Search through millions of homework answers and textbook solutions  

Questions from May 12, 2022

• Assume the low frequency modes of a lattice are simple plane waves so that                                     Is a good approximation, and determine the low tempera-ture behavior of a harmonic...
  May 13, 2022
• 1. Verify this result. {Hint: Introduce the transformation of variables x =β (ε-μ) and note the μ and μ0 are close at small T.} 2. Answer the above question. (Hint: con-sider the average spatial...
  May 13, 2022
• Consider an isomerization process                                                 Where A and B refer to the different ismer states of a molecule. Imagine that the process takes place in a dilute...
  May 13, 2022
• Consider the system described in Exercise 4.15. The canonical partition function is                                                             Where N is the total number of molecules, and q is...
  May 13, 2022
• For the system described in exercises 4.15 and 4.16 we have the canonical partition function                                                               Show that ...
  May 13, 2022
• Consider the ideal Fermi gas of Exercise 4.18 in the high temperature and /or low density regime   (a)   Show that   (b)   Use this result together with the Fermi distribution for {np} to...
  May 13, 2022
• Consider now the ideal Fermi gas of Exercise 4.18 at low temperature and/or high density   (a)   Show that   (b)   Show that   Hence the pressure does not vanish at T =  0. Why?
  May 13, 2022
• (a)   Prove that if the energy eigenvalues of a system can be expressed as a sum of independent contributions E = EA + EB +EC  (e.g. electronic energy, vibrational energy, rota-tional energy) that the...
  May 13, 2022
• State all the necessary assumptions and calculate the entropy and Cp (in units of call per mole-deg) for HBr at 1 atm pressure and 200 C given that here hω is the energy spacing between the ground and...
  May 13, 2022
• Use the information compiled in chapter 8 of Hill’s Introduction to statistical Thermodynamics to calculate the equilibrium constant, K, for the reaction ...
  May 13, 2022
• Consider a one-component gas of non-interacting classical structureless particles of mass m at a temperature T.   (a)   Calculate exactly the grand canonical partition function,   for this system as...
  May 13, 2022
• Show that for positive j and zero field (e. e., H→0+), the lowest energy of the lsing model with a cubic lattice is                                                 and that this lowest energy...
  May 13, 2022
• Make the correspondence concrete. In par-ticulare, derive the precise relationships between the para-meters in the Ising magnet and those in the lattice gas that make the canonical partition function...
  May 13, 2022
• Verify this result. By Taylor expansion, analyze the precise from of m for temperatures near 2Dj/KB and show that the critical exponent β  in this mean filed theory has the value ½. Show that at T =...
  May 13, 2022
• Verify this result. Further, show that with this choice for ∆H, the first-order perturbation ap-proximation for the free energy corresponds to ϱ= ϱ NF, and that kB T In ϱMF is actually an upper bound...
  May 13, 2022
• Consider the one-dimensional I sing model with an external magnetic field. With suitably reduced variables, the canonical partition function is                        (a)   Show that by summing...
  May 13, 2022
• In this exercise, consider the Ising magnet that is isomorphic to the two-state quantal system. Suppose that the electric field, is a random fluctuationg field with a Gaussian probability...
  May 13, 2022
• Consider a one-dimensional Ising magnet in a fluctuating magnetic field, h. The particular model to be considered has the partition function                               with SN+1 = S1. Note that...
  May 13, 2022
• Describe a model system for which you can conceive of circumstances where Newtonian trajectories will be quasi-ergodic. One example might be a system of spherical particles jammed into a fixed volume...
  May 13, 2022
• Construct and describe an algorithm which employs a random number generator and samples states of the Ising magnet consistent with the distribution                                    where Mv is the...
  May 13, 2022
• Describe a Monte Carlo alogorithm that samples canonical equilibrium for a model with                 Where the primed sum is over nearest neighbors on a square lattice and the spins, si can take...
  May 13, 2022
• Run this Monte Carlo code on a computer. Evaluate the average of the spin (for every hundredth step in the trajectory, the code lists the value of the fraction of up spins). Demonstrate that the...
  May 13, 2022
• Consider an Ising magnet at a temperature T and another at a different temperature T’ . Show that the difference in free energy per unit temperature between the two can be computed by averaging ...
  May 13, 2022
• Consider an analogous problem encountered in solution theory. Two solutes are dissolved in a fluid of 400 solvent molecules to simulate a solution at low solute concentrations. Suppose the solutes...
  May 13, 2022
• Construct a Metropolis Monte Carlo code for a two-dimensional Ising magnet with 20 x 20 = 400 Spins and periodic boundary conditions. Run the code and collect statistics from the trajectory to analyze...
  May 13, 2022