A neutral water molecule (H2O) in its vapor state has an electric dipole moment

Question

A neutral water molecule (H2O) in its vapor state has an electric dipole moment of magnitude 6.2E-30 C.m.

a. How far apart are the molecule's centers of positive and negative charge?

b. If the molecule is placed in an electric field of 1.5E4 N/C, what maximum torque can the field exert on it?

c. How much work must an external agent do to rotate this molecule by 180 degrees in this field, starting from its fully aligned position, for which theta=0 ?

d. Compare this energy to a typical thermal energy kT when the molecule is part of a gas at a temperature 400K.

Explanation / Answer

(a) To determine how far apart the charges in the H2O dipole are from
the center of the molecule, proceed as follows: Consider that the
dipole moment (p) is proportional to the separation (d) of the
charges (q) in H2O as follows: p=q.d, so the separation is d = p/q.

But what is q? In terms of the fundamental unit of charge, e, (which
has the value e = 1.6 x 10^-19 Coulombs), the water molecule has a
charge of (q = 10e), i.e. there are 10 electrons and 10 protons. So
then, d = p/q = (6.2x10^-30 C.m)/(10.1.6x10-19 C) = 3.9x10^-12 m.
This is 3.9 picometers. VERY small. What we know then are that the
poles of the molecule (i.e. Hydrogen (+ pole) and Oxygen (- pole))
are very close together. Thus the covalent bond of a polar water
molecule is very strong. This gives you a good idea of what kind of
molecule we are dealing with. However, this separation has no
bearing on microwave heating, as microwave heating is due to rotation
(twisting, d doesn't change), and not vibration (which would change d
by small perturbations.) It is just an interesting calculation
I wanted to show you.

(b) Next, finding the maximum torque produced by the oscillating
electric field is easy: the maximum torque occurs when the dipole
moment is at right angles to the electric field produced by the
microwave oven (i.e. at an angle of 90 degrees.) Then the vector
cross product for torque is reduced from (T = p x E) to simply
(T = pE).

We then have Torque = T = (6.2e-30 C.m)x(1.5x10^4 N/C) which yields a
value of 9.3x10^-26 Newton.meters. I'll leave it up to you if you
want to convert this value to foot-pounds. (To help you, there are
approximately 2.2 pounds per Newton on Earth, and a meter is ~3
feet).

(c) Finally, to find the energy transferred to a molecule of water by
the oscillating electric field, via rotating the molecule by 180
degrees, you must proceed as follows:

The work due to a torque is defined as the difference in potential
energy between the two rotated positions, so the Energy transferred =
H(180) - H(0) and remember that the Hamiltonian is H=-p.E=-pEcos(w),
where w is the angle of rotation. So
Work = (-p*E*cos(180)) - (-p*E*cos(0))
which equals
Work = 2*p*E = 2(6.2x10^-30 C.m)(1.5x10^4 N/C)

Plug this into your calculator, and you'll get a value of
1.9 x 10^-25 Newton.meters.

(For reference, a Newton.meter = 1 Joule. And a Watt = 1 Joule per
second.)

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