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# Use a computer to study the entropy, temperature, and heat ISBN: 9780201380279 40

## Solution for problem 24P Chapter 3

An Introduction to Thermal Physics | 1st Edition

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Problem 24P

Use a computer to study the entropy, temperature, and heat capacity of an Einstein solid, as follows. Let the solid contain 50 oscillators (initially), and from 0 to 100 units of energy. Make a table, analogous to Table 3.2, in which each row represents a different value for the energy. Use separate columns for the energy, multiplicity, entropy, temperature, and heat capacity. To calculate the temperature, evaluate ΔU/ΔS for two nearby rows in the table. (Recall that U = qϵ for some constant ϵ.) The heat capacity (ΔU/ΔT) can be computed in a similar way. The first few rows of the table should look something like this:

 q Ω S/k kT/ϵ C/Nk 0 1 0 0 — 1 50 3.91 .28 .12 2 1275 7.15 .33 .45

(In this table I have computed derivatives using a “centered-difference” approximation. For example, the temperature .28 is computed as 2/(7.15 – 0).) Make a graph of entropy vs. energy and a graph of heat capacity vs. temperature. Then change the number of oscillators to 5000 (to “dilute” the system and look at lower temperatures), and again make a graph of heat capacity vs. temperature. Discuss your prediction for the heat capacity, and compare it to the data for lead, aluminium, and diamond shown in below Figure. Estimate the numerical value of ϵ, in electron-volts, for each of those real solids.

Figure: Measured heat capacities at constant pressure (data points) forone mole each of three different elemental solids. The solid curves show the heatcapacity at constant volume predicted by the model used in Section 7.5, with thehorizontal scale chosen to best fit the data for each substance. At sufficiently hightemperatures, CV for each material approaches the value 3R predicted by theequipartition theorem. The discrepancies between the data and the solid curvesat high T are mostly due to the differences between CP and CV. At T = 0 alldegrees of freedom are frozen out, so both CP and CV go to zero. Data from Y. S.Touloukian, ed., Thermophysical Properties of Matter (Plenum, New York, 1970). Step-by-Step Solution:
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##### ISBN: 9780201380279

Since the solution to 24P from 3 chapter was answered, more than 269 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 24P from chapter: 3 was answered by , our top Physics solution expert on 07/05/17, 04:29AM. This textbook survival guide was created for the textbook: An Introduction to Thermal Physics , edition: 1. An Introduction to Thermal Physics was written by and is associated to the ISBN: 9780201380279. This full solution covers the following key subjects: heat, capacity, temperature, Table, data. This expansive textbook survival guide covers 10 chapters, and 454 solutions. The answer to “Use a computer to study the entropy, temperature, and heat capacity of an Einstein solid, as follows. Let the solid contain 50 oscillators (initially), and from 0 to 100 units of energy. Make a table, analogous to Table 3.2, in which each row represents a different value for the energy. Use separate columns for the energy, multiplicity, entropy, temperature, and heat capacity. To calculate the temperature, evaluate ?U/?S for two nearby rows in the table. (Recall that U = q? for some constant ?.) The heat capacity (?U/?T) can be computed in a similar way. The first few rows of the table should look something like this:q?S/kkT/?C/Nk0100—1503.91.28.12212757.15.33.45(In this table I have computed derivatives using a “centered-difference” approximation. For example, the temperature .28 is computed as 2/(7.15 – 0).) Make a graph of entropy vs. energy and a graph of heat capacity vs. temperature. Then change the number of oscillators to 5000 (to “dilute” the system and look at lower temperatures), and again make a graph of heat capacity vs. temperature. Discuss your prediction for the heat capacity, and compare it to the data for lead, aluminium, and diamond shown in below Figure. Estimate the numerical value of ?, in electron-volts, for each of those real solids.Figure: Measured heat capacities at constant pressure (data points) forone mole each of three different elemental solids. The solid curves show the heatcapacity at constant volume predicted by the model used in Section 7.5, with thehorizontal scale chosen to best fit the data for each substance. At sufficiently hightemperatures, CV for each material approaches the value 3R predicted by theequipartition theorem. The discrepancies between the data and the solid curvesat high T are mostly due to the differences between CP and CV. At T = 0 alldegrees of freedom are frozen out, so both CP and CV go to zero. Data from Y. S.Touloukian, ed., Thermophysical Properties of Matter (Plenum, New York, 1970).” is broken down into a number of easy to follow steps, and 318 words.

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