An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Chapter 8.1, Problem 10P
To determine
Calculation and graphical representation of second virial coefficient for gas of molecules and graphical representation of data for nitrogen.
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Let 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.
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.
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.
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- The Clausius-Clapeyron relation 5.47 is a differential equation that can, in principle, be solved to find the shape of the entire phase-boundary curve. To solve it, however, you have to know how both L and ~V depend on temperature and pressure. Often, over a reasonably small section of the curve, you can take L to be constant. Moreover, if one of the phases is a gas, you can usually neglect the volume of the condensed phase and just take ~V to be the volume of the gas, expressed in terms of temperature and pressure using the ideal gas law. Making all these assumptions, solve the differential equation explicitly to obtain the following formula for the phase boundary curve:This result is called the vapor pressure equation. Caution: Be sure to use this formula only when all the assumptions just listed are valid.arrow_forwardConsider a large system of N indistinguishable, noninteracting molecules (perhaps in an ideal gas or a dilute solution). Find an expression for the Helmholtz free energy of this system, in terms of Z1, the partition function for a single molecule. (Use Stirling's approximation to eliminate the N!) Then use your result to find the chemical potential, again in terms of Z1.arrow_forwardProblem 6.33. Calculate the most probable speed, average speed, and rms speed for oxygen (O₂) molecules at room temperature.arrow_forward
- rork 28 the ofnly Problem 1.31. Imagine some helium in a cylinder with an initial volume of 1 liter and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 liters, in such a way that its pressure rises in direct proportion to its volume. (a) Sketch a graph of pressure vs. volume for this process. (b) Calculate the work done on the gas during this process, assuming that there are no "other" types of work being done. (c) Calculate the change in the helium's energy content during this process. (d) Calculate the amount of heat added to or removed from the helium during this process. (e) Describe what you might do to cause the pressure to rise as the helium еxpands. Problem 1.33. An ideal gas is made to undergo the cyclic process shown in Figure 1.10(a). For each of the steps A, B, and C, determine whether each of the following is positive, nogative, or zero: (a) the work done on the gas; (b) the change in the energy content of the gas; (c) the heat…arrow_forwardRecall Problem 1.34, which concerned an ideal diatomic gas taken around a rectangular cycle on a PV diagram. Suppose now that this system is used as a heat engine, to convert the heat added into mechanical work. (a) Evaluate the efficiency of this engine for the case V2 = 3V1 , P2 = 2P1. (b) Calculate the efficiency of an "ideal" engine operating between the same temperature extremes.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_forward
- 3.2. The simulation of parameter-distributed processes is connected with discretization in space and time. The distribution of changes in the temperature x of a heated at the front massive long metal piece is described by the following partial differential equation: Əx(z,t) F x(z,t) a[x(z,t) - 0,]+b- Əz? where 0, is the ambient temperature, a and b are constants. Derive the discrete model by applying discretization first with respect to z (z; = i.Az) and after that with respect to t (tx = k.At), using backward finite differences for the corresponding derivatives.arrow_forwardStatical Mechanics (Thermal and Statical Physics) Instruction: Write ALL the solutions of this (necessary or and not direct answer). Write also the equations that are needed to solve for a certain problem. Thank you. Problem: Now, we have the number of microstates and in between E and E + ∆E in isolated system of N particles in the volume V is given by: (Please see the image attached) Where a,b, c are constants. Note: Answer also letter A-Darrow_forwardI need help with C) my answer of-192 is in correct Suppose that while pumping up a bike tire, we fairly rapidly compress 1900 cm3 of air from atmospheric pressure and room temperature to a pressure of about 5 atm (which is about 60 psi above atmospheric pressure, which is what a tire gauge would read). (a) What is this packet of air’s volume as it enters the tire? The volume of the packet of air is _601.85cm3. (b) What is its final temperature? The final temperature is 475 K. (c) How much work did we do to compress it? The work done to compress air is _______J.arrow_forward
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