Investigating Microstates And Macrostates Using Random Number Generator

Simulating a Wiggly-Eye Experiment

In previous years, students in PVB301 performed a prac in which they examined microstates and macrostates for a system of wiggle-eyes in a plastic organizer box. Each compartment in the box held one wiggle-eye, which was trapped in the compartment by the closed lid of the box. Students would shake the box, then would record the state of the wiggle-eyes in each of the 18 compartments of the organizer, denoting a right-side-up eye as 1 and an upside-down eye as 0. The stu- dents repeated this process 100 times. (The lab description is available for download on Blackboard if you want to consult it.)

As the caption of Figure 1 implies, we can define the microstates and macrostates for this system as follows: Microstate: A specific pattern of eyes-up and eyes-down. Macrostate: A specific total of eyes-up (or eyes-down.) This year, we will examine the same physics without doing a prac. Instead of using a box of wiggle-eyes to define a macrostate, we will use the random number generator of a spreadsheet program. You can use Excel, OpenOffice, etc. to accomplish this. Here, we will refer the relevant commands in Excel. You can simulate a wiggly-eyes experiment by using the RAND() function in Excel. Generate an array of 18 random numbers between 0 and 1, then round them to a single digit. This 18-member array defines a single microstate (equivalent to one trial in the prac). The associated macrostate is the sum of the individual states in the microstate. Your spreadsheet should look something like this:

Part a (7 marks) Generate histograms showing the distribution of macrostates for systems of 10, 100, and 1000 trials. Normalize the histograms so that they show probability rather than raw counts. In each case, calculate the average macrostate value and its standard deviation. Comment on whether you think the experiment on actual wiggly-eyes would produce the same result or not.

Part b (3 marks) Calculate the number of microstates associated with each macrostate. Find the probability associated with the macrostate in which all the wiggly-eyes are right-side-up, and the probability of the macrostate with half of the eyes right-side-up and half of the eyes upside-down.

2 Entropy and equilibrium in coupled systems of Einstein oscillators Consider two Einstein solids, denoted as A and B, that are coupled together such that they can exchange energy. The total energy in the system is held constant, so that at any point Utotal = UA + UB, and we denote the number of oscillators in each solid as NA and NB, respectively. If we distribute a fixed amount of energy, qtotal, between the two oscillators,

Analyzing the Distribution of Macrostates for Different Trials

we can define the multiplicity of the coupled solids as ?total = ?A?B, where the multiplicity of an individual solid is given as ?(N, q) = q + N − 1 q = (q + N − 1)! q!(N − 1)! (Note that we have not explicitly discussed the notation used in this expression. Visiting the Wikipedia entry for “binomial coefficient” will provide a quick overview.) Spreadsheets have a built-in function for handling the math required in this expression. In Excel, q + N − 1 q = COMBIN(q+N-1,q). We will use this function to investigate a system of two coupled Einstein solids, in particular looking at the entropy S = kB ln ? associated with each of the individual solids, as well as the net entropy for the coupled system. A detailed description of the spreadsheet setup, as well as the physics underpinning it, is given in the paper “A different approach to introducing statistical mechanics” (Am. J. Phys. 65(1), 26, 1997). This paper is available as a preprint at http://physics.weber.edu/schroeder/statmech.pdf Briefly, you should set up a spreadsheet that looks something like the one shown below

and use it to generate plots showing the individual and total entropies for the oscillators and system as well as the multiplicity of the combined system, similar to the ones shown below

Figure 4: Plots showing the individual and total entropies for the oscillators and system (left) and the multiplicity of the combined system (right) for a system with NA = 150, NB =150 and qtotal=300. The data are plotted as a function of qA, the energy in solid A, and the entropy is displayed in units of Bolzmann’s constant.

Part a (8 marks) Consider three different systems, as defined by NA, NB and qtotal (set qtotal=NA+NB). For each system, produce a set of plots like the ones shown above. The first plot will show SA, SB and Stotal as a function of qA, and the second will show ?total as a function of qA. Be sure to indicate the values of NA, NB and qtotal that you have used to generate each set of plots. Please consider at least one system where NA > NB, and at least one system where the converse is true. Use values of Ntotal (and threefore qtotal) on the order of 10-500 or so and make sure to present your data nicely (label axes, include units, scale axes appropriately.)

Exploring Microstates and Macrostates

Part b (2 marks) What were the limitations of using a spreadsheet program for this calculation? How large a system was it possible to consider? Discuss the following implications of this model. What state is implied at the point of maximum multiplicity for the coupled system? How does this relate to the concepts we have been learning in this course?

3 Quasistatic adiabatic processes on an ideal gas Part a (4 marks) Starting from the result given by equipartition of energy, U = 1 2 fN kBT, and considering a quasistatic adiabatic compression for which dW = −P dV , demonstrate that V Tf/2 = constant and that V γP = constant, where, γ = f + 2 f . Show all of your steps, and offer explanation where needed. Part b (3 marks) We can use this result to consider compression within the cylinder of a diesel engine, which we will treat as an adiabatic quasistatic process. First, show that T V γ−1 = constant. Now use this expression to examine the compression stroke. If the temperature of the air (γ=1.4) is 20 oC before compression and the compression ratio is 15:1, what is the temperature of the air after compression? Why don’t diesel engines need spark plugs to ignite the fuel?

