An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 5.3, Problem 48P
To determine
To Show:
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.
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?
1-5. In a strictly steady state situation, both the ions and the electrons will follow
the Boltzmann relation
n; = no exp (-4,6/KT,)
For the case of an infinite, transparent grid charged to a potential 6, show that
the shielding distance is then given approximately by
ne 1
AD
€o KT, KT,.
Show that Ap is determined by the temperature of the colder species.
Chapter 5 Solutions
An Introduction to Thermal Physics
Ch. 5.1 - Prob. 1PCh. 5.1 - Consider the production of ammonia from nitrogen...Ch. 5.1 - Prob. 3PCh. 5.1 - Prob. 4PCh. 5.1 - Consider a fuel cell that uses methane (natural...Ch. 5.1 - Prob. 6PCh. 5.1 - The metabolism of a glucose molecule (see previous...Ch. 5.1 - Derive the thermodynamic identity for G (equation...Ch. 5.1 - Sketch a qualitatively accurate graph of G vs. T...Ch. 5.1 - Suppose you have a mole of water at 25C and...
Ch. 5.1 - Suppose that a hydrogen fuel cell, as described in...Ch. 5.1 - Prob. 12PCh. 5.1 - Prob. 13PCh. 5.1 - Prob. 14PCh. 5.1 - Prob. 15PCh. 5.1 - Prob. 16PCh. 5.1 - Prob. 17PCh. 5.2 - Prob. 18PCh. 5.2 - In the previous section 1 derived the formula...Ch. 5.2 - Prob. 20PCh. 5.2 - Is heat capacity (C) extensive or intensive? What...Ch. 5.2 - Prob. 22PCh. 5.2 - Prob. 23PCh. 5.3 - Go through the arithmetic to verify that diamond...Ch. 5.3 - Prob. 25PCh. 5.3 - How can diamond ever be more stable than graphite,...Ch. 5.3 - Prob. 27PCh. 5.3 - Calcium carbonate, CaCO3, has two common...Ch. 5.3 - Aluminum silicate, Al2SiO5, has three different...Ch. 5.3 - Sketch qualitatively accurate graphs of G vs. T...Ch. 5.3 - Sketch qualitatively accurate graphs of G vs. P...Ch. 5.3 - The density of ice is 917kg/m3. (a) Use the...Ch. 5.3 - An inventor proposes to make a heat engine using...Ch. 5.3 - Below 0.3 K the Slope of the 3He solid–liquid...Ch. 5.3 - Prob. 35PCh. 5.3 - Effect of altitude on boiling water. (a) Use the...Ch. 5.3 - Prob. 37PCh. 5.3 - Prob. 38PCh. 5.3 - Prob. 39PCh. 5.3 - The methods of this section can also be applied to...Ch. 5.3 - Suppose you have a liquid (say, water) in...Ch. 5.3 - Ordinarily, the partial pressure of water vapor in...Ch. 5.3 - Assume that the air you exhale is at 35C, with a...Ch. 5.3 - Prob. 44PCh. 5.3 - Prob. 46PCh. 5.3 - Prob. 47PCh. 5.3 - Prob. 48PCh. 5.3 - Prob. 49PCh. 5.3 - The compression factor of a fluid is defined as...Ch. 5.3 - Prob. 51PCh. 5.3 - Prob. 52PCh. 5.3 - Repeat the preceding problem for T/Tc=0.8.Ch. 5.3 - Prob. 54PCh. 5.3 - Prob. 55PCh. 5.4 - Prove that the entropy of mixing of an ideal...Ch. 5.4 - In this problem you will model the mixing energy...Ch. 5.4 - Suppose you cool a mixture of 50% nitrogen and 50%...Ch. 5.4 - Suppose you start with a liquid mixture of 60%...Ch. 5.4 - Suppose you need a tank of oxygen that is 95%...Ch. 5.4 - Prob. 62PCh. 5.4 - Everything in this section assumes that the total...Ch. 5.4 - Figure 5.32 shows the phase diagram of plagioclase...Ch. 5.4 - Prob. 65PCh. 5.4 - Prob. 66PCh. 5.4 - Prob. 67PCh. 5.4 - Plumbers solder is composed of 67% lead and 33%...Ch. 5.4 - What happens when you spread salt crystals over an...Ch. 5.4 - What happens when you add salt to the ice bath in...Ch. 5.4 - Figure 5.35 (left) shows the free energy curves at...Ch. 5.4 - Repeat the previous problem for the diagram in...Ch. 5.5 - If expression 5.68 is correct, it must be...Ch. 5.5 - Prob. 74PCh. 5.5 - Compare expression 5.68 for the Gibbs free energy...Ch. 5.5 - Seawater has a salinity of 3.5%, meaning that if...Ch. 5.5 - Osmotic pressure measurements can be used to...Ch. 5.5 - Because osmotic pressures can be quite large, you...Ch. 5.5 - Most pasta recipes instruct you to add a teaspoon...Ch. 5.5 - Use the Clausius–Clapeyron relation to derive...Ch. 5.5 - Prob. 81PCh. 5.5 - Use the result of the previous problem to...Ch. 5.6 - Prob. 83PCh. 5.6 - Prob. 84PCh. 5.6 - Prob. 85PCh. 5.6 - Prob. 86PCh. 5.6 - Sulfuric acid, H2SO4, readily dissociates into H+...Ch. 5.6 - Prob. 88PCh. 5.6 - Prob. 89PCh. 5.6 - When solid quartz dissolves in water, it combines...Ch. 5.6 - When carbon dioxide dissolves in water,...Ch. 5.6 - Prob. 92P
