# 04.02 The Microscopic View of Entropy

### Principles of Probability.

This is supplemental Material from "Molecular Driving Forces, K.A. Dill, S. Bromberg", Garland Science, New York:NY, 2003, Chapter 1. See the next three screencasts. This content is useful for graduate level courses that go into more depth or for students interested in more background on probability.

### Principles of Probability I, General Concepts

Principles of Probability I, General Concepts, Correlated and Conditional Events. (msu.edu, 17min) (Flash)
Comprehension Questions:
1. Estimate the probability of pulling an king from a randomly shuffled deck of 52 cards.
2. A coin is flipped 5 times. Estimate the probability that heads is observed three of the 5 times.
3. A die (singular of dice) is a cube with the numbers 1-6 inscribed on its 6 faces. If you roll the die 7 times, what is the probability that 5 will be observed on all 7 rolls?

### Principles of Probability II, Binomial and Multinomial Coeff.

Principles of Probability II, Counting Events, Permutations and Combinations. This part discusses the binomial and multinomial coefficients for putting particles in boxes. The binomial and multinomial coefficient are used in section 4.2 to quantify configurational entropy. (msu.edu, 16min) (Flash) You might like to check out the sample calculations below before attempting the comprehension questions.
Comprehension Questions:
1. Write the formulas for the binomial coefficient, the multinomial coefficient, and the multinomial with repetition.
2. Ten particles are distributed between two boxes. Compute the number of possible ways of achieving 7 particles in Box A and 3 particles in Box B.
3. Note that the binomial distribution is a special case of the multinomial distribution where the number of categories is 2. Also note that the total number of events for a multinomial distribution is given by M^N where M is the number of categories (aka. outcomes, e.g. boxes) and N is the number of objects (aka. trials, e.g. particles). The probability of a particular observation is given by the number of combinations divided by the total number of events. Compute the probability of observing 7 particles in Box A and 3 Particles in Box B.
4. Ten particles are distributed between three boxes. Compute the probability of observing 7 particles in Box A, 3 particles in Box B, and zero particles in Box C.
5. Ten particles are distributed between three boxes. Compute the probability of observing 3 particles in Box A, 3 particles in Box B, and 4 particles in Box C.

### Principles of Probability III, Distributions

Principles of Probability III, Distributions, Normalizing, Distribution Functions, Moments, Variance. This screencast extends beyond material covered in the textbook, but may be helpful if you study statistical mechanics in another course. (msu.edu, 15min) (Flash)

### Molecular Nature of S: Configurational Entropy I.

Connecting Microstates, Macrostates, and the Relation of Entropy to Disorder (uakron.edu, 14min). For small systems, we can count the number of ways of arranging molecules in boxes to understand how the entropy changes with increasing number of molecules. By studying the patterns, we can infer a general mathematical formula that avoids having to enumerate all the possible arrangements of 10^23 molecules (which would be impossible within several lifetimes). A surprising conclusion of this analysis is that entropy is maximized when the molecules are most evenly distributed between the boxes (meaning that their pressures are equal). Is it really so "disordered" to say that all the molecules are neatly arranged into equal numbers in each box? Maybe not in a literary world, but it is the only logical conclusion of a proper definition of "entropy." It is not necessary to watch the videos on probability before watching this one, but it may help. And it might help to re-watch the probability videos after watching this one. Moreover, you might like to see how the numbers relate to the equations through sample calculations (uakron, 15min). These sample calculations show how to compute the number of microstates and probabilities given particles in Box A, B, etc and also the change in entropy.

Comprehension Questions:

1. What is the number of total possible microstates for: (a) 2 particles in 2 boxes (b) 5 particles in 2 boxes (c) 10 particles in 3 boxes.

2. What is the probability of observing: (a) 2 particles in Box A and 3 particles in Box B? (b) 6 particles in Box A and 4 particles in Box B? (c) 6 particles in Box A and 4 particles in Box B and 5 particles in Box C?

### Molecular Nature of S: Configurational Entropy II.

Relating the microscopic perspective on entropy to macroscopic changes in volume (uakron.edu, 11min) Through the introduction of Stirling's approximation, we arrive at a remarkably simple conclusion for changes in entropy relative to the configurations of ideal gas molecules at constant temperature: ΔS = Rln(V2/V1). This makes it easy to compute changes in entropy for ideal gases, even for subtle changes like mixing.

Comprehension Questions:

1. Estimate ln(255!).

2. A system goes from 6 particles in Box A and 4 particles in Box B to 5 particles in each. Estimate the change in S(J/K).

3. A system goes from 6 moles in Box A and 4 moles in Box B to 5 moles in each. Estimate the change in S(J/mol-K).