Show that for an ideal gas 12 c(N)E 3N/2-1 VN, where c(N) is N-dependent proportionality constant.
B.2 The multiplicity of a monatomic ideal gas is given by 2 = f(N)VN U3N/2, where V is the volume occupied by the gas, U its internal energy, N the number of particles in the gas and f(N) a complicated function of N. [2] (i) Show that the entropy S of this system is given by 3 S = Nkg In V + ŽNkg In U + g(N), where g(N) is some function of N. (ii) Define the temperature T...
2) Next week, we will show that the partition function for a monatomic ideal gas is given by Q(N,V,T) - 1 ( 2mk,T 30/2 ? N 422) VN where m is the mass of the gas molecules and h is Planck's constant. Derive expressions for the pressure and energy from this partition function.
Statistical_Mechanics 2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D. It is contained in an area of dimensions L x L. The particle mass is m. (a) Find the density of states D(e) N/L2 (b) Find the Fermi energy as a function of the particle density n = (c) Find the total energy as a function of the Fermi energy ef. (d) Find the chemical potential u as a function of n and T....
Ideal Bose gas (a) Consider a 2D ideal Bose gas with density of state D (e) = DoL2, show that Bose- Einstein condensation is not possible in such a gas. (b) Consider a 4D ideal Bose gas with density of state D(e) = DOL6, find the Bose- Einstein condensation temperature in terms of Do, n = N/La, and a dimensionless integral FM A = (6) ex 1 12 Ideal Bose gas (a) Consider a 2D ideal Bose gas with density...
• (6.45) For a monatomic ideal gas, derive LO S = = Nkr In .N • And DE V и — = -krT in AN Nvo I TV Partition Function for an Ideal Gas . For one particle Z=e-E(s)/kp7 Vo = ve = (v2nimkot) • For N particles 1/VN V ve)
r the recurrence relation o. Consider T(n) = Vn T(Vn) + n a. Why can't you solve this with the master theorem? b. S t involves a constant C, tell me what it is in terms of T(O), T(1), or whatever your inequality by induction. Show the base case. Then show the how that T( n)= 0(n lg ig n). First, clearly indicate the inequality that you wish to hen proceed to prove the inductive hypothesis inductive case, and clearly...
MODEL 2: THE IDEAL GAS LAW Gases demonstrate all of the following properties V x 17p VT Boyle's Law Charles's law Avogadro's Law CRITICAL THINKING QUESTIONS GUM OV which of the following incorporates all the gas behavior observed VONTP V." V Decall that yox is the same thing as y e x in which is a proportionality constant Rewrite the correct expression from CTQ5 with an equals sign, using the symbol Ras the proportionality constant. 7. Starting with your expression...
Problem 1: Ideal Gas Law Problem 1. The ideal gas law states PV nRT where P, V, and T are the pressure, volume and absolute temperature; n is the number of moles of gas; and R is the the ideal gas constant. Consider a 1-gallon canister of gas at a pressure of 1 atm. Answer the following questions: 1. How much energy would be needed to increase the pressure of the closed canister to 50 psi without changing its volume?...
2, For each of these sets. A={3n : n E N), B = {r E R : x2 < 7), and C = {x E R : x < 12), (i) Is the set bounded above? Prove your answer.] ( .] ii) Is the set bounded below? Prove your answer answer the following questions:
Show that Discrete Fourier transform matrix W /Vn is a unitary matrix, where e2j/n and i = V-1, W = 1 1 1 1 1 (n-1) w-2(n-1) w-2 1 z- w~(n-1) -2(n-1) (n-1)(n-1) Show that Discrete Fourier transform matrix W /Vn is a unitary matrix, where e2j/n and i = V-1, W = 1 1 1 1 1 (n-1) w-2(n-1) w-2 1 z- w~(n-1) -2(n-1) (n-1)(n-1)