Kaplan MCAT General Chemistry Review

where h is a proportionality constant known as Planck’s constant, equal to 6.626 × 10^-34 J • s, and f (sometimes designated as the Greek letter , which looks a lot like the English letter v; sometimes you’ll hear students say, “Eee equals eich-vee”) is the frequency of the radiation.

BOHR MODEL OF THE HYDROGEN ATOM

Danish physicist Niels Bohr, in 1913, used the work of Rutherford and Planck to develop his model of the electronic structure of the hydrogen atom. Starting from Rutherford’s findings, Bohr assumed that the hydrogen atom consisted of a central proton around which an electron traveled in a circular orbit and that the centripetal force acting on the electron as it revolved around the nucleus was the electrical force between the positively charged proton and the negatively charged electron.

MCAT Expertise

When we see a formula in our review or on Test Day, we need to focus on ratios and relationships rather than the equation as a whole. This simplifies our “calculations” to a conceptual understanding, which is usually enough to lead us to the right answer.

Bohr used Planck’s quantum theory to correct certain assumptions that classical physics made about the pathways of electrons. Classical mechanics postulates that an object revolving in a circle, such as an electron, may assume an infinite number of values for its radius and velocity. The angular momentum (L = mvr) and kinetic energy (KE = ½mv²) of the object, therefore, can take on any value. However, by incorporating Planck’s quantum theory into his model, Bohr placed restrictions on the value of the angular momentum of the electron revolving around the hydrogen nucleus. Analogous to quantized energy, the angular momentum of an electron, Bohr predicted, is quantized according to the following equation:

where h is Planck’s constant and n is the quantum number, which can be any positive integer. Because the only variable is the quantum number, n, the angular momentum of an electron changes only in discrete amounts with respect to the quantum number.

Bohr then related the permitted angular momentum values to the energy of the electron to obtain the following equation:

where R_H is the experimentally determined Rydberg constant, equal to 2.18 × 10^-18 J/electron. Therefore, like angular momentum, the energy of the electron changes in discrete amounts with respect to the quantum number. A value of zero energy was assigned to the state in which the proton and electron are separated completely, meaning that there is no attractive force between them. Therefore, the electron in any of its quantized states in the atom will have a negative energy as a result of the attractive forces between the electron and proton; hence the negative sign in the previous energy equation. Now, don’t let this confuse you, because ultimately, the only thing the energy equation is saying is that the energy of an electron increases the further out from the nucleus that it is located. Remember that as the denominator (n², in this case) increases, the fraction gets smaller. However, here we are working with negative fractions that get smaller as n² increases. As negative numbers get smaller, they move to the right on the number line, toward zero. So, even though the absolute value is getting smaller (e.g., -8, -7, -6, etc.), its true value is increasing. Think of the concept of quantized energy as the change in gravitational potential energy that you experience when you ascend or descend a flight of stairs. Unlike a ramp, on which you could take an infinite number of steps associated with a continuum of potential energy changes, a staircase only allows you certain changes in height and, as a result, allows only certain discrete (quantized) changes of potential energy.

Key Concept

At first glance, it may not be clear that the energy (E ) is directly proportional to the principle quantum number (n) in this equation. Take notice of the negative charge, which causes the values to approach zero from a greater negative value as n increases (thereby increasing the energy). The negative sign is as important as n’s place in the fraction when it comes to determining proportionality.