Homework 10: Do problem 6.2 in the text. Solution: Part (a)

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1 Homework 10: Do problem 6. in the text Solution: Part (a) The two boxes will come to the same temperature as each other when they are in thermal euilibrium. So, the final temperature is an average of the two initial temperatures. T f T ia + T ib Part (b) Use E. (6.7), with N A N B N, V A V B V, and E tot E A + E B and where A 1 and B Calculate S S f,tot S i,tot. S S f S i ( ) ( )] [ ( ) ( )] k B [N ln(e 1f ) + ln(v ) + N ln(e f ) + ln(v ) k B N ln(e 1i) + ln(v ) + N ln(e i) + ln(v ) [ k B N ln(e 1f ) + N ln(v ) + N ln(e f ) + ln(v ) N ln(e 1i) N ln(v ) ] N ln(e i) N ln(v ) Nk B [ln(e 1f ) + ln(e f ) (ln(e 1i ) + ln(e i ))] Nk B ln(e 1f E f ) ln(e 1i E i ) ( ) Nk E1f E f B ln E 1i E i Recall that E 1i Nk BT 1i and E i Nk BT i. Also, since the two systems end up in thermal euilibrium, E 1f E f Nk BT f Nk (T 1i + T i ) B Substituting, we have 1

2 ( ) S E Nk f B ln E 1i E i ( S Nk Nk (T 1i + T i ) B B ln Nk BT 1i Nk BT 1i S ( k (T1i + T i ) ) B ln 4T 1i T i ) Part (c) Use the hint in the text. Plot S making the substitution X T i,1 T i,. You will see that S does in fact remain greater than or eual to 0. (See plot below) Part (d) Solve your euation S 0 for X T i,1 T i,. (See plot below)

3 In[55]: S kb Nb Log Ti1 Ti 4 Ti1 Ti ; Ti1 Ti X; SX X_ : S FullSimplify SX X kb 1; Nb 1; Plot SX X, X, 0, 10, Frame True, FrameLabel "X", " S", LabelStyle Medium Solve SX X 0, X Out[58] 1 X Log 4 X Out[60] S Out[61] X 1 X 1 mean that S 0 when Ti1 Ti

4 A.1 Isolated Einstein solids List all of the possible microstates of an Einstein solid with the specified number of oscillators, N, and a given amount of energy E hf. Compare your answers in the list to the number of microstates given by ( ) + N 1 Ω(N, ) (a) N, 4 (b) N 4, (c) N 1, anything (d)n anything, 1 Calculate the multiplicity of an Einstein solid with 0 oscillators and 0 units of energy (dont make a table ofall of the microstates!) Solution: Part (a) N, 4 #1 # # Counting how many arrangements there are in the table, we see that there are 15 states. Using the Binomial 4

5 coefficient instead, we have ( ) ( ) N ( ) 6 6! 4 4!! This is the same answer as writing them out by hand! Part (b) N 4, #1 # # # Counting how many arrangements there are in the table, we see that there are 10 states. Using the Binomial coefficient instead, we have ( ) ( ) N ( ) 5 5!!! This is the same answer as writing them out by hand! 5

6 Part (c) N 1, anything There is only one state with each possible energy so multiplicity is only 1. ( ) N + 1 ( ) if N 1 ( ) 1 Part (d) N anything, 1 If 1, then we can put that energy in any oscillator. If we have N oscillators, then we have N choices. ( ) ( ) N + 1 N ( ) N N 1 Part (e) N 0, 0 ( ) Ω 0 ( )

7 A. Two Einstein solids in contact Consider a system of two Einstein solids, A and B, each containing 10 oscillators, sharing a total of 0 units of energy. Assume that the solids are weakly coupled, and that the total energy is fixed. (a) How many different macrostates are available to the system? (b) How many different microstates are available to the system? (c) Assuming that this system is in thermal euilibrium, what is the probablility of finding all the energy in solid A? (d) What is the probability of finding exactly half of the energy in solid A? (e) If, instead of 10 oscillators each, solids A and B had 5 and 15 oscillators respectively, at euilibrium how would the energy most likely be distributed among the two solids? (f) Under what circumstances would this system exhibit irreversible behavior? Solution: Part (a) We need to find the number of microstates, where A + a B 0 There are 1 different values of ( a, b ) ranging from values 0 to 0 that meet the constraint of A + B 0. Part (b) The total number of microstates is ( ) ( ) N ( ) Part (c) The number of microstates for having all the energy in solid A ( A 0) is Ω( a 0) 1. Divide this by the total number of microstates found in part (b) to get the probability, P (all in A)

8 Part (d) To find the probability of finding exactly half of the energy in solid A, we need to first find the number of microstates for having this configuration ( ) Ω 1/,1/ Ω 1/,A Ω 1/,B 10 ( ) P (exactly 1/ in A) Ω 1/,1/ Ω total % Part (e) Make a table of values for the total number of microstates corresponding to the different arrangments of A + B 0, where N A 5 and N B 15. The state with the highest number of microstates will be the state that the two solids will tend towards. (See table below) We find that the energy is most likely to be distributed with A 4 and B 16. Part (f) If the system s initial state is not the macrostate with the highest entropy, it will spontaneously move to that state and it won t come back i.e. it s irreversible. 8

9 Part (e) table 9

Physics 172H Modern Mechanics

Physics 172H Modern Mechanics Instructor: Dr. Mark Haugan Office: PHYS 282 haugan@purdue.edu TAs: Alex Kryzwda John Lorenz akryzwda@purdue.edu jdlorenz@purdue.edu Lecture 22: Matter & Interactions, Ch.