Absolutely Small - Michael D. Fayer [60]
What is really being asked is what is the probability of finding the electron a certain distance from the nucleus when I add together all possible directions? The way to state this question is, what is the probability of finding the electron in a thin spherical shell with the spherical shell having a radius of r? As r gets bigger, the volume of the thin spherical shell increases, which for some distance, offsets the fact that the wavefunction is decreasing. To understand the roll of the thin spherical shell, consider a number of hollow rubber balls each with the same thickness rubber wall. A ball with a small radius (small r) will have less rubber in the wall than a ball with a large radius. If you just went in a single straight line from the center of the ball to the wall, and where you hit the wall you asked what is the thickness of the rubber, it would be independent of the radius of the ball. But it is clear that a large hollow ball has more rubber in its wall then a small ball.
The surface area of a sphere is 4πr2, where r is the sphere’s radius. If you multiply this by the wall thickness, you have the volume of the rubber in the ball. Now it is clear that a larger ball has a lot more rubber in the wall than a small ball. If you double the radius, the amount of rubber increases by a factor of 4. Another important fact is that as r goes to zero, the amount of rubber in the ball goes to zero because the surface area, 4πr2, goes to zero. Asking if an electron is a distance r from the nucleus is like asking how much rubber is in the wall of a ball of radius r. It is necessary to account for the increasing surface area as the radius increases.
THE RADIAL DISTRIBUTION FUNCTION
The radial distribution function is exactly what we need to take into account the three-dimensional nature of an atom. As r is increased and we look in all directions to find the electron, we must include a factor of 4πr2. The radial distribution function is a plot of the probability of finding the electron a distance r from the nucleus for all directions. As discussed in Chapter 5, the Born interpretation of the wavefunction says that the probability of finding a particle in some region of space is proportional to the absolute value squared of the wavefunction. Here we want the probability of finding the electron in a thin spherical shell that has radius r. This is the radial distribution function, which is given by 4πr2|Ψ|2. The vertical lines mean absolute value. For the functions we are dealing with, we just need to square the wavefunction.
Figure 10.4 displays the radial distribution function for the 1s state of the hydrogen atom. The distance that has the maximum probability is not the center of the atom because the volume of the spherical shell goes to zero as r goes to zero. The vertical line shows the location of the maximum in the probability distribution. It is r = 0.529 Å. This is an important and interesting number. In Bohr’s old quantum theory of the hydrogen atom, the 1s state had the elec tron going in a circular orbit with a radius of 0.529 . This distance is called the Bohr radius and is given the symbol a0. What we see from the correct quantum mechanical treatment of the hydrogen atom is that the electron is a probability amplitude wave with the distance for the maximum probability equal to the Bohr radius a0. This is not a coincidence. The Bohr radius is actually a fundamental constant. It is given by
FIGURE 10.4. A plot of the radial distribution function for the 1s orbital as a function of r, the distance from the proton. The radial distribution function is the probability of finding the electron in a thin spherical shell a distance r from the proton. The radial distribution function takes into account that the electron can be found in any direction radially outward from the proton. The distance r is in Å, which is 10-10m.
where all of the parameters were given above when the Rydberg constant was defined in terms of fundamental constants. In fact, the energy levels of the hydrogen atom can be written