Physical Chemistry , 1st ed.

b.For the Be^3 nucleus, the solution to the problem is along similar lines but

now the nuclear charge for the beryllium atom must be included explicitly.

For Z4, the integrals being evaluated are

P

a

4

3

3





2 

0

d



0

sin d 

0.250Å

0

r^2 e^8 r/adr

Note that the upper limit on the rintegral is now 0.250 Å. These expressions

integrate to yield

P

a

6

3

4



 2  (^2) e^8 r/a



8

r^2 a



r

3

a

2

2



2

a

5

3

6

 0 0.250Å

which yields

P

(0.52

6

9

4

Å)^3 

 2 2[(2.28
10 ^2 )(0.00690)Å^3

(1)(5.78 

10 ^4 )Å^3 ]

P.728 or 72.8%

This is to be expected, since the larger nuclear charge pulls the single electron

in closer to the nucleus. Therefore, there is a 72% probability of finding a

 1 selectron within 0.250 Å of a Be^3 nucleus.

Radial probability plots for  2 s, 2 p, 3 s, 3 p, 3 d,...are shown in Figure

11.20. For each wavefunction having quantum numbers nand , there are

n1 points along a spherical radius where the probability of finding an

electron becomes exactly zero. These points are nodes.Specifically, these are

radial nodes,since we are considering the total electron probability at a spher-

ical shell at each value of the radius.

Although ssubshells are spherically symmetric, individual p,d,f,...sub-

shells are not and do have angular dependence. There are several ways of con-

veying the angular dependence of subshells. One common way is to draw an

outline within which the probability of the electron’s appearance is 90%. It is

easiest to use the real form of the wavefunctions to illustrate this behavior.

Figure 11.21 shows the 90% boundary surfaces of real (that is, nonimaginary)

pand dsubshells of hydrogen. It is these angular distributions of the subshells

that lend the “dumbbell” and “rosette” descriptions to the pand dorbitals.

There are several things to note about these plots. First, for each orbital, dif-

ferent axes are used to illustrate the plot, which means that the orbitals point

in different directions in spaceeven though they look very similar. Each section

of the plots is labeled with a plus or a minus to indicate the sign of the wave-

function in that region. Next, for each porbital there is one plane that is tan-

gent to all electron probability. As an example, for the pzorbital, the xyplane

is the plane of exactly zero electron probability. For the pxorbital, the yzplane

has zero electron probability. For the dorbitals, there are two planes where the

electron probability is zero. These are examples ofangular nodes(also called

nodal planes or nodal surfaces). Figure 11.22 shows some of the angular nodes

for pand dorbitals. For the dz^2 orbital, the nodal surface is a two-dimensional

cone. For quantum number , there will be angular nodes. Combining an-

gular nodes with radial nodes, there will be a total ofn1 nodes (both radial

and angular) for any wavefunction n,.

362 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom