Physical Chemistry , 1st ed.

From first law considerations, as mentioned earlier,

dU 0

for this system. But when we separate the system into subsystems, each of

which has its own internal energy, we can include the surface energy. We have

dUdUIdUIIdU  0 (22.9)

Note something very interesting about this equation. It requires that the total

internal energy,dU, be zero; it does not require that dUfor either region, or

for the interface, be unchanging. By defining a natural variable expression for

each region and using the dU value for the interface (equation 22.8), we can

rewrite equation 22.9 as

(TIdSIpIdVI) (TIIdSIIpIIdVII) (dA)  0

We will leave out the explicit multiplication signs in the next few equations. To

simplify the above expression, we note that the temperatures of each region

must be equal (that is,TIequals TII), and that any infinitesimal entropy change

by one region ought to be balanced by an equal and opposite infinitesimal en-

tropy change by the other region. (This is the same thing as saying that dS

0 for the entire system under these conditions.) Mathematically, then, the T dS

terms cancel. What is left, after rearranging, is

dApIdVIpIIdVII 0 (22.10)

Notice that we are notassuming that the pressures in each region are the same!

This is a key point. We will note, however, that if the volume of region I

changes, then the volume of region II must change by the same amount but in

the opposite direction. That is, as one region grows, the other gets smaller, and

by equal magnitudes (and vice versa, but always by equal magnitudes). The

mathematical way of expressing this is

dVIdVII (22.11)

Since we are interested in region I (the liquid), we will substitute for dVIIto

eliminate it. Equation 22.10 becomes

dApIdVIpIIdVI 0

and, after rearranging terms,

dA(pIIpI)dVI 0 (22.12)

In equation 22.12, we have factored out the dVIvariable from two of the terms.

Equation 22.12 is interesting because we have not presumed that the pressures

in the two regions are equal, and in doing so have derived an equation that re-

lates their difference with the surface tension! We can algebraically rearrange

equation 22.12 to get

(pIpII) dVIdA

(Notice how the two pressures have switched their relative orders, because of

the algebra.) We can combine the two differentials to get

(pIpII) 

V

A

I

 (22.13)

which relates the pressure difference on either side of an interface with the sur-

face tension and how the area of the liquid changes with volume. Equation

772 CHAPTER 22 Surfaces