Question

Which of the quantum numbers governs (a) the shape of an orbital, (b) the energy of an orbital, (c) the spin properties of the electron, (d) the spatial orientation of the orbital?

Short answer

(a) The shape of an orbital is governed by the angular momentum quantum number (l). (b) The energy of an orbital is determined by both the principal quantum number (n) and the angular momentum quantum number (l), although in hydrogen atoms, only n affects the energy. (c) The spin properties of the electron are governed by the spin quantum number (ms). (d) The spatial orientation of the orbital is governed by the magnetic quantum number (ml).

Step-by-step solution

(a) Shape of an orbital)

The shape of an orbital is determined by the angular momentum quantum number (l). The different values of l give rise to different types of orbitals: (l=0) represents s orbitals, which are spherical; (l=1) represents p orbitals, which are dumbbell-shaped; (l=2) represents d orbitals, which are more complex in shape; and (l=3) represents f orbitals, which are even more complex.

(b) Energy of an orbital)

The energy of an orbital depends on both the principal quantum number (n) and the angular momentum quantum number (l). In a hydrogen atom, the energy of an orbital depends only on the principal quantum number (n). However, in multi-electron atoms, both n and l determine the energy of an orbital due to electron-electron interactions.

(c) Spin properties of the electron)

The spin properties of the electron are determined by the spin quantum number (ms). The electron can have either of two possible spin values, +1/2 or -1/2, which correspond to the electron's "spin-up" or "spin-down" states.

(d) Spatial orientation of the orbital)

The spatial orientation of the orbital in an atom or molecule is determined by the magnetic quantum number (ml). The magnetic quantum number can have integer values from -l to +l, where l is the angular momentum quantum number. Different values of ml represent different orientations of the same type of orbital (s, p, d, or f) in space.