Statistical Physics

6.3 Entropy of Rubber 85Fig. 6.1. A simplified model of rubber. A rubber molecule is modeled by a chainconsisting of rigid rods. The rigid rods are shown as arrows with circles at bothends. The rods are assumed always to be parallel to the x-axis6.3 Entropy of RubberWe shall now calculate the entropy of this model system, and obtain a relationbetween force and length for the system. We assume that the total number ofrods is N. Therefore, the maximum length of the chain is Na.AsshowninFig. 6.1, we treat the rods as vectors. When all the rods points in the positivex-direction, the length is at its maximum. In an ordinary state of the system,each rod can point in either the positive or the negative direction. Whenthere are N + rods pointing in the positive direction and N − (= N − N + )rodspointing in the negative direction, the total length is x =(N + − N − )a. Wecalculate the entropy when the total length is x. This is given by the logarithmof the number W of microscopic states that realize this length x. This numberis equal to the number of ways of choosing N + objects from N. NotingthatN + = Na+ x2aand N − = Na− x2a, (6.2)we obtain W (x):The entropy is then given byW (x) = N C N+ = N!N + !N − ! ≃N NN N++ N N−−. (6.3)S (x) =k B ln W (x) ≃ k B [N ln N − N + ln N + − N − ln N − ]( N= k B[N ln N −2 2a)+ x ( Nln2 2a)+ x= k B N( N−2 2a)− x ( Nln2 2a)]− x[ln 2 − 1 (1+ x ) (ln 1+ x )2 Na Na− 1 2(1 − x ) (ln 1 − x ) ] . (6.4)Na Na