An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Textbook Question
Chapter 2.6, Problem 37P
Using the Same method as in the text, calculate the entropy of mixing for a system of two monatomic ideal gases, A and B, whose relative proportion is arbitrary. Let N be the total number of molecules and let x be the fraction of these that are of species B. You should find
Check that this expression reduces to the one given in the text when
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I have answered b but I got 4.33 Kg for A and not correct
(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 ×10 e690 microstates? Express your answer as a multiple of kB.
The entropy of the solid is 1590.92kB.
The natural variables for the internal energy U are the entropy S and the volume V.
This means that, if you know S and V, you can find U(S, V) and simple expressions for T
and P.
Supposed instead that you know U(T, V).
Show that this leads to the following expression for P:
-/),
dT
+ f(V)
T
aV
T2
where f(V) is an arbitrary function of V.
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 2 Solutions
An Introduction to Thermal Physics
Ch. 2.1 - Prob. 1PCh. 2.1 - Prob. 2PCh. 2.1 - Prob. 3PCh. 2.1 - Prob. 4PCh. 2.2 - For an Einstein solid with each of the following...Ch. 2.2 - Prob. 6PCh. 2.2 - Prob. 7PCh. 2.3 - Prob. 8PCh. 2.3 - Use a computer to reproduce the table and graph in...Ch. 2.3 - Use a computer to produce a table and graph, like...
Ch. 2.3 - Use a computer to produce a table and graph, like...Ch. 2.4 - Prob. 12PCh. 2.4 - Fun with logarithms. (a) Simplify the expression...Ch. 2.4 - Write e1023 in the form 10x, for some x.Ch. 2.4 - Prob. 15PCh. 2.4 - Prob. 16PCh. 2.4 - Prob. 17PCh. 2.4 - Prob. 18PCh. 2.4 - Prob. 19PCh. 2.4 - Suppose you were to shrink Figure 2.7 until the...Ch. 2.4 - Prob. 21PCh. 2.4 - Prob. 22PCh. 2.4 - Prob. 23PCh. 2.4 - Prob. 24PCh. 2.4 - Prob. 25PCh. 2.5 - Prob. 26PCh. 2.5 - Prob. 27PCh. 2.6 - How many possible arrangements are there for a...Ch. 2.6 - Consider a system of two Einstein solids, with...Ch. 2.6 - Prob. 30PCh. 2.6 - Fill in the algebraic steps to derive the...Ch. 2.6 - Prob. 32PCh. 2.6 - Use the Sackur-Tetrode equation to calculate the...Ch. 2.6 - Prob. 34PCh. 2.6 - According to the Sackur-Tetrode equation, the...Ch. 2.6 - For either a monatomic ideal gas or a...Ch. 2.6 - Using the Same method as in the text, calculate...Ch. 2.6 - Prob. 38PCh. 2.6 - Compute the entropy of a mole of helium at room...Ch. 2.6 - For each of the following irreversible process,...Ch. 2.6 - Describe a few of your favorite, and least...Ch. 2.6 - A black hole is a region of space where gravity is...
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- Prove that the entropy of mixing of an ideal mixture has aninfinite slope, when plotted vs. x, at x = 0 and x = l.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_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
- (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.arrow_forwardProblem 1: Describe a situation in which the entropy of a container of gas is constant. In other words, come up with your own problem where the answer is that AS = 0.arrow_forwardBy considering the number of accessible states for an ideal two-dimensional gas made up of N adsorbed molecules on a surface of area A, obtain an expression for the entropy of a system of this kind. Use the entropy expression to obtain the equation of state in terms of N, A, and the force per unit length F. What is the specific heat of the two-dimensional gas at constant area?arrow_forward
- Plot 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_forward1.50 moles of a monatomic ideal gas goes isothermally from state 1 to state 2. P1 = 2.8×105 Pa, V1 = 88 m3, and P2 = 6.6×105 Pa. What is the volume in state 2, in m3? Your answer needs to have 2 significant figures, including the negative sign in your answer if needed. Do not include the positive sign if the answer is positive. No unit is needed in your answer, it is already given in the question statement.arrow_forwardA mass m of water is heated reversibly from temperature T1 to T2 at a constant pressure of P. In this problem, we are going to determine an expression for the change in entropy, ΔS.Assume we can heat the given water infinitesimally slowly so that the process is reversible. Therefore, heat in any infinitesimal step is given by the following: dQ = mc dT, where c is the specific heat and is constant. Part (a) Write an expression for the change in entropy ΔS for the system. Part (b) Calculate the change in entropy in cal/K for a sample of water with mass m = 1.9 kg and changing temperature from T1 = 22.9°C to T2 = (22.9+10)°C. The specific heat c of water is 1,000 cal/kg/K.arrow_forward
- a. Find an appropriate expression for the change in entropy in the following two cases: 1) S=S(T, V) 2) s= S(T, P) Where: S is entropy, T is temperature, V is volume, P is pressure b. Prove the following two themodynamie property relationships (똥),-() 8C, Where: T, P. V are temperature, pressure and volume, respectively. C, and C, are specific heats at constant volume and constant pressure, respectively.arrow_forwarddU =T dS – P dV Re-arrange the thermodynamic identity by solving for dS. Assume we are applying this new expression to an ideal gas, re-write the du term in terms of the appropriate heat capacity and temperature. Re-write the coefficient of the dV term in terms of volume. Connect these “classical" expressions for entropy to the "modern" interpretation where we are concerned with the uncertainty associated with position and momentum of the particles in an ideal gas. Answer with sentences. Now, re-write the expression in terms of enthalpy so you can connect entropy with enthalpy. Expalin why this final expression makes sense.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_forward
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