An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Textbook Question
Chapter 5.4, Problem 56P
Prove that the entropy of mixing of an ideal mixture has an infinite slope, when plotted vs. x, at
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(a) What is the entropy of an Einstein solid with 4 atoms and an energy of 18ε? Express your answer as a multiple of kB .
The entropy of the solid is ____ kB.(b) What is the entropy of an Einstein solid in a macropartition that contains 9 ×10690×10690 microstates? Express your answer as a multiple of kB.
The entropy of the solid is ____ kB.
A highly non-ideal gas has an entropy given by S=aNU/V, where the internal energy, U is a function of T.Find the pressure, the expression for the heat capacity at constant volume, and the chemical potential.
Problem 3:
Consider an Einstein solid with N oscillators and total energy U = qe, in the limit N,q » 1
(with no assumptions made about the relative size of N and q).
+ N°
(9 +N\9
a) Starting with this formula, find an expression for the entropy of an Einstein solid as a
function of N and q. Explain why factors omitted from the formula have no effect on the
entropy.
b) Derive an expression for the temperature of the solid, as a function of N and q. Simplify
your expression as a much as possible.
c) Invert the result of part (c) to get the energy U as a function of temperature T. As always,
simplify the final result as much as possible.
d) Show that, in the high temperature limit (q » N), the heat capacity is C = Nkg. (Hint:
when x is small, e* = 1+ x.) Is this the result you would expect? Explain.
e) Plot energy U vs. temperature T using dimensionless variables, Cy/(Nkg) vs. t =
kgT/e, for t in the range from 0 to 2. Discuss your prediction for the heat capacity at low
temperature…
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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- For either a monatomic ideal gas or a high-temperature Einstein solid, the entropy is given by Nk times some logarithm. The logarithm is never large, so if all you want is an order-of-magnitude estimate, you can neglect it and just say S - Nk. That is, the entropy in fundamental units is of the order of the rv number of particles in the system. This conclusion turns out to be true for most systems (with some important exceptions at low temperatures where the particles are behaving in an orderly way). So just for fun, make a very rough estimate of the entropy of each of the following: this book (a kilogram of carbon compounds); a moose (400 kg of water); the sun (2 x 1030 kg of ionized hydrogen).arrow_forwardStarting with the Clausius Inequality, ∂S ≥ ∂q/T, can you prove that, under conditions of constant pressure and entropy, for the total entropy to increase, ∂H ≤ 0 J?arrow_forwardConsider a system in thermal equilibrium, having energies 0 and 'E'. Find the partition function (Z) and Entropy (S) of the system.arrow_forward
- Consider a particle is confined in a three-level system. The probability of finding the particle in first level is 0.38, for second level 0.36, and for third level it is 0.26. What is the entropy of the system. (a) 6.5x104 J/K (b) 4.7x104 m² -kg s.K (c) 1.5x10 m kg s.K (d) -1.5x103 J/Karrow_forwardPlot the function dS/dT for a two-level system, the temperature coefficient of its entropy, against kT/ε. Is there a temperature at which this coefficient passes through a maximum? If you find a maximum, explain its physical origins.arrow_forwardFor one component gas that is confined in a box with volume V. We can get the entropy of the gas as S= Nk, in- where N is the total a² number of atoms, a is the radius of the atom. Can you guess how it is obtained?arrow_forward
- Consider a case of n particles and two compartments, if n1 is the number of particles in one compartment and remaining n2=n-n1 particles in other compartment, then the number of microstates in the macrostate (n1, n-n1) or thermodynamic probability isarrow_forward(a) A 5.0-kg rock at a temperature of 20 °C is dropped into a shallow lake also at 20 °C from a height of 1.0 × 103 m . What is the resulting change in entropy of the universe? (b) If the temperature of the rock is 100 °C when it is dropped, what is the change of entropy of the universe? Assume that air friction is negligible (not a good assumption) and that c = 860 J/kg · K is the specific heat of the rock.arrow_forwardFor one component gas that is confined in a box with volume V. V We can get the entropy of the gas as S = Nk, In where N is the total number of atoms, a is the radius of the atom. Can you guess (or work out) how it is obtained?arrow_forward
- What is the entropy S of this system at a given temperature T, asT→∞, and T→0. We expect the entropy S to be 0 at T= 0. Does entropy increase as the temperature T increases?arrow_forwardTwo kilograms of water at 0 degree Celsius is heated to 100 degree Celsius. Compute its entropy in change. The temperature of the water is increased reversibly in a series of infinitesimal steps, in each of which the temperature is raised by an infinitesimal amount dT. The heat required is dQ =mc dTarrow_forwardThe partition function of a hypothetical system is given by In Z = «TªV where a is a constant. Evaluate the mean energy E, the pressure P, and the entropy S.arrow_forward
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