Orbitals radial probability functions

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. 

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. 

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... 

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. 

A Figure 6.19 shows the radial probability densities for the Is, 2s, and 3s orbitals of hydrogen as a function of r, the distance from the nucleus—each resulting curve is the radial probability function for the orbital. Three features of these plots are noteworthy the number of peaks, the number of points at which the probability function goes to zero (called nodes), and how spread out the distribution is, which gives a sense of the size of the orbital... 

Interpret radial probability function graphs for the orbitals. (Section 6.6)... 

Solution. The most probable radius can be obtained from the highest peak in the radial distribution function, because this function is a measure of the probability of finding an electron in a volume element at a certain distance from the nucleus. Because the radial distribution function for a I s orbital has a single peak, the radius at which this peak occurs can be calculated by taking the first derivative of the function with respect to r and setting it equal to zero. For a Is orbital, R(r) and the first derivative of the radial probability function are... 

Figure 6.17 I Graphs of the radial probability function are shown for hydrogen orbitals with w = 1, 2, and 3. The increase in orbital size with increasing n is obvious. More subtle variations for orbitals with the same n value are also important in understanding the sequence of orbital energies.

Figure 1.28 (a) A useful measure of the electron density in a li orbital is obtained by first dividing the orbital into successive thin spherical shells of thickness dr. (b) A plot of the probability of finding a li electron in each shell, called the radial probability function, as a function of distance from the nucleus shows a maximum at 52.9 pm from the nucleus. Interestingly, this is equal to the Bohr radius. 

Figure 1.31 (a) The radial part of the wavefunction for the 2p and 3p hydrogen atomic orbitals, (b) The radial probability function for the orbitals shown in (a). 

Using the radial probability function, calculate the average values of r for the Is and 2s orbitals of the hydrogen atom. How do these compare with the most probable values (Hint You will need to consult a standard table of integrals for this problem.)... 

The magnitude and shape of such a mean-field potential is shown below [21] in figure B3.1.4 for the two 1 s electrons of a beryllium atom. The Be nucleus is at the origin, and one electron is held fixed 0.13 A from the nucleus, the maximum of the Is orbital s radial probability density. The Coulomb potential experienced by the second electron is then a function of the second electron s position along the v-axis (coimecting the Be nucleus and the first electron) and its distance perpendicular to the v-axis. For simplicity, this second electron... 

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