Homework Answers
Part a). The entropy change associated with the given states = 21 + 0 + 0 = 21
Part b). The entropy change associated with the given states = 20 + 1 + 0 = 21
Part c). The entropy change from (a) to (b) = 21 - 21 = 0, i.e. the system is at equilibrium.
Part d). The entropy change associated with the given states = 17 + 2 + 2 = 21
Part e). The entropy change associated with the given states = 17 + 4 + 0 = 21
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Question 14. Consider the following 3-state model: a system of 21 non-interacting particles that can occupy...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T and a particle reservoir at chemical potential fl. Assume that we can neglect different spin orientations of the fermions. Each particle can be in one of three single-partiele states with energies 0, A and 2A. (a) Find the grand partition function of the system. (b) Find the mean number of particles and mean energy of the system. (C) Find the most probable microstate of...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-partic...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-particle energy levels, with energies ε,-0, ε,-2e and ε,-3. The system is in thermal equilibrium at temperature T. Suppose the particles are bosons with integer spin. i) How many states do you expect this system to have? Justify your answer [2 marks] (ii) Make a table showing, for each state of this system, the energy of the state, the number of particles (M, M,, N) with...
need help with thermodynamics A system consists of N weakly interacting particles, each of which can...
need help with thermodynamics
A system consists of N weakly interacting particles, each of which can be in either of two states with respective energies e and 2. where e1 2 1. Without explicit calculation, make a qualitative plot of the mean energy U the entropy S of the system as a function of its temperature T. What is in the limit of very low and very high temperatures? What is S in the limit of very low and very...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state system consisting of N non-interacting particles. Each particle i can be in one of an infinite number of states designated by an integer, n; = 0,1,2, .... The energy of particle i is given by a = en; where e is a constant. Note: you may need the series sum Li-ori = a) If the particles are distinguishable, compute QIT,N) and A(T,N) for this...
6. Consider a quantum system of N particles with only three possible states to oc- cupy...
6. Consider a quantum system of N particles with only three possible states to oc- cupy for each particle. The energy values of these states are equal to 0, €and €3, respectively. (a) [10 points) You observe that the probability to sample eash state is P = 0.9. P2 = 0.09 and p = 0.01 at T = 300 K. What are the energies and c? Recall that the probability to occupy state is proportional to where k= 1.38 x...
Consider a system of 1000 particles that can only have two energies, E1 and E2 with...
Consider a system of 1000 particles that can only have two energies, E1 and E2 with E2>E1. The difference in the energy between these two values is delta E= E2-E1. Asume g1=g2=1. a) graph the number of paricle n1, n2 in states of E1 E2 as a function of kbT/deltaE, where kb is Boltzmann constant. Explain the result b) at what value kbT/deltaE do 750 of the particles have the energy E1
6. Consider a quantum system of N particles with only three possible states to oc cupy...
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4. A particle moves in a periodic one-dimensional potential, V(x a)-V(x); physically, this may represent the motion of non-interacting electrons in a crys- tal lattice. Let us call n), n - 0, +1, t2,...
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Consider a system of 1000 particles that can only have two energies, E1 and E2, with...
Consider a system of 1000 particles that can only have two energies, E1 and E2, with E2 > E1. The difference in the energy between these two values IS Δ E2-E1. Assume g1 = g2-1. . Assume ga2- a) Graph the number of particles, nı and n2, in states E1 and E2 as a function of kBT AE, where ks is Boltzmann's constant. Explain your result. b) At what value of kBT /AE do 750 of the particles have the...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T and a particle reservoir at chemical potential fl. Assume that we can neglect different spin orientations of the fermions. Each particle can be in one of three single-partiele states with energies 0, A and 2A. (a) Find the grand partition function of the system. (b) Find the mean number of particles and mean energy of the system. (C) Find the most probable microstate of...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-particle energy levels, with energies ε,-0, ε,-2e and ε,-3. The system is in thermal equilibrium at temperature T. Suppose the particles are bosons with integer spin. i) How many states do you expect this system to have? Justify your answer [2 marks] (ii) Make a table showing, for each state of this system, the energy of the state, the number of particles (M, M,, N) with...
need help with thermodynamics
A system consists of N weakly interacting particles, each of which can be in either of two states with respective energies e and 2. where e1 2 1. Without explicit calculation, make a qualitative plot of the mean energy U the entropy S of the system as a function of its temperature T. What is in the limit of very low and very high temperatures? What is S in the limit of very low and very...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state system consisting of N non-interacting particles. Each particle i can be in one of an infinite number of states designated by an integer, n; = 0,1,2, .... The energy of particle i is given by a = en; where e is a constant. Note: you may need the series sum Li-ori = a) If the particles are distinguishable, compute QIT,N) and A(T,N) for this...
6. Consider a quantum system of N particles with only three possible states to oc- cupy for each particle. The energy values of these states are equal to 0, €and €3, respectively. (a) [10 points) You observe that the probability to sample eash state is P = 0.9. P2 = 0.09 and p = 0.01 at T = 300 K. What are the energies and c? Recall that the probability to occupy state is proportional to where k= 1.38 x...
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4. A particle moves in a periodic one-dimensional potential, V(x a)-V(x); physically, this may represent the motion of non-interacting electrons in a crys- tal lattice. Let us call n), n - 0, +1, t2, particle located at site n, with (n'In) -Sn,Let H be the system Hamiltonian and U(a) the discrete translation operator: U(a)|n) - [n +1). In the tight- binding approximation, one neglects the overlap of electron states separated by a distance larger than a, so that where is...
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