List and count the ways of arranging three particles. Now consider four particles. What is the general formula for the number of arrangements versus the number of particles? (ANS. 2N) How many arrangements correspond to having two particles in box A and one in box B? What is the probability of {2H, 1T}? (ANS. 3/8) How many arrangements correspond to {2H, 2T}? {3H, 2T}? {4H, 2T}? {3H, 3T}? (ANS. N!/[(N – m)!m!])
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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While I have provided the final answers for each of the problem statements, I need help with how to get to those problem statements. Can you please help me in that area with trying to answer as best as you can? Thank you
Question:
Call placement of a particle in box A, “heads” and placement in box B, “tails.” Given one particle, there are two ways of arranging it, H or T. For two particles, there are four ways of arranging them, {HH, HT, TH, TT}. We can treat the microstates by considering each particle in order. For example, {H T H H} means the first particle is in box A, the second in box B, the third in box A and the fourth in box A.
- List and count the ways of arranging three particles. Now consider four particles. What is the general formula for the number of arrangements versus the number of particles? (ANS. 2N)
- How many arrangements correspond to having two particles in box A and one in box B? What is the probability of {2H, 1T}? (ANS. 3/8)
- How many arrangements correspond to {2H, 2T}? {3H, 2T}? {4H, 2T}? {3H, 3T}? (ANS. N!/[(N – m)!m!])
- List the macrostates and corresponding number of microstates for an eight-particle, two-box system. What portion of all microstates are parts of either 5:3, 4:4, or 3:5 macrostates? (ANS. 71%)
- What is the change of entropy in going from a 5:3 macrostate to a 4:4 macrostate? (ANS. 3.08E-24 J/K)
- Use Stirling’s approximation to estimate the change of entropy in going from a distribution of 50.1% of 6.022E23 in box A to a distribution of 50.001%, and from 50.001% to 50.000%. (ANS. 1.2E18 J/K)
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