A monatomic particle at rest can be in either of two energy levels, separated by an energy ε.Consider a dilute gas of N such particles at a fixed temperature T.a) Write down the probability for a single particle to be moving with momentum h❘k and in theexcited energy level.b) Derive the canonical partition function of a single particle.c)[You may use the Gaussian integral:୮dx x²e-x²y²=4y3and approximate the density of states in three dimensions as g(k)dk ≈k²dk]Hence, or otherwise, find the internal energy per particle of this gas and comment on howit behaves at large temperature

it appears to be a multipart question on a cantilever beam with a point load. Here's how to approach…

Answered: A monatomic particle at rest can be in…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Consider the function v(1,2) =( [1s(1) 3s(2) + 3s(1) 1s(2)] [x(1) B(2) + B(1) a(2)] Which of the following statements is incorrect concerning p(1,2) ? a. W(1,2) is normalized. Ob. The function W(1,2) is symmetric with respect to the exchange of the space and the spin coordinates of the two electrons. OC. y(1,2) is an eigenfunction of the reference (or zero-order) Hamiltonian (in which the electron-electron repulsion term is ignored) of Li with eigenvalue = -5 hartree. d. The function y(1,2) is an acceptable wave function to describe the properties of one of the excited states of Lit. Oe. The function 4(1,2) is an eigenfunction of the operator S,(1,2) = S;(1) + S,(2) with eigenvalue zero.
3. The classical partition function of a gas of noninteracting indistinguishable particles is written as exp{- N! 2m Z= where N is the number of particles of mass m, r, and p, are the position and the momentum of the ith particle, B = 1/(kpT), and Tis the temperature of the gas. The volume of the gas is V. (a) Find the analytic expression of the partition function of the gas. (b) Obtain the total mean energy E of the gas from the partition function. (c) Obtain the entropy S of the gas from the partition function and the total mean energy. Lexp(-x³xdx = Va Hint:
please answer c) only 2. a) A spinless particle, mass m, is confined to a two-dimensional box of length L. The stationary Schrödinger equation is - +a) v(x, y) = Ev(x, y), for 0 < r, y < L. The bound- ary conditions on ý are that it vanishes at the edges of the box. Verify that solutions are given by 2 v(1, y) sin L where n., ny = 1,2..., and find the corresponding energy. Let L and m be such that h'n?/(2mL²) = 1 eV. How many states of the system have energies between 9 eV and 24 eV? b) We now consider a macroscopic box (L of order cm) so that h'n?/(2mL?) ~ 10-20 eV. If we define the wave vector k as ("", ""), show that the density of states g(k), defined such that the number of states with |k| between k and k +dk is given by g(k)dk, is Ak 9(k) = 27 c) Use the expression for g(k) to show that at room temperature the partition function for the translational energy of a particle in a macroscopic 2-dimensional box is Z1 = Aoq, where 2/3 oq = ng = mk„T/2nh?. Hence show that the average…
What is y(x) given that d²y dx² +4e-2x = 0 where you have the boundary conditions: dy dx y = yo (a constant) at You have to integrate twice as this is a second order differential equation so you need two boundary conditions to get the complete solution. (Most people are better at differentiating then integrating so check your answer by differentiating it!) = q (a constant) at x = 0 x = 0
Please
Answer in 90 minutes please. Count as double the questions if necessary.
Consider a system of N distinguishable particles with two energy levels for each particle, a ground state with energy zero and an excited state with energy ɛ > 0. What is the average energy per particle as the system temperature T → ∞ ?
Suppose there are 30 identical blue balls and 15 identical red balls in a bag. Without looking inside the bag, take a ball out of it and write down its color. Repeat this 10 times. What is the probability that 6 of these 10 balls are blue in each of the following situations? 1) What if we return the ball to the bag after each selection and then make the next selection? 2) What if we do not return the ball to the bag after each selection and then make the next selection?
Problem 2) Consider the following Maxwell Boltzmann distribution of molecular speeds: P(v) = 4( m 27kBT. mp² v²e 2kgT To calculate average values for say f(v) (function of v) one just integrates f(v) with P(v)dv from zero to infinity = P(v)f(v)dv, where signifies average of f(v). Of course, the distribution should be normalized: P(v)dv=1, (is a requirement for any probability distribution). a) Check the last equation. b) Calculate the average of v. c) Calculate the average of v². d) Calculate from c) the RMS value of the speed. e) Calculate the most probable value of v. f) Square the results of b, d and e and rank them from smallest to the largest value.

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