In Bohr’s model, however, the electron was assumed to be at this distance 100% of the time, whereas in the Schrödinger model, it is at this distance only some of the time. Thus the most probable radius obtained from quantum mechanics is identical to the radius calculated by classical mechanics. This peak corresponds to the most probable radius for the electron, 52.9 pm, which is exactly the radius predicted by Bohr’s model of the hydrogen atom.įor the hydrogen atom, the peak in the radial probability plot occurs at r = 0.529 Å (52.9 pm), which is exactly the radius calculated by Bohr for the n = 1 orbit. Because the surface area of each shell increases more rapidly with increasing r than the electron probability density decreases, a plot of electron probability versus r (the radial probability) shows a peak. (d) If we count the number of dots in each spherical shell, we obtain the total probability of finding the electron at a given value of r. (c) The surface area of each shell, given by 4πr2, increases rapidly with increasing r. The density of the dots is therefore greatest in the innermost shells of the onion. (b) A plot of electron probability density \Psi^2 versus r shows that the electron probability density is greatest at r = 0 and falls off smoothly with increasing r. (a) Imagine dividing the atom’s total volume into very thin concentric shells as shown in the onion drawing. Most important, when r is very small, the surface area of a spherical shell is so small that the total probability of finding an electron close to the nucleus is very low at the nucleus, the electron probability vanishes ( Figure 3.6.1d).įigure 3.6.1: Most Probable Radius for the Electron in the Ground State of the Hydrogen Atom. Because the surface area of the spherical shells increases more rapidly with increasing r than the electron probability density decreases, the plot of radial probability has a maximum at a particular distance ( Figure 3.6.1d). In contrast, the surface area of each spherical shell is equal to 4π r 2, which increases very rapidly with increasing r ( Figure 3.6.1c). Recall that the electron probability density is greatest at r = 0 ( Figure 3.6.1b), so the density of dots is greatest for the smallest spherical shells in part (a) in Figure 3.6.1. In effect, we are dividing the atom into very thin concentric shells, much like the layers of an onion ( Figure 3.6.1a), and calculating the probability of finding an electron on each spherical shell. In contrast, we can calculate the radial probability (the probability of finding a 1s electron at a distance r from the nucleus) by adding together the probabilities of an electron being at all points on a series of x spherical shells of radius r 1, r 2, r 3,…, r x-1, r x. At very large values of r, the electron probability density is very small but not zero. ![]() ![]() The probability density is greatest at r = 0 (at the nucleus) and decreases steadily with increasing distance. The 1 s orbital is spherically symmetrical, so the probability of finding a 1 s electron at any given point depends only on its distance from the nucleus. Because \Psi^2 gives the probability of finding an electron in a given volume of space (such as a cubic picometer), a plot of \Psi^2 versus distance from the nucleus ( r) is a plot of the probability density. One way of representing electron probability distributions was illustrated previously for the orbital of hydrogen. In contrast to his concept of a simple circular orbit with a fixed radius, orbitals are mathematically derived regions of space with different probabilities of containing an electron.
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