An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Question
Chapter 2.1, Problem 1P
(a)
To determine
To Find: All possible outcomes in flipping four fair coins
(b)
To determine
To Find: Different macrostates and their probabilities in flipping four coins.
(c)
To determine
To Find: The multiplicity of each macrostate using the combinatorial.
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Use the Maxwell distribution to calculate the average value of v2 for the molecules in an ideal gas. Check that your answer agrees with equation 6.41 (Attached).
Use a computer to reproduce the table and graph in Figure 2.4: two Einstein solids, each containing three harmonic oscillators, with a total of six units of energy. Then modify the table and graph to show the case where one Einstein solid contains six harmonic oscillators and the other contains four harmonic oscillators (with the total number of energy units still equal to six). Assuming that all microstates are equally likely, what is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?
Based from the sample prob. 2.10. Then answer PRACTICE EXERCISE 2.10 with Complete solutions
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) Make a diagram showing how many distinct ways (how many microstates, the multiplicity) there are of putting q = 2 indistinguishable objects in N = 3 boxes. Assuming that all microstates are equally probable, what is the probability that both objects are in the left-most box? What is the correct formula for the mulitiplicity as a function of N and q? b) Make a diagram showing how many distinct ways (the multiplicity) there are of putting q = 2 distinguishable objects in N= 3 boxes. Assuming that all microstates are equally probable, what is the probability that both objects are in the left-most box? Label the two objects R and G. What is the correct formula for the mulitiplicity as a function of N and q? Below are the diagrams, started for you. Complete the diagrams. distinguishable indistinguishable RG •. !R !Garrow_forwardSuppose we know exactly two arbitrary distributions p(x|ωi) and priors P(ωi) in a d-dimensional feature space. (a) Prove that the true error cannot decrease if we first project the distributions to a lower dimensional space and then classify them. (b) Despite this fact, suggest why in an actual pattern recognition application we might not want to include an arbitrarily high number of feature dimensions.arrow_forwardProblem 1: This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf, and so on. a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving 1-x the formula by long division. Prove it by first multiplying both sides of the equation by (1 – x), and then thinking about the right-hand side of the resulting expression. b) Evaluate the partition function for a single harmonic oscillator. Use the result of (a) to simplify your answer as much as possible. c) Use E = - дz to find an expression for the average energy of a single oscillator. z aB Simplify as much as possible. d) What is the total energy of the system of N oscillators at temperature T?arrow_forward
- In a source-free region, show that dE a*E V²E – po at afarrow_forwardT04.2 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_forwardQuestion d only. Please do it step by step and show how you do the integration for complex exponentialsarrow_forward
- solve and explain in detail: (a) Can we observe diffusion for point particles? Point particles are defi ned as particles of zero radius. (b) A particle is diffusing freely between -infinity < x < +infinity. Calculate <x(t) > and< x^2(t) >. (Hint 1: You can calculate this by integrating diffusion equation. Hint2: Average velocity of the diffusing particle is zero. Hint 3: Probability of fi nding the particle at in nity is zero) (c) Suppose you can track the movement of a single particle in a given system. This means you can fi nd the position of a single particle at every moment. Then, how would you use the results of part (b) to find whether the movement of that particle is diffusion.arrow_forwardTo perform sensitivity analysis involving an integer linear program, it is best to use the same approach as you would for a linear program. use LP Relaxation. make multiple computer runs. use the dual prices very cautiously.arrow_forwardAssume that a certain type of cyclone has a particle cut diameter of 6 microns; (a) calculate the overall efficiency of this cyclone on the particles of Problem 4.4 and (b) calculate the overall efficiency of two of these cyclones placed in series.arrow_forward
- write the solution step by step and clearly.arrow_forwardConsider N identical harmonic oscillators (as in the Einstein floor). Permissible Energies of each oscillator (E = n h f (n = 0, 1, 2 ...)) 0, hf, 2hf and so on. A) Calculating the selection function of a single harmonic oscillator. What is the division of N oscillators? B) Obtain the average energy of N oscillators at temperature T from the partition function. C) Calculate this capacity and T-> 0 and At T-> infinity limits, what will the heat capacity be? Are these results consistent with the experiment? Why? What is the correct theory about this? D) Find the Helmholtz free energy from this system. E) Derive the expression that gives the entropy of this system for the temperature.arrow_forwardConsider a van der Waal's gas that undergoes an isothermal expansion from volume V₁ to volume V₂. Calculate the change in the Helmholtz free energy. 2.2 (a) (b) From the theory of thermodynamics, with T and V T independent, ()₁ = T()-p. Show that the change in internal energy is AU = a(-1/2).arrow_forward
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