Stat Mech : Balls in a box counting problem

• Opticmist
In summary, the problem is about arranging V balls inside a box with N blue balls and V-N red balls, such that V/2 balls are on each side. The formula for calculating the number of arrangements that accommodate p number of blue balls on the left hand side includes four terms: selecting p blue balls, arranging them on the left side, arranging N-p blue balls on the right side, and arranging all red balls inside the box. The fourth term is necessary because the red balls are not identical, so rearranging them in the remaining spots can result in unique arrangements.

Homework Statement

I already have the solution to the problem. Just need some help deciphering the logic behind it.

There are V balls, which are identical except for their color. N of them are blue and V-N are red. We place the balls inside a box so that V/2 balls are on each side.

How many arrangements(states) accommodate a p number of blue balls on the left hand side of the box.

---

The Attempt at a Solution

Actual Solution:

$$\Gamma = \frac{N!}{p! (N-P)!} (\frac{V}{2})^p (\frac{V}{2})^{N-p} (V-N)!$$

The explanation given for each term is as follows:

1) Pick p blue balls to occupy LHS
2) Spatially arrange p blue balls in LHS
3) Spatially arrange N-p blue balls on RHS
4) Spatially arrange all red balls inside the box.

Question:
I get the first three terms. I feel that the last term should not be there.

Since all red balls are identical, my feeling is that rearranging them in the remaining spots is not going to give any unique arrangements. Can someone please explain why the 4th term needs to be there?

Thanks!

The balls apparently aren't indistinguishable. If they were, then the first factor shouldn't be there either because any selection of p blue balls out of N would be the same.

Makes perfect sense now. Thanks!

1. What is the "balls in a box" counting problem in statistical mechanics?

The "balls in a box" counting problem is a classic problem in statistical mechanics that involves determining the number of ways that a certain number of identical particles (or "balls") can be arranged in a given volume or "box". This problem is important in understanding the behavior of gases and other systems of particles.

2. How is the "balls in a box" counting problem related to entropy?

The "balls in a box" counting problem is directly related to entropy, which is a measure of the disorder or randomness in a system. In statistical mechanics, entropy is related to the number of microstates (or possible arrangements) that a system can have. The more microstates a system can have, the higher its entropy.

3. What is the difference between distinguishable and indistinguishable particles in the "balls in a box" counting problem?

In the "balls in a box" counting problem, distinguishable particles are those that can be identified from one another, while indistinguishable particles are identical and cannot be differentiated. This distinction is important in determining the number of ways that the particles can be arranged in the box, and it can affect the overall entropy of the system.

4. How does the number of particles and the volume of the box affect the solutions to the "balls in a box" counting problem?

The number of particles and the volume of the box are key factors in solving the "balls in a box" counting problem. As the number of particles increases, the number of possible arrangements also increases, leading to a higher entropy. Similarly, as the volume of the box increases, there are more possible locations for the particles, resulting in a higher number of microstates and a higher entropy.

5. What are some real-world applications of the "balls in a box" counting problem?

The "balls in a box" counting problem has many real-world applications, particularly in the fields of chemistry, physics, and engineering. It can be used to understand the behavior of gases, the diffusion of molecules, and the properties of materials. Additionally, it is a fundamental concept in understanding thermodynamics and the laws of entropy.

• Precalculus Mathematics Homework Help
Replies
10
Views
826
• Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
Replies
4
Views
1K
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
32
Views
2K
Replies
3
Views
1K
Replies
9
Views
1K
Replies
3
Views
851
• Precalculus Mathematics Homework Help
Replies
5
Views
933
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K