Concept explainers
The Ising model can be used to simulate other systems besides ferromagnets; examples include antiferromagnets, binary alloys, and even fluids. The Ising model of a fluid is called a lattice gas. We imagine that space is divided into a lattice of sites, each of which can be either occupied by a gas molecule or unoccupied. The system has no kinetic energy, and the only potential energy comes from interactions of molecules on adjacent sites. Specifically, there is a contribution of
(a) Write down a formula for the grand partition function for this system, as a function of
(b) Rearrange your formula to show that it is identical, up to a multiplicative factor that does not depend on the state of the system, to the ordinary partition function for an Ising ferromagnet in the presence of an external magnetic field B, provided that you make the replacements
(c) Discuss the implications. Which states of the magnet correspond to low-density states of the lattice gas? Which states of the magnet correspond to high-density states in which the gas has condensed into a liquid? What shape does this model predict for the liquid-gas phase boundary in the
Want to see the full answer?
Check out a sample textbook solution- What is the number density of conduction electrons in gold, which is a monovalent metal? Use the molar mass and density provided in Appendix Farrow_forwardConsider two immiscible liquids such as water and oil. If a spherical oil molecule of radius r is taken out of the oil phase and placed in the water phase, the unfavorable energy of this transfer is proportional to the area of the solute (oil) molecule newly exposed to the solvent (water) multiplied by the interfacial energy, i, of the oil-water interface. The interfacial energy of the bulk cyclohexane-water interface is i = 50 mJ m-2, and the radius of a cyclohexane molecule is 0.28 nm. Using Boltzmann distribution, estimate the solubility of cyclohexane in water at 25 C in units of mol L-1.The concentration of water in water phase is 55.5 mol L-1.arrow_forwardThe intensities of spectroscopic transitions between the vibrational states of a molecule are proportional to the square of the integral ∫ψv′xψvdx over all space. Use the relations between Hermite polynomials given in Table 7E.1 to show that the only permitted transitions are those for which v′ = v ± 1 and evaluate the integral in these cases.arrow_forward
- In solid KCI the smallest distance between the centers of a. potassium ion and a chloride ion is 314 pm. Calculate the length of the edge of the unit cell and the density of KCI, assuming it has the same structure as sodium chloride.arrow_forwardA triatomic molecule can have a linear configuration, as does CO2 (as shown), or it can be nonlinear, like H2O (as shown). Suppose the temperature of a gas of triatomic molecules is sufficiently low that vibrational motion is negligible. What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant, (a) if the molecules are linear and (b) if the molecules are nonlinear? At high temperatures, a triatomic molecule has two modes of vibration, and each contributes 1/2 R to the molar specific heat for its kinetic energy and another 1/2 R for its potential energy. Identify the high-temperature molarspecific heat at constant volume for a triatomic ideal gas of (c) linear molecules and (d) nonlinear molecules. (e) Explain how specific heat data can be used to determine whether a triatomic molecule is linear or nonlinear. Are the data as shown sufficient to make this determination?arrow_forwardA hydrogen atom of mass 1.67 x 10 27 kg is attached to a very large'protein by a bond that behaves much like a spring. (a) If the vibrational frequency of the hydrogen is 1.0 x 10 14 Hz, what is the “effective” force constant of this spring-type bond? (b) If the total vibrational energy is kT (k is Boltzmann’s constant), approximately what is the classical amplitude of vibration at room temperature? By comparison, the diameter-of a hydrogen atom is about 10 -10 m ?arrow_forward
- Consider a heterojunction between two semiconductormaterials, A and B, whose electron affinities are equal. Their energy band gaps are EgA= 1.0 eV and EgB= 2.0 eV. The cubic lattice constant may be assumed to be a = 6.00Å for each material, and the dielectric constant to be 10ε0 throughout the heterojunction. In all cases you may assume flat-band conditions within each material. (a) Draw the energy band edge diagram and compute ΔEC and ΔEV for this heterojunction, assuming that the electron affinity rule is valid. (b) Now suppose that both materials have a “midgap” energy at the exact midpoint of their energy band gaps, and that in each material a deviation of the Fermi level from the “midgap” energy at a heterojunction interface produces a charge density at the interface of –σ(EF-Emidgap) with σ = 1×1014q/cm2·eV. Assume that these charges in each material are separated from each other by one cubic lattice constant at the heterojunction interface. By requiring overall charge neutrality…arrow_forwardA rigid tank of volume V = 0.014 m3 contains carbon monoxide at a temperature of T0 = 25° C and a pressure of P0 = 9.00 × 105 Pa. This molecule should be treated as a diatomic ideal gas with active vibrational modes. Part (a) In this model, how many degrees of freedom does each molecule of carbon monoxide have? Part (b) The temperature of the gas increases by 10° C. Select the process that has occurred from the choices below. Part (c) Calculate the pressure of the gas in pascal at this increased temperature. Part (d) Calculate the change to the internal energy of the gas in joules. Part (e) Calculate the change in the entropy of the gas in joules per kelvin. I know you cannot answer all parts however manyuo can will helparrow_forward
- Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage LearningPhysics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning