Question

Infrared spectroscopy takes advantage of the ability of covalent bonds in a molecule to absorb light of certain energy. To assign particular peaks in an IR spectrum to a certain type of bond, covalent bonds are modeled as springs between two atoms. Hooke’s Law tells us that the energy (U) of the bond is a function of temperature and the distance between the atoms, so U = U(T,x). The change in volume upon stretching a bond is negligible, so the work done on the bond is given by w = F dx.

a. Write out an expression for the differential change in entropy, starting with the first/second laws of thermodynamics and the total derivative of U(T,x). Assume the stretching of a bond through infrared light exposure is reversible.

b. If the process occurs at constant temperature T and Hooke’s Law says that U = ½ kx²  (where k is referred to as the spring constant) and F = 4x, give the expression for DS when stretching a bond from x₁ to x₂.

Answer 1

a.

From first law of thermodynamics,

Here,

Therefore,

Also, we know,

Therefore, we get,

The work done:

The volume change is negligible, so we ignore the the pdV part. So we get

From second law,

(for a reversible process)

For irreversible,

Assembling all of this in the first law,

dividing both sides by T

This is the expression for entropy.

b.

Using the functional form of U,

Also, given

The process occurs at constant temperature, so

So the expression for entropy becomes,

The entropy for streching from x1 to x2:

One last thing. The work done due to streching of a spring is given by: so F is the force of streching. By Hooke's law, this streching force is given by:

Comparing with , we get k=4

Therefore,