In lecture, we discussed the stretching of a chain molecule from a (short) end-t

Question

In lecture, we discussed the stretching of a chain molecule from a (short) end-to-end distance L_1 to a (long) end-to-end distance L_2 at fixed temperature T. Assuming the chain molecule behaves like an "ideal polymer", we calculated differences in energy and entropy, Delta E = E(L_2) - E(L_1) and Delta S = S(L_2) - S(L_1), respectively. Imagine that the chain molecule is held at constant tension f(L_1). Suddenly, the tension is increased to f(L_2) and the polymer extends spontaneously until it reaches a length L_2. This process occurs spontaneously at fixed T, f, N (N is the number of atoms in the chain molecule). Which auxiliary function is convenient to analyze this process?

Explanation / Answer

The combined first and second laws of thermodynamics state how an increment of mechanical work f dx done on the system can produce an increase in the internal energy dU or a decrease in the entropy dS: f dx = dU T dS (1)

Clearly, the relative importance of the entropic contribution increases with temperature T, and this provides a convenient means of determining experimentally whether the material’s stiffness in energetic or entropic in origin.

The retractive force needed to hold a rubber band at fixed elongation will increase with increasing temperature, as the increased thermal agitation will make the internal structure more vigorous in its natural attempts to restore randomness.

But the retractive force in specimen — which shows little entropic elasticity — will decrease with temperature, as thermal expansion will act to relieve the internal stress. In contrast to the instantaneous nature of the energetically controlled elasticity, the conformational or entropic changes are processes whose rates are sensitive to the local molecular mobility.

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