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.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_forwardProblem 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_forwardShow that the one-particle partition function Z₁ for a 2D ideal gas confined to area A is: A 2²/1 Z₁ = Sarrow_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_forwardUse a computer to reproduce the table and graph in Figure 2.4: two Einstein solids, each containing three harmonic oscillators, with a total of six units of energy. Then modify the table and graph to show the case where one Einstein solid contains six harmonic oscillators and the other contains four harmonic oscillators (with the total number of energy units still equal to six). Assuming that all microstates are equally likely, what is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?arrow_forwardLet Ω be a new thermodynamic potential that is a “natural” function of temperature T, volume V, and the chemical potential μ. Provide a definition of Φ in the form of a Legendre transformation and also write its total differential, or derived fundamental equation, in terms of these natural variables.arrow_forward
- 5. Non-harmonic gas: let us re-examine the generalized ideal gas introduced in the previous section, using statistical mechanics rather than kinetic theory. Consider a gas of N non-interacting atoms in a d-dimensional box of "volume" V, with a kinetic energy H =EAPI". where i, is the momentum of the ith particle.arrow_forwardPressure and bulk modulus of an electron gas. (a) Derive a relation connecting the pressure and volume of an electron gas at 0 K. Hint: Use the result of Problem 1 and the relation between er and electron concentration. The result may be writ- ten as p =(U/V). (b) Show that the bulk modulus B = –V(dp/əV) of an electron gas at 0 K is B = 5p/3 = 10U/9V. (c) Estimate for potassium, using Table 1, the value of the electron gas contribution to B.arrow_forwardFrom the thermodynamic identity dU = t do –p dV +µ dN, for a closed system of fixed composition, show that (*) --(). - p, ƏV V which is a standard relation for any PVT system. (2). - (*), do Hint: Start with the Helmholtz free energy to derive the relationship which is one of the Maxwell relations of thermodynamics.arrow_forward
- Problem 1: Maxwell relations a) Functions encountered in physics are generally well enough behaved that their mixed partial derivatives do not depend on which derivative is taken first. For instance, -P = aslvN S,N Using the smoothness assumption, we find д гди- where each av is taken with S fixed, each is taken with V fixed, and N is always held fixed. From the thermodynamic identity (for U) you can evaluate the partial derivatives in parentheses a nontrivial identity called a Maxwell relation. Go through the derivation of this relation step by step. Derive each Maxwell relation for the thermodynamic identities discussed in the text (for U,H,F,G, not necessarily in this order), as shown below: a) lav S,N as/y.N b) s/ P.N N*() d) - (), c) ат/у av ат/ р N You might find the following equations helpful: dU = TdS – PdV + µdN dF = -SdT – PdV + µdN dG = -SdT + VdP + µdN dH = TdS + VdP + µdNarrow_forwardCalculate the change in Gibbs and Helmholtz free energies for one mole of an ideal gas that undergoes a change in pressure from 2.0 bar to 1.0 bar at a constant temperature of 298.2K. Clearly show and state the partial derivative (sensitivity) in G that you need to integrate.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_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