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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
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Check out a sample textbook solution- A molecule in a gas undergoes about 1.0 × 109 collisions in each second. Suppose that (a) every collision is effective in deactivating the molecule rotationally and (b) that one collision in 10 is effective. Calculate the width (in cm-1) of rotational transitions in the molecule.arrow_forwardThe Earth's atmosphere is composed of about 78 percent nitrogen, 21 percent oxygen, 0.9 percent argon, and 0.1 percent other gasses. To find out why these gasses are "trapped" in the earth's atmosphere, consider a projectile with mass m that is about to launch vertically upward from earth. a. Ignore air resistance, show that the projectile can only escape the magnetic pull of the earth if it is launched vertically upward with a kinetic energy greater than mgRearth, where g = 9.80 m/s? and the earth's radius Rearde = 6378 km. b. Compute the temperature required by a nitrogen molecule (molar mass 28.0 g/mol) and an oxygen molecule (molar mass 32 g/mol) to achieve the average translational kinetic energy required to escape earth? c. Repeat part (b) for the moon, for which g = 1.63 m/s? and Rmoon = 1740 km. d. Present your conclusion on the atmosphere of earth and moon based on the results from parts (b) and (c).arrow_forwardThe potential energy of two atoms in a diatomic molecule is approximated by U(r) = a/r12-b/r6, where r is the spacing between atoms and a and b are positive constants. Suppose the distance between the two atoms is equal to the equilibrium distance found in part A. What minimum energy must be added to the molecule to dissociate it - that is, to separate the two atoms to an infinite distance apart? This is called the dissociation energy of the molecule. Express your answer in terms of the variables a and b. For the molecule CO, the equilibrium distance between the carbon and oxygen atoms is 1.13\times 10-10m and the dissociation energy is 1.54\times 10-18J per molecule. Find the value of the constant a. Express your answer in joules times meter in the twelth power. Find the value of the constant b. Express your answer in joules times meter in the sixth power.arrow_forward
- One description of the potential energy of a diatomic molecule is given by the Lennard–Jones potential, U = (A)/(r12) - (B)/(r6)where A and B are constants and r is the separation distance between the atoms. For the H2 molecule, take A = 0.124 x 10-120 eV ⋅ m12 and B = 1.488 x 10-60 eV ⋅ m6. Find (a) the separation distance r0 at which the energy of the molecule is a minimum and (b) the energy E required to break up theH2 molecule.arrow_forwardMy physics class has turned to online due to the COVID-19 and I am having trouble with some of the word problems my teacher gave me to work on. She presented me with: I am contemplating a career change. If I had 500g of stolen gold jewelry, how much energy would I have to add to it to melt it (so that it was not longer recognizable)? Pretend that it is pure gold. Would you be able to show me how to accomplish this problem so I can see the steps? Thank you for your time. John Paytonarrow_forwardInterstellar space is quite different from the gaseous environments we commonly encounter on Earth. For instance, a typical density of the medium is about 1 atom cm−3 and that atom is typically H; the effective temperature due to stellar background radiation is about 10 kK. Estimate the diffusion coefficient and thermal conductivity of H under these conditions. Compare your answers with the values for gases under typical terrestrial conditions. Comment: Energy is in fact transferred much more effectively by radiation.arrow_forward
- An expensive vacuum system can achieve a pressure as low as 1.00 x 10-8 N/m² at 21.5 °C. How many atoms N are there in a cubic centimeter at this pressure and temperature? The Boltzmann constant k = 1.38 x 10-23 J/K. O 2.46 x 106 atoms O 2.44 x 104 atoms O 2.46 x 105 atoms O 2.46 x 107 atoms O 2.46 x 108 atomsarrow_forwardA gas consists of three atoms with access to three different quantum states with the same energy. How many different microstates can be formed from these quantum levels for the case of the Fermi-Dirac gas, where the atoms are differentiated but only one atom is allowed in each state.arrow_forwardYou are using a microscope to view a particle of lycopodium powder suspended in a drop of water on a microscope slide. As water molecules bombard the particle, it "jitters" about in a random motion (Brownian motion). The particle's average kinetic energy is the same as that of a molecule in an ideal gas (K = 3KBT). The particle (assumed to be spherical) has a density of 250 kg/m³ in water at 17ºC. 2 (a) If the particle has a diameter d, determine an expression for its rms speed in terms of the diameter d. (Enter your answer as a multiple of d-³/2. Assume is in m/s and d-3/2 is in m-3/2. Do not include units in your answer.) Vrms d-3/2 Vrms (b) Assuming the particle moves at a constant speed equal to the rms speed, determine the time required for it to travel a distance equal to its diameter. (Enter your answer as a multiple of d5/2. Assume t is in seconds and d5/2 is in m5/2. Do not include units in your answer.) d5/2 t = (c) Evaluate the rms speed (in mm/s) and the time (in ms)…arrow_forward
- I'm hoping for a good explanation of how to do this. I'm also wondering why it matters if the configuration is linear or nonlinear? A triatomic molecule can have a linear configuration, as does CO2 (Figure a), or it can be nonlinear, like H2O (Figure b). Suppose the temperature of a gas of triatomic molecules is sufficiently low that vibrational motion is negligible. (a) What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant (R) if the molecules are linear?Eint/nT = ? (b) What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant (R) if the molecules are nonlinear?Eint/nT = ? At high temperatures, a triatomic molecule has two modes of vibration, and each contributes 0.5R to the molar specific heat for its kinetic energy and another 0.5R for its potential energy. (c) Identify the high-temperature molar specific heat at constant volume for a triatomic ideal gas of the linear molecules. (Use…arrow_forwardIn 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_forwardn=1 11. The ground state energy of a free Fermi gas comprising N particles in a box of volume V in terms of Part = N the Fermi energy Ep is NORT A) NEF B) NEF A) PR B) P₁ = e C) P₁₁ = e D) Pn=f NEF D) NEF E) 2 N 12. For a given macrostate of a system, let P be the probability that the system is in the microstate 14 >. The corresponding Gibbs entropy is given by S-k En PinPn. We can obtain the familiar Boltzmann entropy S = k In from the given Gibbs entropy if the probability distribution is, (♫ is the number of accessible microstates), = hw WEE = Ef- nt & hun) 1 A) E B) E2 2 N ~/M D) E E) Independent of energy E + 2 E) P₁ = 2² = 13.For a system with linear dispersion E(k)= hvk, in three dimensions, the density of states at energy E depends on energy as + 1 h mk + / buk 2n=N M/M. rim Ef = 1/1/₂ n = q S = -x En Pulupn. S = Kenn -* & Plukk € (K) = hvkarrow_forward
- Physics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage LearningPhysics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning