Shapes of p orbitals

To obtain pictures of the orbital ip = R0< >, we would need to combine a plot of R with that of 0, which requires a fourth dimension. There are two common ways to overcome this problem. One is to plot contour values of ip for a plane through the three-dimensional distribution as shown in Figures 3.8a,c another is to plot the surface of one particular contour in three dimensions, as shown in Figures 3.8b,d. The shapes of these surfaces are referred to as the shape of the orbital. However, plots of the angular function 0 (Figure 3.7) are often used to describe the shape of the orbital ip = RQ because they are simple to draw. This is satisfactory for s orbitals, which have a spherical shape, but it is only a rough approximation to the true shape of p orbitals, which do not consist of two spheres but rather two squashed spheres or doughnut shapes. 

The sizes of orbitals increase with increasing n and the true shapes of p orbitals are diffuse, as shown in Figure 5-26. The directions of p, d, and / orbitals, however, are easier to visualize in drawings such as those in Figures 5-23, 5-24, and 5-25 therefore, these slender representations are usually used. 

Figure 2.2 (a) Shape of p orbitals (top three) and (h) d orbitals (lower five). 

What happens with the second principal energy level (n = 2) Here we find two sublevels, 2s and 2p. Like Is in the first principal energy level, the 2s orbital is spherical in shape but is larger in size and higher in energy. It also holds a maximum of two electrons. The second type of orbital is designated by 2p. The 2p sublevel consists of three orbitals 2p, 2py, and 2p. The shape of p orbitals is quite different from the s orbitals, as shown in Figure 10.9. 

The orbitals occupied by p, d, and/electrons have three-dimensional shapes different from those of the electrons. There are three p orbitals, starting with n = 2. Each p orbital has two lobes like a balloon tied in the middle. The three p orbitals are arranged in three perpendicular directions, along the x, y, and z axes around the nucleus (see Figure 5.7). As with s orbitals, the shape of p orbitals is the same, but the volume increases at higher energy levels. 

Example The electron configuration for Be is Is lsfi but we write [He]2s where [He] is equivalent to all the electron orbitals in the helium atom. The Letters, s, p, d, and f designate the shape of the orbitals and the superscript gives the number of electrons in that orbital. 

Figure 1.3 Representations of s, p, and d orbitals. The s orbitals are spherical, the p orbitals are dumbbell-shaped, and four of the five d orbitals are cloverleafshaped. Different lobes of p orbitals are often drawn for convenience as teardrops, but their true shape is more like that of a doorknob, as indicated.

The shapes and orientations of p orbitals are shown in Figure 6.7. Notice that—... 

Figure 13.2 Combination of three atomic p orbitals to form three n molecular orbitals in the allyl radical. The bonding n molecular orbital is formed by the combination of the three p orbitals with lobes of the same sign overlapping above and below the plane of the atoms. The nonbonding n molecular orbital has a node at C2. The antibonding n molecular orbital has two nodes between Cl and C2, and between C2 and C3. The shapes of molecular orbitals for the allyl radical calculated using quantum mechanical principles are shown alongside the schematic orbitals.

Figure 13.5 Formation of the n molecular orbitals of 1,3-butadiene from four isolated p orbitals. The shapes of molecular orbitals for 1,3-butadiene calculated using quantum mechanical principles are shown alongside the schematic orbitals.

The value of the phase constant a does not affect the size or shape of the orbit, but it does affect the position of the orbit as a function of time. If at time t = t number of different identical oscillators have q = 0 but values of q lying between v and — v the representative points will lie on a straight line coincident with the p axis. In time, these points will describe their individual orbits but remain collinear. 

The second quantum number, the orbital angular momentum quantum number I, is generally related to the shape of the orbital and depends upon n, taking integral values from 0 to n — 1. The different values are always referred to by letters s for I =0, p for I = d for I = 2, and / for / = 3. 

We now examine the right side of the Oh character table to see what AOs on A are available to match these requirements. Within an j, p, d manifold, there are none for Tlg or Tltl. For T2k and Tiu we have, respectively (drt., dx=, dy.) and (pSJ pt pzj. If we look back at our previous treatment of a bonding in octahedral AB6 we find that the Tu, set was also needed there. In view of the need for strong a bonds and the fact that the set of p orbitals is very well shaped to form a bonds, this is normally their primary role. We are then left with only the type of A—B n bonding. 

FIGURE 3.16 One s-orbital and three p-orbitals blend into four sp3 hybrid orbitals pointing toward the vertices of a tetrahedron. This exploded view shows the shapes of the orbitals. The arrows show their orientations. 

P Some students believe that / V I the shape of an orbital shows how the electron moves about the atom. An orbital defines the region in which the electron moves with an indeterminable motion. 

To summarize at this point, it is reiterated that wavefunction tjr r,6,three-dimensional shape of each orbital can be represented by a contour surface, on which every point has the same value off. The three-dimensional shapes of nine hydrogenic orbitals (2s, 2p, and 3d) are displayed in Fig. 2.1.5. In these orbitals, the nodal surfaces are located at the intersections where f changes its sign. For instance, for the 2p orbital, the yz plane is a nodal plane. For the 3d y orbital, the xz and yz plane are the nodal planes. 

The second quantum number describes the shape of the orbital as s, p, d, f or g. These shapes do not describe the electron s path but rather are mathematical models showing the probability of the electron s location. The s and p orbital shapes are shown in Figure 8.9, but descriptions of the d and f orbitals are reserved for more advanced texts. 

The individual shells are made up of a number of swfrshells in which the electrons have different spatial arrangements (the shapes of the orbitals differ). The number of subshells available in a given shell is equal to the shell number. So shell 1 comprises only one subshell (its type is designated s). Shell 2 is made up of two subshells (an s type subshell of the same shape as the Is subshell and a second type designated p type this is the 2p subshell). Shell 3 comprises three subshells, an s type (3s), a p type (3p), and type designated d,... 

Azimuthal quantum number (1)—This number describes the shape of the orbital. The azimuthal quantum number can have values, from 0 to n-1, and these values correspond to certain orbital shapes. While the value can theoretically have a value as high as 6, we will see later that no values higher than 3 are found. The values that do exist, 0, 1, 2, and 3, correspond to particular shapes and are commonly designated as s, p, d, and /orbitals, respectively. In our house analogy, this quantum number would correspond roughly to the City. That is, it is a bit more specific than the State, but it still doesn t tell us exactly where the house is. 

The orbital angular momentum quantum number, , determines the shape of the orbital. Instead of expressing this as a number, letters are used to label the different shapes of orbitals, s orbitals have f = 0, and p orbitals have - 1. 

All this explains why the shape of an orbital depends on the orbital angular quantum number, t. All s orbitals ( = 0) are spherical, all p orbitals ( - 1) are shaped like a figure eight, and d orbitals ( = 2) are yet another different shape. The problem is that these probability density plots take a long time to draw—organic chemists need a simple easy way to represent orbitals. The contour diagrams were easier to draw but even they were a little tedious. Even simpler still is to draw just one contour within which there is, say, a 90% chance of finding the electron. This means that all s orbitals can be represented by a circle, and all p orbitals by a pair of lobes. 

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