A4 The number of microstates of a system composed of N consituents scales as Ω(E, V...
1. (10 pts) Consider a system of N classical, independent harmonic oscil- lators. In the microcanonical ensemble, calculate Ω(E) and Ω(B) exactly. From them, calculate the entropy S(E, N) and temperature T in the large N limit. 2. (10 points) Consider the same system as in problem 1. Calculate the average energy and entropy starting from the canonical ensemble.
3. Entropy of physical systems (a) Consider two interacting systems A and B. Calculate the number of microstates and the entropy S of the combined system if i. (1 point) ΩΑ-2 and Ω13 3; ii. (1 point) A0204 and 2B 103x104 (b) A 20 Ω resistor is held at constant temperature of 300 K. A 10 A current is passed through the resistor for a minute i. (1 point) What is the change in entropy of the resistor? ii. (1...
The following relationships regarding entropy (S) and the number of microstates in a system (W) are true at 0 K: a. S = 0, W = 0 b. S = 1, W = 0 c. S = 0, W = 1 d. S = 213.7, W = 7.69 x 1034 e. S = -213.7, W = 7.69 x 1034
02) Consider a hydrostatic system represented by the thermodynamic variables volume V, pressure P and temperature T. a) Consider entropy S = S(T, V) and derive the equation TdS. Tas = Cvat +T (1) dV. V Show that this equation can be written as follows BT TdS = CydT + PDV where Cv is the thermal capacity at constant volume, B is the isobaric expansiveness and K is isothermal compressibility: b) Consider a gas described by the equation of state...
4. 10 points A monoatomic gas obeys the van der Waals equation: N²a P= NT V - Nb V2 where N is the number of particles and a and b are known constants and t = kbT. The gas has a heat capacity Cy = 3N/2 in the limit V +0. a) Using the thermodynamic identities and the equation of state prove that acv = 0. av т (3 pts) b) Use the result of part a) to determine the...
Calculate the internal energy, entropy and the equation of states for a system composed of N indistinguishable and non-interacting particles with the partition function (9) given below (a, b and C are constants): q = C(kt)}(V – Nb) eVKY
АЗ The microcanonical entropy of some gas is Sm Bin [V-Vo)Nf(E,N] , where V is the volume of the system and Vo is a constant. Find the equation of state. (10 pts
This is a Physical Chemistry problem. Thank you in advance! 1. Consider a system composed of a collection of N molecules that can distribute so that there are ni molecules in the state with energy E, n2 molecules in the state with energy E2, n3 molecules in the state with energy E..etc. The number of ways, W, in which molecules can distribute this way is N! N! W П "! n,in In,.. II Suppose that this system is undergoing the...
3. Derive the following relationship between the Helmholtz free energy F and the partition function Z for a system of N particles: (a) Starting with the thermodynamic definition F-U-TS, substitute the statistical mechanics results which give U and S in terms of occupation numbers n, state energies e and the most probable number of microstates t* to find, (b) Write out texplicitly in terms of occupation numbers using Stirling's approxima- tion (check the Lagrange multiplier derivation of the Boltzmann distribution)...
(TOTAL MARKS: 25) QUESTION 4 (15 marks] Q4(a) Assume 4 fermionic particles (N=N,+NA+N, -4) populate 3 degenerate energy levels E <E, <E, with 8, = 4,8, = 3.8, = 2 and N, 2N, 2N, What are the possible macrostates of this system ? (3 marks) (l) For each macrostate found at (), count the number of possible microstates using sketches showing the quantum state occupation number in each energy level. (7 marks) (H) Retrieve your results at (ii) if the...