tailieunhanh - Introduction to Thermodynamics and Statistical Physics phần 3

Như vậy, năng lượng của một liên kết ¡¢ ½} ω a ¡n + 1 ¢ cho l = 2, (1,137) En, l =} ωb n + 1 l = b 2 n = 0, 1, 2, . Chuỗi được tổ chức dưới một F căng thẳng. Hiện hLi chiều dài trung bình của chuỗi trong giới hạn của T nhiệt độ cao được cho bởi hLi = N | . Problems Set 1 a Show that a is given by a a 2 . fFa 1 tanh JTj b Show that in the limit of high temperature the spring constant is given approximately by k -4T_. Na2 Na 19. A long elastic molecule can be modelled as a linear chain of N links. The state of each link is characterized by two quantum numbers l and n. The length of a link is either l a or l b. The vibrational state of a link is modelled as a harmonic oscillator whose angular frequency is Wa for a link of length a and Wb for a link of length b. Thus the energy of a link is _ J Wa n 2 for l a n 1 .Wb n for l b n 0 1 2 The chain is held under a tension F. Show that the mean length L of the chain in the limit of high temperature T is given by L b NFWWai2-i 2S O 2 Wb W Wb Wa 2 ụ v where S 1 t . 20. The elasticity of a rubber band can be described in terms of a onedimensional model of N polymer molecules linked together end-to-end. The angle between successive links is equally likely to be 0 or 180 . The length of each polymer is d and the total length is L. The system is in thermal equilibrium at temperature T. Show that the force f required to maintain a length L is given by f d tanh 1 Nd 21. Consider a system which has two single particle states both of the same energy. When both states are unoccupied the energy of the system is Eyal Buks Thermodynamics and Statistical Physics 27 Chapter 1. The Principle of Largest Uncertainty zero when one state or the other is occupied by one particle the energy is e. We suppose that the energy of the system is much higher infinitely higher when both states are occupied. Show that in thermal equilibrium at temperature T the average number of particles in the level is . . 2 . Ni -------------------rr 2 exp fi e - p where p is the chemical potential and fi 1 t. 22. Consider an array of N two-level particles. Each one can be in one of two states having energy E and . 2 respectively. The numbers of particles in states 1 and 2 are 1 .