Figure 20-33 gives the force magnitude F versus stretch...

Figure $20-33$ gives the force magnitude $F$ versus stretch distance $x$ for a rubber band, with the scale of the $F$ axis set by $F_{s}=1.50 \mathrm{~N}$ and the scale of the $x$ axis set by $x_{s}=3.50 \mathrm{~cm}$. The temperature is $2.00^{\circ} \mathrm{C}$. When the rubber band is stretched by $x=1.70 \mathrm{~cm},$ at what rate does the entropy of the rubber band change during a small additional stretch?

Key Concepts

Thermodynamic Entropy

Entropy is a fundamental state function in thermodynamics that quantifies the amount of disorder or the number of available microstates in a system. In processes that involve changes in mechanical variables (such as stretching), the entropy can change due to the redistribution of internal energy and disorder, even if no heat is exchanged with the environment. This concept is key in understanding how mechanical work can influence thermal properties.

Free Energy and Its Differential Relations

Free energy, particularly the Helmholtz free energy, is used to describe the trade-off between energy and entropy in a system at constant temperature. Its differential, dF = -S dT + Y dx (where Y is the generalized force), shows how changes in temperature or mechanical displacement (stretching) affect the system's free energy, and it connects to the entropy through partial derivatives. This framework allows one to relate mechanical work, such as stretching a rubber band, to changes in entropy.

Maxwell Relations

Maxwell relations are a set of equations derived from the equality of mixed second partial derivatives of thermodynamic potentials like the free energy. For systems undergoing deformation, a Maxwell relation can relate the change in entropy with respect to a mechanical parameter (such as stretch distance) at constant temperature to the temperature dependence of the associated force. This relation is instrumental in determining how stretching (or strain) affects the system's entropy.

Entropic Elasticity

Entropic elasticity describes the behavior of materials like rubber where elasticity primarily arises from changes in entropy rather than changes in internal energy. When a rubber band is stretched, the configurational entropy of its polymer chains decreases, which is directly linked to the macroscopic force and the work done during stretching. Understanding entropic elasticity is key to explaining why mechanical deformations in elastomers are closely tied to thermodynamic properties such as entropy change.

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