Intermolecular Forces and Potential Energy

2. Intermolecular Forces and Potential Energy

Consider the earth’s mass m E (whose origin is at its center). Newton’s law of gravita- tion states that the force F exerted by the earth towards its origin on another mass m located at

a distance r is given by the relation F = C mm /r 2 E , where C is the gravitational constant. In vector form

Fr ()

Cmmr /| r = 3 E | ,

where C = 6.67

2 ×10 –2 Nm kg , and Fr ( ) in units of N. The force exerted on a unit mass by

the earth, i.e., its gravitational acceleration, gr ()

= 3 Cm

E r /| r | . (If Fr ( ) is an attractive force, r it carries a negative sign, since it acts towards the origin. Typically, g <0, since it is attractive r

towards the earth, and, in order to move a mass away from the earth through a distance dr , work must be done to overcome the earth’s attractive force.) Therefore, the work done upon a mass m, i.e., the work input to raise that mass, is given by

r δφ = δW = – r Fr () ⋅ dr . (54)

We see from Eq. (54) that δW/ dr = – Fr ( ) . Using the relation for the gravitational accelera- tion, the work performed to raise a unit mass is

w = W/m = φ g = –Cm E /r + C 1 ,

where φ g is known as the gravitational potential. As r →∞, . φ g →C 1 so that C 1 =0. Therefore,

Figure 16: Schematic illustration of a water molecule.

φ g = –Cm E /r, (55) r

and d φ g / dr represents the gravitational force exerted on a unit mass. The energy stored in a mass under the influence of the earth’s gravitational field grows with an increase in the dis-

tance r. This gravitational potential energy is similar to the energy contained within a raised weight that induces it to fall unless it is constrained. Similarly, work must be performed to

move in a charge of Q c coulombs through an electrical potential. Likewise, if a molecule A is located at an origin and molecule B is situated at a dis- tance l removed from it, the potential energy stored within the molecule can be determined if

the characteristics of the force field are known. Alternatively, if the potential is known, the force exerted by a molecule on another can be determined (as illustrated above by the deriva- r

tive –d φ g / dr ). The Lennard–Jones’ (LJ) (6-12 law) empirical approach for like molecular pairs, such as the homonuclear molecular pair O 2 –O 2 , furnishes the intermolecular potential

energy in the form

(56) where ε represents the characteristic interaction energy between molecules, i.e., the maximum

12 Φ(l) = 4ε ((l 6

0 /l) – (l 0 /l) ),

attraction energy or minimum potential energy Φ min ( ε=Φ min ≈ 0.77 k B T c , with k B denoting the Boltzmann constant and T c the critical temperature), l 0 represents the distance at which the

potential is zero (cf. Figure 17 ) and is approximately equal to the characteristic or collision diameter σ of a molecule at which the potential curve shown in Figure 17 is almost vertical.

Tables A-3 tabulate σ and ε/k B (in K) for many substances. The term k B is called Boltzmann constant (= ¯R /N = 1.33x10 -26 Avog (kJ /molecule K)). In order to calculate the minimum po- tential energy l min , Eq. (56) can be differentiated with respect to l and set equal to zero. From

this exercise l /l

min 0 =2 =1.1225, and the corresponding value of Φ min = ε. Hence,

12 Φ(l)/ Φ 6

min = 4((l 0 /l) – (l 0 /l) ),

22 ,J; forcex10 0 0.5 1 1.5 2 2.5 3

Potenialx10 -10

Figure 17: Calculated LJ potential and force field for nitrogen molecules. Figure 18 presents a plot of the nondimensional intermolecular potential with respect to l/l 0 . If

we approximate an ideal gas as that gas where | φ (l limit )| ≈ 0.01| φ min |, then we obtain l limit /l 0 = 3.075 (from Eq. (57)) . Note that we are comparing attractive potential of gases with those of

maximum potential (i.e of liquids/solids). A better definition will be given in Chapter 6. The interaction force between the molecules is given by the relation F(l) =–d Φ/dl, so that F(l)/ Φ min

= (4/l

0 )(12(l 0 /l) –6(l 0 /l) 7 ). The maximum attractive force occurs at l max /l 0 = 1.2445, and the

corresponding force |F max | = 2.3964 | Φ min |l 0 . Therefore,

(58) It is seen from Eqs. (58) and (53) that gravitational forces are proportional to masses (inde-

F(l)/

| = –0.599(–12(l /l) 13 +6 (l /l) |F 7 max 0 0 ).

pendent of the chemical composition) and inverse of distance square while the LJ forces are inversely proportional 1/l 7 , and depends upon the chemical composition of the masses. As- suming l 0 to equal σ, for molecular nitrogen σ = l 0 = 3.681 Å and ε/k B = 91.5 K. Using the

value of k

B = 1.38 ×10 J molecule –1 K –1 , Φ and F can be determined for given values of l 0 /l. Results are presented for molecular nitrogen in Figure 17 .

If the molecules are spaced relatively far apart, the attractive force is negligible. Ideal gases fall into this regime. As the molecules are brought closer together, although the attractive forces increase, the momentum of the moving molecules is high enough to keep them apart. As the intermolecular distance is further decreased, the attractive forces become so strong that the matter changes phase from gas to liquid. Upon decreasing this distance further, the forces experienced by the molecules become negligible (i.e., d Φ/dl = 0 or Φ is maximized),

and the matter is now a solid in which the molecules are well–positioned. From Eq. (58) we see that the attractive force F(l)

∝ (l 3 ) –7/3 has units of approximately (volume) –2 . This concept can be used in developing van der Waals’ equation of state (see

Chapter 6). The LJ relation assumes the force field to be spatially symmetric around the mole- cule, an assumption which is valid over a wide range of conditions for gases such as O 2 , N 2 , and He and the other noble gases. However, this is not necessarily true for polar molecules Chapter 6). The LJ relation assumes the force field to be spatially symmetric around the mole- cule, an assumption which is valid over a wide range of conditions for gases such as O 2 , N 2 , and He and the other noble gases. However, this is not necessarily true for polar molecules

NH 3 . For the sake of illustration we will assume the LJ relation to also hold for polar gases.