Kaplan MCAT General Chemistry Review

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, m_l, m_s. 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 m_l. 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 m_l, 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/n_i² - 1/n_f²]. 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