Kaplan MCAT General Chemistry Review

The second quantum number is called the azimuthal (angular momentum) quantum number and is designated by the letter l. The second quantum number refers to the shape and number of subshells within a given principal energy level (shell). The azimuthal quantum number is very important because it has important implications for chemical bonding and bond angles. The value of n limits the value of l in the following way: For any given value of n, the range of possible values for l is 0 to (n–1). For example, within the first principal energy level, n = 1, the only possible value for l is 0; within the second principal energy level, n = 2, the possible values for l are 0 and 1. A simpler way to remember this relationship is that the n-value also tells you the number of possible subshells.

Therefore, there’s only one subshell in the first principal energy level; there are two subshells within the second principal energy level; there are three subshells within the third principal energy level, and so on. The subshells also go by names other than the integer value of l: The l = 0 subshell is also known as the s subshell; the l = 1 subshell is also known as the p subshell; the l = 2 subshell is known as the d subshell; and finally, the l = 3 subshell is the f subshell. You’re probably more used to working with these letter names than with the integer values.

Key Concept

For any principal quantum number n, there will be n possible values for l.

The maximum number of electrons that can exist within a given subshell is equal to 4l + 2. The energies of the subshells increase with increasing l value; however, the energies of subshells from different principal energy levels may overlap. For example, the 4s subshell will have a lower energy than the 3d subshell. This is why, ultimately, the image of increasingly larger eggshells falls short of adequately serving as an analogy.

Magnetic Quantum Number

The third quantum number is the magnetic quantum number and is designated m_l. The magnetic quantum number specifies the particular orbital within a subshell where an electron is highly likely to be found at a given moment in time. Each orbital can hold a maximum of two electrons. The possible values of m_l are the integers between -l and +l, including 0. For example, the s subshell, with its l value = 0, limits the possible m_l value to 0, and since there is a single value of m_l for the s subshell, there is only one orbital in the s subshell. The p subshell, with its l value = 1, limits the possible m_l values to -1, 0, +1, and since there are three values for m_l for the p subshell, there are three orbitals in the p subshell. The d subshell has five orbitals, and the f subshell has seven orbitals. The shape of the orbitals, as the number of orbitals, is dependent upon the subshell in which they are found. The s subshell orbital is spherical, while the three p subshell orbitals are each dumbbell shaped along the x-, y-, and z-axes. In fact, the p orbitals are often referred to as p_x, p_y, and p_z. The shapes of the orbitals in the d and f subshells are much more complex, and the MCAT will not expect you to answer questions about their appearance. Of course, any discussion of orbital shape must not allow for a literal interpretation of the term, since we are using the term to describe “densities of probabilities” for finding electrons in regions of space surrounding the nucleus.

Key Concept

For any value of l, there will be 2l + 1 possible values for m_l. For any n, this produces n² possible values of m_l (i.e., n² orbitals).

Spin Quantum Number

The fourth quantum number is called the spin quantum number and is denoted by m_s. In classical mechanics, an object spinning about its axis has an infinite number of possible values for its angular momentum. However, this does not apply to the electron, which has two spin orientations designated +½ and -½. Whenever two electrons are in the same orbital, they must have opposite spins.

Key Concept

For any value of n, there will be a maximum of 2n² electrons (i.e., two per orbital).

In this case, they are often referred to as paired. Electrons in different orbitals with the same m_s values are said to have parallel spins.

The quantum numbers for the orbitals in the second principal energy level, with their maximum number of electrons noted in parentheses, are shown in Table 1.2.

Table 1.2

ELECTRON CONFIGURATION AND ORBITAL FILLING