In nature, almost all elements exist as two or more isotopes, and these isotopes are usually present in the same proportions in any sample of a naturally occurring element. The presence of these isotopes accounts for the fact that the accepted atomic weight for most elements is not a whole number. The masses listed in the periodic table are weighted averages that account for the relative abundance of various isotopes. See Figure 1.2 for the relative abundances in nature of the first several elements. Hydrogen, which is very abundant, has three isotopes.
Key Concept
Bromine is listed in the periodic table as having a mass of 79.9 amu. This is an average of the two naturally occurring isotopes, bromine-79 and bromine-81, which occur in almost equal proportions. There are no bromine atoms with an actual mass of 79.9 amu.
Figure 1.2
Example: Element Q consists of three different isotopes, A, B, and C. Isotope A has an atomic mass of 40 amu and accounts for 60 percent of naturally occurring Q. The atomic mass of isotope B is 44 amu and accounts for 25 percent of Q. Finally, isotope C has an atomic mass of 41 amu and a natural abundance of 15 percent. What is the atomic weight of element Q?
Solution: 0.60(40 amu) + 0.25(44 amu) + 0.15(41 amu) = 24.00 amu + 11.00 amu + 6.15 amu = 41.15 amu
The atomic weight of element Q is 41.15 g/mol.
Bohr’s Model of the Hydrogen Atom
We’ve come a long way from J. J. Thomson’s 1904 “plum pudding” model of the atom as negatively charged “corpuscles” (what we call electrons) surrounded by a density of positive charge that others (but not Thomson himself) likened to free-moving negatively charged “plums” suspended in a positively charged “pudding.” We kid you not; we couldn’t make this stuff up if we tried. In 1910, Ernest Rutherford provided experimental evidence that an atom has a dense, positively charged nucleus that accounts for only a small portion of the atom’s volume. Eleven years earlier, Max Planck developed the first quantum theory, proposing that energy emitted as electromagnetic radiation from matter comes in discrete bundles called quanta. The energy value of a quantum, he determined, is given by this equation:
E = hf
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
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 RH 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.
Bohr came to describe the structure of the hydrogen atom as a nucleus with one proton forming a dense core around which a single electron revolved in a defined pathway of a discrete energy value. Transferring an amount of energy exactly equal to the difference in energy between one pathway, or orbit, and another, resulted in the electron “jumping” from one pathway to a higher energy one. These pathways or orbits had increasing radii, and the orbit with the smallest radius in which hydrogen’s electron could be found was called the ground state and corresponded to n = 1. When the electron was promoted to a higher energy orbit (one with a larger radius), the atom was said to be in the excited state. Bohr likened his model of the hydrogen atom to the planets orbiting the sun, in which each planet traveled along a (roughly) circular pathway at set distances (and energy values) with respect to the sun. In spite of the fact that Bohr’s model of the atom was overturned within the span of two decades, he was awarded the Nobel Prize in Physics in 1922 for his work on the structure of the atom and to this day is considered one of the greatest scientists of the 20th century.