ATOMIC SCALE STRUCTURES
PERIODIC TABLE
2.4 PRIMARY BONDS
2.4.1 Ionic Bonding
The most common type of bond in a compound containing both electropositive and electronegative elements is theionic bond. This bond involveselectron transferfrom the electropositive atom to the electronegative atom. A high⌬EN between the atoms favors the formation of ionic bonds. Some of the more common ionic compounds involve the electropositive Group I elements (Na and Li)combining with electronegative elements from either Group VII (Cl or F)or Group VI (O)to form alkali halides (LiF, NaCl)and oxides (Li2O and Na2O).
As shown in Figure 2.4–1, an ionic bond is formed in NaCl when the single electron from the nearly empty valence band of the Na atom (1s22s22p63s1)is transferred to the nearly filled valence band of the Cl atom (1s22s22p63s23p5). The result is the formation of a Na cation, Na⫹, and a Cl anion, Cl⫺, each with a completely filled and therefore stable valence electron shell. Once charge transfer has occurred, a force of attraction develops between the ions. The magnitude of the force is a function of both the valence of the ions and their separation:
Fa共x兲 ⫽兩 Z1Z2兩 q2
40x2 (2.4–1)
where Ziis the valence of the ion, q is the charge of an electron,0is the permittivity of vacuum共⫽ 8.85 ⫻ 10⫺12C2/N-m2兲, and x is the separation distance between the ions. A force of the form given in Equation 2.4–1 is known as acoulombic force.
The coulombic force draws the ions together until their filled electron shells begin to overlap. Until the point of overlap, the electrons associated with the cation are indepen-dent of those of the anion. When the electron shells of the ions begin to impinge upon one another, their electrons begin to interact and they can no longer be considered to be independent. The Pauli exclusion principle requires that some of the interacting electrons be promoted to higher energy levels so that no two electrons will have the same four quantum numbers. Since this process requires the energy of the system to increase, a repulsive force develops in order to minimize the overlap of the electron shells of adjacent
FIGURE 2.4–1 An example of an ionic bond showing electron transfer from Na to Cl to form the Na⫹cation and Cl⫺anion pair.
ions. An alternate explanation for the development of a repulsive force is that the ions occupy a finite amount of space (the hard sphere model) and resist being forced to occupy a smaller volume.
The magnitude of the repulsive force increases rapidly as the ions are forced closer and closer together. Expressing this in one of many possible formalisms:
Fr共x兲 ⫽ ⫺K
xm (2.4–2)
where K and m are constants and m⬎ 2 (a common experimental value for m is 12).
Comparison of Equations 2.4–1and 2.4–2 shows that the repulsive force is dominant at small values of x and the attractive force dominates at larger separation distances.
The equilibrium separation distance x0 can be found by setting the sum of the forces equal to zero. That is, equilibrium occurs at the value of x for which
Fa共x兲 ⫹ Fr共x兲 ⫽ 0 ⫽兩 Z1Z2兩 q2 40x2 ⫺ K
xm (2.4–3)
Note that Equation 2.4–3 is valid only in the directions in which ions are in contact with one another. Figure 2.4–2a shows the relationship between the competing forces and x0. This figure, and others like it, will be referred to as the bond-force curve. Since x0
represents the average center-to-center distance between atoms, it is also known as the bond length.
Consider the energy changes that occur during the formation of an ionic compound.
The three important factors are the energy, or work, necessary to create the ions from neutral atoms, the work done by the attractive force (work done by the system) in drawing the ions together from an infinite separation distance, and the work done against the repulsive force (work done on the system) in bringing the ions together.
The energy required to remove an electron from an isolated neutral atom is referred to as its ionization potential. The ionization potential of Na BNa⫹ is 5.14 eV. The energy released when an isolated neutral electronegative atom gains an electron is termed its electron affinity. The electron affinity of ClBCl⫺ is 4.02 eV. Therefore, the net energy (work) required to create the pair of isolated Na⫹and Cl⫺ions is 1.12 eV.
The attractive force does work as it draws the ions together from an infinite separation:
Ua共x兲 ⫽冕x⬁Fadx⫽⫺兩 Z41Z20兩 qx2 (2.4–4) A similar integration for the repulsive force yields
Ur共x兲 ⫽冕x⬁Frdx⫽ xCn (2.4–5)
where n⫽ m ⫺ 1and C ⫽ K兾n. The summation of the three work terms gives an expression for the net work done by the system in bringing two atoms from an infinite separation to a separation distance x:
U共x兲 ⫽ Ui⫺ 兩 Z1Z2兩 q2 40x ⫹ C
xn (2.4–6)
where Uiis the 1.12 eV of work discussed above. A negative value of U共x兲 indicates that the compound is more stable than the isolated atoms. Figure 2.4–2b is a schematic illustration of the relationship between U and x and will be referred to as either the bond-energy curveor the bond-energy well. The minimum of the bond-energy curve corresponds to the equilibrium separation distance x . The bond-energy curve contains a
FIGURE 2.4–2 (a) The bond-force curve showing the location of the equilibrium separation distance x0. Note the ap-proximately linear slope of the total force curve in the vicinity of x0. (b) The bond-energy curve for the ionic compound NaCl showing the location of the equilibrium separation distance x0.
wealth of information concerning the properties of the ionic solid. After investigating the other types of atomic bonds, we will return to the bond-energy curve to extract this information.
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EXAMPLE 2.4–1
Describe the electron transfer process that occurs in the formation of the ionic compound Li2O.
Solution
Appendix B shows that the electron configuration for Li is 1s22s1and for O it is 1s22s22p4. Thus, Li has one valence electron and is electropositive. Oxygen has six valence electrons and is electro-negative. Li can obtain a filled valence shell by transferring its lone valence electron to the electroneg-ative O atom. This results in the formation of a Li⫹ion and an O⫺ion. The O⫺ion, however, still does not contain a filled valence shell. If a second Li atom transfers its lone valence electron to the O⫺ion, the result is a stable group of ions composed of two Li⫹and one O2⫺ (i.e., Li2O).
0
Fa(x) = Attractive force xo
Ft(x) = Total force
Fr(x) = Repulsive force x
Force (arbitrary units)Energy (arbitrary units)
0
(a)
(b)
Repulsion energy Ur
x Ui
Total energy Ut
Attractive energy Ua
Na+ Cl–
xo
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EXAMPLE 2.4–2
Use the concept of equilibrium separation distance x0together with the mathematical relationship among force, distance, and energy to demonstrate that the minimum point on the bond-energy curve corresponds to x0.
Solution
Energy, or work, is the product of force and distance. As shown in Equation 2.4–4, the total energy is
U共x兲 ⫽冕x⬁F dx
Alternatively we might write F⫽dU
dx
Since equilibrium is defined as the point at which the total force is zero, we may write F⫽dU
dx⫽ 0 at equilibrium
dU兾dx represents the slope of the bond-energy curve, and the slope is zero at the bottom of the bond-energy curve. Hence, x0corresponds to the minimum of the bond-energy curve.