An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Chapter 5.1, Problem 8P
Derive the
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Take U = U(T, P) and V = V (T, P)
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.
. An ideal classical gas composed of N particles, each of mass m, is enclosed in a vertical cylinder of
height L placed in a uniform gravitational field (of acceleration g) and is in thermal equilibrium;
ultimately, both N and L → ∞. Evaluate the partition function of the gas and derive expressions for
its major thermodynamic properties. Explain why the specific heat of this system is larger than that
of a corresponding system in free space.
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
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- 3.2 When the number of particles changes in a thermodynamic transformation, it is important to use the correct form of entropy for an ideal gas, as given by the Sacker-Tetrode equation (2.49). (a) Use the Sacker-Tetrode equation to calculate A(V, T) and G(P, T) for an ideal gas. Show, in particular, that A(V,T) = NkT[In(nλ³) — 1], where it is the density, and λ = √2h²/mkT is the thermal wavelength. (b) Obtain the chemical potential for an ideal gas from (2A/3N)v.7 and (ƏG/AN)P.T. Show that you get the same answer μ = KT ln(nλ³).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_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
- From 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_forwardConsider 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.arrow_forward3.2 Show that the entropy maximum principle encountered in the study of isolated systems [eq. (2.50)] is equivalent to the statement that the non- flow exergy of an isolated system is destined to decrease or, at best, remain unchanged. 33 Conuidarrow_forward
- Consider a monoatomic ideal gas, with Zint 1. The partition function is then 2Tm \ 3N/2 VN zmonoatomic ideal gas N! h?B Use F = -kT In Z, the Stirling approximation In N! = N In N to derive the chemical potential of the monoatomic ideal gas as a function of T, N and V. You may want to N and the appropriate partial derivative compare your result with what you got in Weekly Practice 9. (a) Take the atomic mass of Xenon to be 131 (Xenon has 8 different stable isotopes and many more metastable ones). What is the chemical potential for pure Xe gas at 1 atm and T = 300 K? Use the ideal gas law and give the answer in eV. (b) Repeat the computation from part (a) if Xe is only 1% (by number density or, equivalently, partial pressure) of a mixture of different gasses. Note: if pure Xenon is allowed to come in contact with the gas in part (b), the net flow of Xenon atoms should be into the mixture. This tells you that your answer to part (b) should be smaller than your answer to part (a).arrow_forwardLet the system be one mole of argon gas at room temperature and atmospheric pressure. Compute the total energy (kinetic only, neglecting atomic rest energies), entropy, enthalpy, Helmholtz free energy, and Gibbs free energy. Express all answers in SI units.arrow_forwardI need the answer as soon as possiblearrow_forward
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