Problems 麦克斯韦关系

9.9 Rubber bands are entropic springs. 橡胶带是熵弹簧。

Experiments show that the retractive force f of polymeric elastomers as a function of temperature T and expansion L is approximately given by f(T,L) = aT(L−L0) where a and L0 are constants.
(a) Use Maxwell’s relations to determine the entropy and enthalpy, S(L) and H(L), at constant T and p.
(b) If you adiabatically stretch a rubber band by a small amount, its temperature increases, but its volume does not change. Derive an expression for its temperatureT as a function of L, L0, a, and its heat capacity C = (∂U/∂T).
中文。实验表明，聚合弹性体的回缩力 f 与温度 T 和膨胀率 L 的函数关系近似为 f(T,L) = aT(L-L0)，其中 a 和 L0 为常数。
(a) 利用麦克斯韦关系确定恒定 T 和 p 时的熵和焓 S(L) 和 H(L)。
(b) 如果将橡皮筋绝热拉伸一小段，其温度会升高，但其体积不变。推导其温度 T 与 L、L0、a 及其热容 C = (∂U/∂T) 的函数关系式。
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9.10 Metal elasticity is due to energy, not entropy. 金属弹性是由能量而非熵引起的。

Experiments show that the retractive force f of a metal rod as a function of temperature T and extension L relative to undeformed length L0 is given by f(T,L) =EaΔL/L0, where ΔL = L[1−α(T−T0)]−L0 = L−Lα(T−T0)−L0. a is the cross-sectional area of the rod, E (which has the role of a spring constant) is called Young’s modulus,and α ≈ 10−5 is the linear expansion coefficient. Compute H(L) and S(L). Is the main dependence on L due to enthalpy H or entropy S?
中文。实验表明，金属杆的回缩力 f 与温度 T 和相对于未变形长度 L0 的延伸率 L 的函数关系为 f(T,L) =EaΔL/L0，其中 ΔL = L[1-α(T-T0)]-L0 = L-Lα(T-T0)-L0。a 是杆的横截面积，E（具有弹簧常数的作用）称为杨氏模量，α ≈ 10-5 是线膨胀系数。计算 H(L) 和 S(L)。对 L 的主要依赖是由于焓 H 还是熵 S？