(3) Let us consider a particle that can take three energies —ɛ, 0, ɛ. This particle is immersed in abath of constant temperature T.(a) Calculate the probability P for this particle to have the energy -ε. Evaluate this probabilityP in the limits of T = 0 and T = ∞o.(b) Calculate the partition function Z and the Helmholtz free energy F.

Based on the information in the image, we can solve for the probability (P) of a particle having…

Answered: (3) Let us consider a particle that can…

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)...

See similar textbooks

Similar questions

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 the excited energy level. b) Derive the canonical partition function of a single particle. c) [You may use the Gaussian integral: ୮ dx x²e-x²y² = 4y3 and 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 how it behaves at large temperature
For an ideal gas of classical non- interacting atoms in thermal equilibrium, the Cartesian component of the velocity are statistically independent. In three dimensions, the probability density distribution of the velocity is: where σ² = kBT m P(Vx, Vy, Vz) = (2nо²)-³/² exp 20² 1. Show that the probability density of the velocity is normalized. 2. Find an expression of the arithmetic average of the speed. 3. Find and expression of the root-mean-square value of the speed. 4. Estimate the standard deviation of the speed.
Want solution of all parts.... ASAP
Show that at high enough temperatures (where KBT » ħw) the partition function of a simple quantum mechanical harmonic oscillator is approximately Z≈ (Bħw)-¹ Then use the partition function to calculate the high temperature expressions for the internal energy U, the heat capacity Cy, the Helmholtz function F and the entropy S.
The partition funetion for the ensemble characterized by constant V, E, and G = µÑ is given to a very good approximation by ø(V, E, µN)=Q(N,V,E)eBHN, where G = µN is the Gibbs energy (µ is the chemical potential and N is the average number of particles). Find an expression for the characteristic thermodynamic function for this ensemble in terms of the partition function ø(V, E, µN).
A. (a) Consider a canonical ensemble having N particle, V volume and at T temperature. Write down the expression of partition function (Q(N,V,T)) of this canonical ensemble in terms of the microstate energy Ej. (b) Write down the expression for Helmhotz free energy (A) and pressure (P) in terms of Q(N,V.T). (c) Now, assume that for a system of dense gas you can write down the Q(N,V,T) as, 1 (2amk,T Q(N,V,T)= N! (V- Nb)" e Treat a and b as constants. Get the expression for pressure (P) in terms of V, a, b, N, kg and T. Rearrange that expression to get a form where in the RHS of the equation will have Nk T. Identify the equation.
Describe the equipartition theorem, which holds in classical (non-quantum) thermodynamics for systems in thermal equilibrium under technical conditions?
The energy spectrum of a particle consists of four states with energies 0,e, 2 e,3 e. Let Z, (T),Z, (T) and Ze(") denote the canonical partition functions for four non- interacting particles at temperature T. The subscripts B, F and C corresponds to bosons, fermions and distinguishable classical particles, respectively. Let y= exp k„T Which one of the following statements is true about ZB (T), ZF (T) and Zc (T)? (a) They are polynomials in y of degree 12,6 and 12, respectively. (b) They are polynomials in y of degree 16,10 and16, respectively (c) They are polynomials in y of degree 9,6 and 12, respectively. (d) They are polynomials in y of degree 12,10 and 16, respectively.
Prove that the free energy of a semi-classical ideal gas is given by the following relationship (ZNY F = -kT In N!
Obtain the expression for the average thermal energy when the partition function is Z: ħo (A) 극청고 2 (B) kzT 1 (C) -k,T 4 (D) 4k,T
The probability p; of occupying an available state j is: Pj 9je-BEj Z where g; is the degeneracy of the state and 1/ẞ=kBT. Derive an expression for the temperature I in terms of the probabilities of occupation of two different states i and k.
Given the internal energy U and entropy S of N weakly interacting particles in a closed system with fixed volume V. U = NkgT² (27In 2) U S = Nkg lnz + T (a) Prove the Helmholtz free energy (b) Prove the Pressure of the system is F = -NkBT ln z P = Nk Tln z) ( T

SEE MORE QUESTIONS

Recommended textbooks for you

College Physics

Physics

ISBN:

9781305952300

Author:

Raymond A. Serway, Chris Vuille

Publisher:

Cengage Learning

University Physics (14th Edition)

Physics

ISBN:

9780133969290

Author:

Hugh D. Young, Roger A. Freedman

Publisher:

PEARSON

Introduction To Quantum Mechanics

Physics

ISBN:

9781107189638

Author:

Griffiths, David J., Schroeter, Darrell F.

Publisher:

Cambridge University Press

College Physics

Physics

ISBN:

9781305952300

Author:

Raymond A. Serway, Chris Vuille

Publisher:

Cengage Learning

University Physics (14th Edition)

Physics

ISBN:

9780133969290

Author:

Hugh D. Young, Roger A. Freedman

Publisher:

PEARSON

Introduction To Quantum Mechanics

Physics

ISBN:

9781107189638

Author:

Griffiths, David J., Schroeter, Darrell F.

Publisher:

Cambridge University Press

Physics for Scientists and Engineers

Physics

ISBN:

9781337553278

Author:

Raymond A. Serway, John W. Jewett

Publisher:

Cengage Learning

Lecture- Tutorials for Introductory Astronomy

Physics

ISBN:

9780321820464

Author:

Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden

Publisher:

Addison-Wesley

College Physics: A Strategic Approach (4th Editio…

Physics

ISBN:

9780134609034

Author:

Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field

Publisher:

PEARSON

SEE MORE TEXTBOOKS