Radial probability function

Figure 10-1. Radial probability functions for typical 4/ and 6s orbitals.

The radial probability function Dni or rRni 2, drawn to the same scale, of the first six hydrogenic orbitals. 

In all the radial probability plots, the electron density, or probability of finding the electron, falls off rapidly as the distance from the nucleus increases. It falls olf most quickly for the 1 orbital by r = Sa, the probability is approaching zero. By contrast, the 3d orbital has a maximum at r = 9ao and does not approach zero until approximately r = 20ao- All the orbitals, including the s orbitals, have zero probability at the center of the nucleus, because Anir R = 0 at r = 0. The radial probability ftinctions are a combination of which increases rapidly with r, and R, which may have maxima and minima, but generally decreases exponentially with r. The product of these two factors gives the characteristic probabilities seen in the plots. Because chemical reactions depend on the shape and extent of orbitals at large distances from the nucleus, the radial probability functions help show which orbitals are most likely to be involved in reactions. 

FIGURE 2-7 Radial Wave Functions and Radial Probability Functions,... 

Fig. 2.4 Radial probability functions for n = 1,2, 3 for the hydrogen atom. The function gives the probability of finding the electron in a spherical shell of thickness dr at a distance r from the nucleus. [From Herzberg, G. Atomic Spectra and Atomic Structure Dover. New York, 1944. Reproduced with permission.]...

So how do s orbitals differ as the value of n changes One way to address this question is to look at the radial probability function, also called the radial probability density, which is defined as the probability that we will find the electron at a specific distance from the nucleus. 

How many maxima would you expect to find in the radial probability function for the 4s orbital of the hydrogen atom How many nodes would you expect in this function ... 

A FIGURE 6.18 Radial probability distributions for the Is, 2s, and 3s orbitals of hydrogen. These graphs of the radial probability function plot probability of finding the electron as a function of distance from the nucleus. As n increases, the most likely distance at which to find the electron (the highest peak) moves farther from the nucleus. 

The radial probabihty function, which we used in Figure 6.18, differs from the probability density. The radial probability function equals the total probability of finding the electron at all the points at any distance r from the nucleus. In other words, to calculate this function, we need to add up the probability densities [iji(r)] over all points located a distance r fiom the nucleus. FIGURE 6.20 compares the probability density at a point ([< <(r)] ) with the radial probability function. 

Let s examine the difference between probability density and radial probability function more closely. FIGURE 6.21 shows plots of [tfr(r)] as a function of r for the 1 2 and 3s orbitals of the... 

As shown in Figure 6.20, the collection of points a distance r from the nucleus is the surface of a sphere of radius r. The probability density at each point on that spherical surface is [i/<(r)]. To add up all the individual probability densities requires calculus and so is beyond the scope of this text. However, the result of that calculation tells us that the radial probability function is the probability density, multiplied by the surface area of the sphere,... 

The radial probability functions in Figure 6.18 provide us with the more useful information because they tell us the probability of finding the electron at all points a distance r from the nucleus, not just one particular point. 

SECTION 6.6 Contour representations are useful for visualizing the shapes of the orbitals. Represented this way, 5 orbitals appear as spheres that increase in size as n increases. The radial probability function tells us the probability that the electron will be found at a certain distance from the nucleus. The wave function for each p orbital has two lobes on opposite sides of the nucleus. They are oriented along the x, y, and z axes. Four of the d orbitals appear as shapes with four lohes around the nucleus the fifth one, the d orbital, is represented as two lobes along the z axis and a doughnut in the xy plane. Regions in which the wave function is zero are called nodes. There is zero probability that the electron will be found at a node. 

Consider the discussion of radial probability functions in A Closer Look in Section 6.6. (a) What is the difference between the probability density as a function of rand the radial probability function as a function of r (b) What is the significance of the term 47rr in the radial probability functions for the s orbitals (c) Based on Figures 6.18 and 6.21, make sketches of what you think the probability density as a function of r and the radial probability function would look like for the 4s orbital of the hydrogen atom. 

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