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 7.4, Problem 46P
(a)
To determine
The Helmholtz free energy.
(b)
To determine
The value of entropy of the system.
(c)
To determine
The value of pressure of the system.
(d)
To determine
The value of free energy the system.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents 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.
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?
calculate the heat capacity of an Einstein solid in the low-temperature limit. Sketch the predicted heat capacity as a function of temperature. (Note: Measurements of heat capacities of actual solids at low temperatures do not confirm the prediction that you will make in this problem.
Chapter 7 Solutions
An Introduction to Thermal Physics
Ch. 7.1 - Prob. 1PCh. 7.1 - Prob. 3PCh. 7.1 - Prob. 4PCh. 7.1 - Show that when a system is in thermal and...Ch. 7.1 - Prob. 7PCh. 7.2 - Prob. 8PCh. 7.2 - Prob. 9PCh. 7.2 - Prob. 11PCh. 7.2 - Prob. 12PCh. 7.2 - Prob. 13P
Ch. 7.2 - Prob. 14PCh. 7.2 - Prob. 15PCh. 7.2 - Prob. 16PCh. 7.2 - Prob. 17PCh. 7.2 - Prob. 18PCh. 7.3 - Prob. 19PCh. 7.3 - Prob. 20PCh. 7.3 - Prob. 21PCh. 7.3 - Prob. 22PCh. 7.3 - Prob. 24PCh. 7.3 - Prob. 25PCh. 7.3 - Prob. 26PCh. 7.3 - Prob. 29PCh. 7.3 - Prob. 32PCh. 7.3 - Prob. 33PCh. 7.3 - Prob. 34PCh. 7.4 - Prob. 37PCh. 7.4 - Prob. 38PCh. 7.4 - Prob. 39PCh. 7.4 - Prob. 40PCh. 7.4 - Prob. 41PCh. 7.4 - Prob. 42PCh. 7.4 - Prob. 43PCh. 7.4 - Prob. 44PCh. 7.4 - Prob. 45PCh. 7.4 - Prob. 46PCh. 7.4 - Prob. 47PCh. 7.4 - Prob. 48PCh. 7.4 - Prob. 49PCh. 7.4 - Prob. 50PCh. 7.4 - Prob. 51PCh. 7.4 - Prob. 52PCh. 7.4 - Prob. 53PCh. 7.4 - Prob. 54PCh. 7.4 - Prob. 55PCh. 7.4 - Prob. 56PCh. 7.5 - Prob. 57PCh. 7.5 - Prob. 58PCh. 7.5 - Prob. 59PCh. 7.5 - Prob. 60PCh. 7.5 - The heat capacity of liquid 4He below 0.6 K is...Ch. 7.5 - Prob. 62PCh. 7.5 - Prob. 63PCh. 7.5 - Prob. 64PCh. 7.6 - Prob. 65PCh. 7.6 - Prob. 66PCh. 7.6 - Prob. 67PCh. 7.6 - Prob. 68PCh. 7.6 - If you have a computer system that can do...Ch. 7.6 - Prob. 70PCh. 7.6 - Prob. 71PCh. 7.6 - Prob. 72PCh. 7.6 - Prob. 73PCh. 7.6 - Prob. 75P
Knowledge Booster
Similar questions
- Find the number density N/V for Bose-Einstein condensation to occur in helium at room temperature (293 K). Compare your answer with the number density for an ideal gas at room temperature at 1 atmosphere pressure.arrow_forwardWhich one of the following statements about the exchange energy of the few lowest excited states of helium, in which the two electrons are in different subshells, is incorrect? Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. b с d e In the absence of the exchange energy, the degeneracy between 1s¹2s¹ and 1s¹2p¹ configurations would only be lifted by relativistic corrections. For a given configuration and L the exchange energy favours (ie the energy is lower for) S-1 compared with S=0 O For a given configuration and L the exchange energy favours states that are spatially anti-symmetric The exchange energy gives larger shifts in the levels than relativistic corrections L, S remain good quantum numbers in the presence of the exchange energyarrow_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_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. a) If the classical theory of specific heat were valid, what would be the thermal energy of one mole of Cu at the temperature T=0D? The Debye temperature for Cu is 340°K. b) Calculate the actual thermal energy according to the Debye theory (use Fig. 3.13), and compare with the classical value obtained above. (For the purpose of this calculation, you may approximate the Debye curve by a straight line joining the origin to the point on the Debye curve at T=0p.) c) What is the order of magnitude of the maximum displacement of a Cu atom at the Debye temperature? Compare this displacement with the interatomic distance. C., cal/g-mol - "K Cu, 343°K Ag. 226°K ■ Pb, 102°K x C, 1860°K 0.5 T/OD Fig. 3.13 Specific heats versus reduced temperature for four substances. Numbers refer to Debye temperatures. Note the high Debye temperature for diamond.arrow_forwardConsider the half oscillator" in which a particle of mass m is restricted to the region x > 0 by the potential energy U(x) = 00 for a O where k is the spring constant. What are the energies of the ground state and fırst excited state? Explain your reasoning. Give the energies in terms of the oscillator frequency wo = Vk/m. Formulas.pdf (Click here-->)arrow_forward
- Consider a black body of surface area 20.0 cm² and temperature 5 000 K. (a) How much power does it radiate? (b) At what wavelength does it radiate most intensely? Find the spectral power per wavelength at (c) the wavelength from part b. For answer to (d-i) present your results in the form of the table below. Expect very high exponents of the order 10+1000 d) 1.00 nm (an x- or y ray), e) 5.00 nm (ultraviolet light or an x-ray), f) 400 nm (at the boundary between UV and visible light), g) 700 nm (at the boundary between visible and infrared light), h) 1.00 mm (infrared light or a microwave) 2πheA hc ehc lik gT P(1), W/m - 1 25 ik;T 1.00 nm 5.00 nm 400 nm 580 nm and i) 10.0 cm (a microwave or radio wave). j) About how much power does the object radiate as visible light? (8 points total) 700 nm 1.00 mm 10.0 cmarrow_forwardU = PV P = AT2 Find F0(U,V,N) and F1(U,V,N) After that use, Gibbs-Duhem to prove dF2=0 and finally apply Euler relation to find S=S(U,V,N)arrow_forwardCalculate the following quantities for a three-dimensional classical ideal gas: (a) (vx), (b) (v?). (c) (v*vx), (d) (vàvy), (e) ((Vx + () (v²v?). bv,)^) (b a constant),arrow_forward
- Consider a material that is well approximated by a black body. The max. emitted irradiance is found at a wavelength of 445 nm. What is the temperature of the black body. Provide your answer in Kelvin.arrow_forwardSuppose someone is running a fever of 102.0° F (average being 98.6° F). How much more power (in Watts) does this person radiate than when this person is at normal human body temperature, assuming the fever causes no swelling or edema, or emaciation? Remember that for thermal radiators, intensity I = sigma T^4; where sigma is the Stefan-Boltzmann constant and T is temperature in Kelvins.arrow_forwardA. (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.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningUniversity Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University Press
- Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningLecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-WesleyCollege Physics: A Strategic Approach (4th Editio...PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON
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