Hydrogen radial probability function

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

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.]...

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. 

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... 

Let s examine the difference between probability density and radial probability function more closely. Figure 6.22 shows plots of [ilf r)f as a function of r for the Is, 2s, and 3s orbitals of the hydrogen atom. You will notice that these plots look distinctly different from the radial probability functions shown in Figure 6.19. 

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 1.31 (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.)... 

Radial probability functions of the hydrogen atom, (The factor is included because the radial volume element is r dr.) The number of nodes increases with both n and 1. 

Figure 8-2. The radial probability functions, r R, of a hydrogen atom for the first two energy levels are shown. Note that the probability functions decay to zero as r approaches infinity.

FIGURE 1.32 The radial distribution function tells us the probability density for finding an electron at a given radius summed over all directions. The graph shows the radial distribution function for the 1s-, 2s-, and 3s-orbitals in hydrogen. Note how the most probable radius icorresponding to the greatest maximum) increases as n increases. 

FIGURE 1.42 The radial distribution functions for s-, p-, and cf-orbitals in the first three shells of a hydrogen atom. Note that the probability maxima for orbitals of the same shell are close to each other however, note that an electron in an ns-orbital has a higher probability of being found close to the nucleus than does an electron in an np-orbital or an nd-orbital. 

Fig. 1-2.—The wave function u, its square, and the radial probability distribution function 47rrVi fo the normal hydrogen atom.

The probability of finding the electron in the ground state of the hydrogen atom between radii r and r + dr is given by D(r)Ar, where D(r) is the radial probability density function shown in Figure 4.5. The most probable distance of the electron from the nucleus is found by locating the maximum in D(r) (see Problem 4.12 below). It should come as no surprise to discover that this maximum occurs at the value r = ao, the Bohr radius. 

Figure 4.5 The radial probability density function for the 1s atomic orbital of the hydrogen atom...

Consider the radial probability density function, D(r), for the ground state of the hydrogen atom. This function describes the probability per unit length of finding an electron at a radial distance between r and r + dr (see Figure 6.5). 

The solvation structure around a molecule is commonly described by a pair correlation function (PCF) or radial distribution function g(r). This function represents the probability of finding a specific particle (atom) at a distance r from the atom being studied. Figure 4.32 shows the PCF of oxygen-oxygen and hydrogen-oxygen in liquid water. 

The plots for hydrogen-like wave functions of radial function R(r) versus r, the distance from the nucleus and the probability distribution function 4jrr2[R(r) 2 versus r are shown... 

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