An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
expand_more
expand_more
format_list_bulleted
Question
Chapter 8.1, Problem 13P
To determine
Verification of the statement that total energy
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Consider an ideal gas containing N atoms in a container of volume Pressure P, and absolute temperature T1 (not to be confused with K. E. T). Use the virtual theorem to derive the equation of state for a perfect gas.
You are studying a gas known as "gopherine" and looking in the literature you find that someone has
reported the partition function for one molecule of this gas,
5/2
AzT
q(V, T) = )
%3D
h?m
Assume that the molecules are independent and indistinguishable. Derive the expressions for the
energy, (E), for this gas. Give your answers in terms of N, kg, T. V and the constants A and B.
O (E) = NkaT
ㅇ (E) =D NkaT
ㅇ (E) %3D NkaT-
O (E) = ANKET -
O (E) = - T
ㅇ (E)=D 쑤-
Problem 1:
This problem concerns a collection of N identical harmonic oscillators (perhaps an
Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf,
and so on.
a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving
1-x
the formula by long division. Prove it by first multiplying both sides of the
equation by (1 – x), and then thinking about the right-hand side of the resulting
expression.
b) Evaluate the partition function for a single harmonic oscillator. Use the result of
(a) to simplify your answer as much as possible.
c) Use E = -
дz
to find an expression for the average energy of a single oscillator.
z aB
Simplify as much as possible.
d) What is the total energy of the system of N oscillators at temperature T?
Knowledge Booster
Similar questions
- An approximate partition function for a gas of hard spheres can be obtained from the partition function of a monatomic gas by by V-b, where b is related to the volume of the W hard spheres. Derive expressions for the energy and the pressure of this system. (Use the following as necessary: b, kg, N, T, and V.) replacing V in Q(N, V, B) = [9(V, P)] (where q(V, B) = (22m) 3/2v) by V N! h²ß (E) (P) =arrow_forwardProblem 1: In statistical mechanics, the internal energy of an ideal gas is given by: N. aNkB 2/3 (3NKB U = U(S,V) = е where a is a constant. 1- Show that the variation of the internal energy is given by: 2 dS - \3V 2 dU = dV \3NkB 2- Using the fundamental relation of thermodynamic dU = T.ds – p. dV, show that the equation of state PV = nRT follows from the first expression of U.arrow_forwardA system containing a single particle in a rigid container at temperature 100 K has a translational partition function of Ztrans = 1.5 × 1012. (a) If the mass of the particle is doubled, what is the new value for ztrans? Report your answer in scientific notation (in Blackboard, 1.23 x 1045 is written 1.23E45).arrow_forward
- Using MATLAB editor, make a script m-file which includes a header block and comments: Utilizing the ideal gas law: Vmol= RT/P Calculate the molecular volume where: R = 0.08206 L-atm/(mol-K) P = 1.015 atm. and T = 270 - 315 K in 5 degree increments Make a display matrix which has the values of T in the first column and Vmol in the second column Save the script and publish function to create a pdf file from the script in a file named "ECE105_Wk2_L1_Prep_1"arrow_forwardFor 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.arrow_forwardPlease answer all parts: Problem 3: There are lots of examples of ideal gases in the universe, and they exist in many different conditions. In this problem we will examine what the temperature of these various phenomena are. Part (a) Give an expression for the temperature of an ideal gas in terms of pressure P, particle density per unit volume ρ, and fundamental constants. T = ______ Part (b) Near the surface of Venus, its atmosphere has a pressure fv= 91 times the pressure of Earth's atmosphere, and a particle density of around ρv = 0.91 × 1027 m-3. What is the temperature of Venus' atmosphere (in C) near the surface? Part (c) The Orion nebula is one of the brightest diffuse nebulae in the sky (look for it in the winter, just below the three bright stars in Orion's belt). It is a very complicated mess of gas, dust, young star systems, and brown dwarfs, but let's estimate its temperature if we assume it is a uniform ideal gas. Assume it is a sphere of radius r = 5.7 × 1015 m…arrow_forward
- Problem 2: Average values Prove that, for any system in equilibrium with a reservoir at temperature T, the average 1 дZ value of the energy is Ē = – z дв In Z, where ß = 1/kT. These formulas can be дв extremely useful when you have an explicit formula for the partition function.arrow_forwarddefine the chemical potential in terms of derivatives of the ĈE energy E and enthalpy H. For a one component system, these are u= ON and Evaluate these expressions for an ideal gas and compare to ON µ = -kT In (kT/PA³ ) from H=| P,s OF and |= 1 TP ON v.rarrow_forwardP2D.8 Use the fact that (∂U/∂V)T = a/Vm2for a van der Waals gas (Topic 1C)to show that μCp,m ≈ (2a/RT) − b by using the definition of μ and appropriaterelations between partial derivatives. Hint: Use the approximation pVm ≈ RTwhen it is justifiable to do so.arrow_forward
- Consider N identical harmonic oscillators (as in the Einstein floor). Permissible Energies of each oscillator (E = n h f (n = 0, 1, 2 ...)) 0, hf, 2hf and so on. A) Calculating the selection function of a single harmonic oscillator. What is the division of N oscillators? B) Obtain the average energy of N oscillators at temperature T from the partition function. C) Calculate this capacity and T-> 0 and At T-> infinity limits, what will the heat capacity be? Are these results consistent with the experiment? Why? What is the correct theory about this? D) Find the Helmholtz free energy from this system. E) Derive the expression that gives the entropy of this system for the temperature.arrow_forward(b) Consider the following heat system on the real line: U - U = 0, XER, 1>0 %3D u(x, 0) = | sin x), rER. i. Use the fundamental solution of the heat equation to write down a solution u to the system above as an integral. ii. Show that the solution u that you have found is bounded by 1.arrow_forwarde. Consider one unit cell and assume the length of the side of the cube is “a”. Remember that “a” is the distance between the centers of two adjacent atoms. How long is “a”, the edge of a unit cell, in terms of radius, r, of an atom? Write also your answer in the summary table.Answer: __________f. Based on the earlier questions, a simple cubic cell has the equivalent of only 1 atom. Recall the volume of sphere with radius, r, is expressed as V = 4/3 πr3. With this information, find the total volume of all the spheres in this unit cell, expressed in terms of r. (Hint: To do this, take the total number of atoms and multiply it by the volume of one atom, with radius, r)Answer: __________arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning
Modern Physics
Physics
ISBN:9781111794378
Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:Cengage Learning