Show that the force f necessary to stretch a strip of rubber

Show that the force f necessary to stretch a strip of rubber isothermally to a length L is related to the Helmholtz free energy F by the relationship below. Do not assume the rubber behaves ideally. F = (partial differential F/partial differential L)_T

Solution

The thermodynamic identity for a rubber band is

dU = T dS + dL, (1)

where T is the temperature, is the tension, U is the internal energy of the rubber band, S is its entropy, and L is its length. This relation follows naturally from theFirst Law of Thermodynamics, combined with the definition of work as the dot product of force and displacement

. We could use the differential relation given in Eq. (1), but since we are working at constant T, the Helmholtz free energy F provides a more useful starting point

dF = dL SdT. (2)

The corresponding Maxwell relation ( S/ L) _T = ( / T) _L , (3)

tells us that we can determine how entropy changes with length at fixed temperature by measuring how the tension changes with temperature at fixed length. At the same time, measurement of the tension reveals how the free energy varies with isothermal changes in length = (F/ L) _T ..... (4)

Thus, measurements of and ( /T)_L , as a function of length, allow us to find F, S, and ultimately U by integration