An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Chapter 2.6, Problem 32P
To determine
To Find: The entropy of the two-dimensional monoatomic ideal gas.
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Now, let's use this property of logarithms to learn something about the number of microstates available to a molecular system. The absolute
entropy of a system is related to the number of microstates available to it via Boltzmann's formula S = kB In W. If a system containing one
mole of an ideal gas has an entropy of 167.7 J/K, how many microstates does it have? Report the order of W, as we have defined it above,
and you should use scientific notation, 1.23E45, and report 3 (three) significant figures.
please do fast
5. Find the expression for the entropy of a single harmonic oscillator.
The multiplicity of an Einstein solid can be approximated as: given png
for N oscillators and q energy units. The internal energy of the system is U=qϵ, where ϵ is some constant. Find an expression for the entropy of an Einstein solid as a function of N and q. Use this expression to derive temperature as a function of energy, U. Take your derivation a step further, and determine the formula for heat capacity using T(U,N). Show that as T goes to ∞, the heat capacity becomes C=Nk. (Consider when x is small, ex~1+x.) Does this make sense? Explain your logic and steps.
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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- (a) Answer the following questions about entropy-volume relationships: (i) For a general system, use the Helmholtz free energy as an intermediary to express the derivative () in terms of a T. derivative of the pressure P with respect to temperature T. (ii) Assuming an ideal gas, evaluate your derivative of P, and finally integrate () to determine the volume dependence of the av entropy S of the classical ideal gas. (iii) Comment on your result, and in particular on how an alternative understanding of the S(V) dependence can be achieved on the basis of spatial multiplicity considerations.arrow_forwardConsider again the system of two large, identical Einstein solids. For the case N = 1023 , compute the entropy of this system (in terms of Boltzmann's constant), assuming that all of the microstates are allowed. (This is the system's entropy over long time scalesarrow_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_forward
- T04.2 Atoms in a harmonic trap We consider Nparticles in one dimension in an external potential, mw2 K(x) = 2 X7. (to)Write the complete Hamiltonian function for the system. Then calculate the number of micro-states MAND) by means of the semiclassical approach. (b)Calculate the entropy in the thermodynamic limit. (c)Calculate the temperature and the work differential based on the result in part (b).arrow_forwardI 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.arrow_forwardProblem #2 For heat exchange between a thermal reservoir at 300 K and a constant volume system containing one mole of monatomic ideal gas: a) Derive the equation for the total change in entropy for a designed initial system temperature Tj. b) Plot AStotal vs. Tsys for the initial system temperature ranging from 160 K to 500 K in increments of 10 K (i.e., Tsys = 160 K, 170 K, ... , 500 K). Use Matlab, Excel or similar plotting software for your plot. Label the plot axes and include units. %Darrow_forward
- 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…arrow_forwardFor 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_forwardA 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.arrow_forward
- By 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_forwarda. 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_forwardHi, could I get some help with this macro-connection physics problem involving isothermal expansion? The set up is: For an isothermal reversible expansion of two moles of an ideal gas, what is the entropy change of the a) gas and b) the surroundings in J/K to 4 digits of precision if the gas volume quadruples, assuming NA = 6.022e23 and kB = 1.38e-23 J/K? Thank you.arrow_forward
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