Entropy-Driven Motors: From Rubber Bands to Molecular Machines · AI Deep Generator #DEEP-482861F2

Physics · thermodynamics

Problem

When you quickly stretch a rubber band and touch it to your lips, it feels warm. When you let it contract, it feels cool. More surprisingly, a stretched rubber band contracts when heated — opposite to the thermal expansion of most materials. Consider an ideal rubber band modeled as a collection of polymer chains. Each chain has NN freely-jointed segments of length aa. The end-to-end distance of a single chain is rr. From polymer physics, the entropy of a single chain is: Schain=−kBr22Na2+S0S_{\text{chain}} = -\frac{k_B r^2}{2Na^2} + S_0 where kBk_B is Boltzmann's constant and S0S_0 is a constant. (a) [10 pt] For a rubber band containing nn polymer chains with total length LL at temperature TT, derive the fundamental thermodynamic relation dU=TdS+FdLdU = TdS + FdL where FF is the tension. Show that for an ideal rubber band (where internal energy UU depends only on temperature), the tension must have the form F=αTLF = \alpha T L where α\alpha is a positive constant. Express α\alpha in terms of the given parameters. (b) [12 pt] Design a thermodynamic cycle for a heat engine using the rubber band. The cycle operates between two thermal reservoirs at temperatures ThT_h and TcT_c (Th>TcT_h > T_c) and between lengths L1L_1 and L2L_2 (L2>L1L_2 > L_1). The cycle consists of: - Process 1→2: Isothermal stretching at ThT_h - Process 2→3: Adiabatic relaxation - Process 3→4: Isothermal compression at TcT_c - Process 4→1: Adiabatic stretching Determine the length ratio L3/L2L_3/L_2 after the adiabatic relaxation. For the special case where L2=2L1L_2 = 2L_1, calculate the efficiency of this engine and compare it to the Carnot efficiency. (c) [15 pt] The microscopic model can be refined. Each polymer chain segment can point in any direction in 3D space. Using random walk statistics, the probability distribution for the end-to-end vector r⃗\vec{r} is: P(r⃗)=(32πNa2)3/2exp⁡(−3r22Na2)P(\vec{r}) = \left(\frac{3}{2\pi Na^2}\right)^{3/2} \exp\left(-\frac{3r^2}{2Na^2}\right) Derive the partition function ZZ for a single chain with its ends fixed at separation rr. Hence calculate the Helmholtz free energy A=−kBTln⁡ZA = -k_B T \ln Z and verify it gives the same force law as part (a). (d) [10 pt] When the rubber band is stretched isothermally, heat flows into it (opposite to gas compression). Calculate the heat capacity at constant length CLC_L and at constant tension CFC_F. Show that CF−CL=−αL2TC_F - C_L = -\alpha L^2 T and explain why CF<CLC_F < C_L (unlike gases where CP>CVC_P > C_V). (e) [13 pt] Real rubber shows deviation from the ideal behavior at large extensions. A better model includes finite chain extensibility: F=nkBTL[L/Lmax⁡1−(L/Lmax⁡)2+ln⁡(1−(L/Lmax⁡)2)]F = \frac{nk_B T}{L}\left[\frac{L/L_{\max}}{1-(L/L_{\max})^2} + \ln(1-(L/L_{\max})^2)\right] where Lmax⁡=nNaL_{\max} = nNa is the maximum possible length. Analyze the behavior as L→Lmax⁡L \to L_{\max}. What is the limiting efficiency of the heat engine as the operating length approaches Lmax⁡L_{\max}? (f) [8 pt] The entropy-driven tension in rubber is analogous to radiation pressure. For blackbody radiation in a cavity of volume VV at temperature TT, the pressure is P=13UVP = \frac{1}{3}\frac{U}{V} where U=σVT4U = \sigma VT^4 (with σ=4π2kB415c3ℏ3\sigma = \frac{4\pi^2 k_B^4}{15c^3\hbar^3}). Show that both systems satisfy (∂F∂T)L=(∂S∂L)T\left(\frac{\partial F}{\partial T}\right)_L = \left(\frac{\partial S}{\partial L}\right)_T (a Maxwell relation), and interpret this connection. (g) [7 pt] Molecular motors in cells (like kinesin) can be modeled as entropic springs. If such a motor has spring constant k=αTk = \alpha T with α=10−3\alpha = 10^{-3} N/(m·K) and operates at body temperature (310 K), calculate the maximum useful work per ATP molecule (which provides 20 kBTk_B T of energy) if the motor takes steps of 8 nm. What fundamental limit does this represent?

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