according to the Bohr model? Justify.

No.Electrons within the same obital angular momentum L have the same energies according to Bohr model.

(b) Is it possible for the electrons to have different energies but the same orbital angular momentum L, according to quantum mechanics? Justify.

Yes. According to quantum mechanics it is possible to have different energies within the same orbital angular momentum L. Energy is determined by N and angular momentum is determined by L. Different value of N are compatible with the same value of L.

Both answers are correct. In quantum mechanics, each N value can have L values up to N-1. Hence N=1 can have only L = 0, but N=2 can have L=0 or n=1, etc. (There is no N=0). All principal quantum numers allow an L=0 suborbital (angular momentum) state.

To answer these questions, we need to understand the Bohr model and quantum mechanics as they apply to hydrogen atoms.

The Bohr model is a simplified model of the hydrogen atom proposed by Niels Bohr in 1913. According to this model, electrons orbit the nucleus in specific, discrete energy levels called shells. Each shell has a specific energy value associated with it. The orbital angular momentum, denoted as L, is determined by the azimuthal quantum number (l), which determines the shape of the electronic orbit.

According to the Bohr model, electrons with the same orbital angular momentum L are assigned the same energy within a given energy level. This means that in a given energy level, the electrons have the same energy regardless of the value of L. Therefore, in the Bohr model, it is not possible for electrons to have different energies but the same orbital angular momentum L.

On the other hand, in quantum mechanics, which is a more accurate description of the behavior of electrons, the energy of an electron in a hydrogen atom is not solely determined by the orbital angular momentum. In addition to the orbital angular momentum, the energy is also influenced by the principal quantum number (n). The principal quantum number determines the shell or energy level in which the electron resides.

In quantum mechanics, different values of n can be compatible with the same value of L. This means that electrons with the same orbital angular momentum can have different energies depending on their principal quantum number.

So, to answer the second question, in quantum mechanics, it is indeed possible for electrons to have different energies but the same orbital angular momentum L. This is because the energy is determined by both the principal quantum number (n) and the orbital angular momentum (L).