Quantum Mechanical Model of Atoms
While Bohr’s model marked a significant advancement in the understanding of the structure of atoms (at least we were no longer talking about plum pudding), his model ultimately proved inadequate to explain the structure and behavior of atoms containing more than one electron. The model’s failure was a result of Bohr’s not taking into account the repulsion between multiple electrons surrounding one nucleus. Modern quantum mechanics has led to a more rigorous and generalized study of the electronic structure of atoms. The most important difference between Bohr’s model and the modern quantum mechanical model is that Bohr postulated that electrons follow a clearly defined circular pathway or orbit at a fixed distance from the nucleus, whereas modern quantum mechanics has shown that this is not the case. Rather, we now understand that electrons move rapidly in extraordinarily complex patterns within regions of space around the nucleus called orbitals. The confidence by which those in Bohr’s time believed they could identify the location (or pathway) of the electron is now replaced by a more modest suggestion that the best we can do is describe the probability of finding an electron within a given region of space surrounding the nucleus. In the current quantum mechanical model, it is impossible to pinpoint exactly where an electron is at any given moment in time, and this is expressed best by the Heisenberg uncertainty principle: It is impossible to simultaneously determine, with perfect accuracy, the momentum and the position of an electron. If we want to assess the position of an electron, the electron has to stop (thereby changing its momentum); if we want to assess its momentum, the electron has to be moving (thereby changing its position).
QUANTUM NUMBERS
Modern atomic theory postulates that any electron in an atom can be completely described by four quantum numbers: n, l, ml, ms. Furthermore, according to the Pauli exclusion principle, no two electrons in a given atom can possess the same set of four quantum numbers. The position and energy of an electron described by its quantum numbers is known as its energy state. The value of n limits the value of l, which in turn limits the values of ml. Think of this like a country: A country has a defined number of states, and each state has a defined number of cities or towns. The values of the quantum numbers qualitatively give information about the orientation of the orbital. As we examine the four quantum numbers more closely, pay attention especially to l and ml, as these two tend to give students the greatest difficulty.
Principal Quantum Number
The first quantum number is commonly known as the principal quantum number and is denoted by the letter n. This is the quantum number used in Bohr’s model that can theoretically take on any positive integer value. The larger the integer value of n, the higher the energy level and radius of the electron’s orbit(al). Within each shell of some n value, there is a capacity to hold a certain number of electrons equal to 2n2, and the capacity to hold electrons increases as the n value increases. The difference in energy between two shells decreases as the distance from the nucleus increases because the energy difference is a function of [1/ni2 - 1/nf2]. For example, the energy difference between the n = 3 and the n = 4 shells is less than the energy difference between the n = 1 and the n = 2 shells. The term shell brings to mind the notion of eggshells, and you’ve probably heard the analogy between n values and eggshells of increasing size. This is fine as long as you don’t extend the analogy to the point that you are thinking about electron pathways as precisely defined orbits. Nevertheless, if thinking about eggshells helps you to remember that the principal quantum number says something about the overall energy of the electron orbitals as a function of distance from the nucleus, then go with it.
Bridge
A larger integer value for the principal quantum number indicates a larger radius and higher energy. This is similar to gravitational potential energy, where the higher the object is above the earth, the higher its potential energy will be.
Azimuthal Quantum Number
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 ml. 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 ml are the integers between -l and +l, including 0. For example, the s subshell, with its l value = 0, limits the possible ml value to 0, and since there is a single value of ml for the s subshell, there is only one orbital in the s subshell. The p subshell, with its l value = 1, limits the possible ml values to -1, 0, +1, and since there are three values for ml 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 px, py, and pz. 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.