Question

The Lennard-Jones potential energy, U(x), is a function of "x" the distance between a pair of molecules. It is frequently called the van der Walls potential between molecules for dipole-dipole interactions. The potential is

U(x) = C12/ x^12 - C6/x^6             [1]

Here we assume for water molecules:

C12 = 9.5 x 104" J•m12

C6 = 1 x 10-76 J•m6.

(a) What is the separation distance (Xs) between two water molecules where the applied force (F) to pull the molecules apart is a maximum. Recall F = dU/dx.

(b) Calculate the maximum force (Fmax) required to separate two water molecules [overcome the cohesive force between the molecules].

(c) If oxygen has a mass of 16 g/mole and hydrogen has a mass of 1 g/mole, using Avodadro's number (6 x 10^23 molecules/mole) and the gravitational force of 9.8 m/s^2, calculate the force of attraction (due to weight) of one water molecule toward Earth.

(d) How does the gravitational force compare to van der Waals force of attraction of two water molecules?

Answer 1

a)

U(x) = C12/x^12 - C6 / x^6

Now F = d/dx (U(x)   = d/dx(C12x^-12 - C6 x^-6 )

                               = -12(C12 x^-13 - ( -) 6 C6 x^-7

                               = -12 C12/x^13 + 6 C6 / x^7

                               = 12(9.5*104/x^13    + 6 *(1*10-76) /x^7

                        F   =   -11856/x^13    - 397/x^7   = - 11856 x^-13   - 396 x^-7

            It ismax ifd/dxF is   = 0

                  dF/dx =   -11856(-13) x ^-14   -(-7) 396 x^-8    = 0

                             = 154128+2772 x^6/x^14    = 0

So 154128 +2772x^6 = 0

2772 x^6 = -154128

x^6    = - 154128/2772 = -55.60

x = (-55.60)^.166666

Calculator shows this is imaginary .

Si it seems data given for c12,c6 is wrong in question

Maximum force is obtained by putting this x in equation of F

c)

for H2O it has molecularweight = 16 +2*1 = 18

mass of one molecule = molar mass/ avogadro number = 18/6*10^23   = 3*10^-23

                       g = 9.8

so the force of attraction (due to weight) of one water molecule toward Earth. = mg

                                                                      =   3*10^-23   *9.8   = 29.4*10^-23 N

d)    vanderwaal's force is much greater than gravitational pull

vanderwaal force as calculated in b)      >   29*10^-23