4 The Carnot cycle The figure below shows a PV diagram for a Carnot cycle: AB is an isothermal expansion at temperature T1, CD is an isothermal compression at temperature T2, and BC and DA are adiabatic expansion and compression, respectively. The gas has a heat capacity ratio of γ.

Part a (1 mark) Write down expressions for the heats associated with the processes indicated by AB and CD. Express the heat as a function of the relevant temperature and volumes. Part b (2 marks) For each segment of the cycle (AB, BC, CD, DA), write down the relations between the quantities of state P and V . For example, for the segment AB, write down the relationship between PA, VA, PB and VB. You do not need to derive the expressions you use, but you do need to name the process (e.g.: isothermal expansion). Part c (2 marks) The expression for efficiency in terms of the magnitude of the heats exchanged with the reservoirs is η = Qin − Qout Qin , which we can express in terms of QAB and QCD as η = QAB + QCD QAB , where the sign change results from the way we have defined the heat flows. Use your expressions for QAB and QCD and the relations between P and V found for each cycle segment in Part b to express the efficiency in terms of the temperatures T1 and T2 as η = 1 − T2 T1 . Hint: show that VB VA = VC VD

5 Thermodynamic potentials Entalpy of mixing (4 marks) Lemonade can be cooled by adding lumps of ice to it. A student discovers that 70g of ice at a temperature of 0 C cools 0.3 kg of lemonade from 28 C to 7 C. (The latent heat of melting ice is 0.33 MJ/kg and the specific heat capacity of water is 4.2 kJ/kg K). Determine: a) the energy gained by the ice in melting (1 mark) b) the energy gained by the melted ice(1 mark) c) the energy lost by the lemonade (1 mark) d) a value for the specific heat capacity of the lemonade(1 mark)

Gibbs free energy of an alloy (9 marks) The stability of a pseudobinary alloy (see lecture week 6) is subject to the minimization of the Gibbs free energy of mixing G m = Hm − T Sm Let us consider the pseudobinary alloy AxB1−xC which crystallizes as a zincblende. The solid solution can be considered as formed by the 5 possible tetrahedra with an atom C in the centre and A or B atoms at the corners A4C , A3BC , A2B2C , A1B3C ,B4C which can be written as A4−nBnC for n=0...4. I Part a (1 marks) Calculate the probability Pn(x)

of each tetrahedron for a composition x in the alloy (x A atoms and 1-x B atoms) in the random approximation Part b (4 marks) For the same composition x 1) Calculate the number of macrostates ? for 1 mole of the alloy 2) Demonstrate that the entropy of mixing is: S m = −N kB(xlnx + (1 − x)ln(1 − x)) Part c (4 marks) 1) give a justification of the formula for the enthalpy of mixing in the random approximation: Hm(x) = X 4 n=0 n(4 − n) · w · Pn(x) where w is the energy required to create a pair AB from AA and BB: w = AB − AA + BB 2 2) Calculate the Gibbs free energy of mixing at T=200 K and T=300 K for a mole of GaxAl1−xAs alloy at x=0.25, assuming w= 0.1330 Kcal/mole. Discuss the stability of the alloy at the two temperatures.

The histograms can be drawn as

The values of each microstate and the standard deviation are calculated and it seems like the same values will be obtained from wiggley-eyes. 

1. The micro states for each state are calculated in the excel and the same is attached as-

2. The spreadsheet with data is attached below-

The entropy curves can be generated as-

The second curve for the multiplicity can be drawn as-

1. The limitations of the spreadsheet program are as follows-
2. Calculated values may differ by standard deviation.
3. Curve start and end point cannot be adjusted to make it more presentable.

Answer 3

Quasistatic process-

1. The given equation is as follows-And  Also, 

1. For the adiabatic process- 

Answer 4

The given Carnot cycle is as follows-

1. The heats associated for the process AB and CD- 

1. The relation between P and V for processes AB, BC, CD and DA-

For process AB- 

For process BC- 

For process CD- 

For process DA- 

1. The efficiency of the cycle can be calculated as-

Answer 5

1. The energy gained from ice melting- 

1. Energy gained by melted ice-

1. The energy lost by lemonade- 

4. Specific heat of lemonade c = 4.2 kJ/kgK 

Gibbs free energy

The given data are as follows-

The Gibbs free energy of mixing-

Now considering the pseudobinary alloy- 

Part a

The probability of each Tetrahydron can be calculated as- 

Part b

1. The number of microstates can be calculated as- 

2. The entropy of mixing- 

Yes, the entropy of the mixing will be depending on the fraction and thus the expression expresses the mixing entropy in correct way. 

Part c

1. The enthalpy of mixing in random approximation can be calculated as- 

Where w is the energy required and can be calculated as- 

The expression is the sum of 5 states of the random number approximation. 

2. The given data are as follows-

T = 200 K and T = 300 K

n = 1 mole

 alloy at x = 0.25 

Now the Gibbs free energy can be calculated as- ................i 

 Now substituting all values in equation i-

For T = 200 K:  

For T = 300 K: 

At 300 K the Gibbs free energy is less as compared to the Gibbs free energy at 200K and thus it will be less stable at 200 K.