# Probabilities of molecules being somewhere

In summary, the probability of finding the molecules in the same distribution as before the partition was punctured is (500/600)^500 * (100/600)^100. This takes into account the equal a priori probabilities and the fact that the total number of molecules in each gas remains the same after equilibrium is achieved.
A box is separated by a partition into two parts of equal volume. The left side of the box contains 500 molecules of nitrogen gas; the right side contains 100 molecules of oxygen gas. The two gases are at the same temperature. The partition is punctured, and equilibrium is eventually attained. Assume that the volume of the box is large enough for each gas to undergo a free expansion and not change temperature.

What is the probability that the molecules will be found in the same distribution as they were before the partition was punctured, that is, 500 nitrogen molecules in the left half and 100 oxygen molecules in the right half?

I have no idea how to start this. Would i do

(500!*100!)/600!

?

Thank you for your question. I can help provide some insights into this scenario.

To start, we can assume that the molecules in each gas are identical and indistinguishable, meaning that we cannot tell one nitrogen molecule from another or one oxygen molecule from another. This assumption is important because it means that the distribution of molecules within each gas does not matter, as long as the total number of molecules in each gas remains the same.

Next, we can apply the principle of equal a priori probabilities. This means that before the partition was punctured, there was an equal chance of any molecule being on the left or right side of the box, regardless of its type (nitrogen or oxygen). So, the probability of finding a nitrogen molecule on the left side was 500/600, and the probability of finding an oxygen molecule on the right side was 100/600.

Now, when the partition is punctured and equilibrium is achieved, the molecules will have a chance of moving from one side to the other. However, the total number of molecules on each side will remain the same (500 nitrogen and 100 oxygen). This means that the probability of finding a nitrogen molecule on the left side is still 500/600, and the probability of finding an oxygen molecule on the right side is still 100/600.

Therefore, the probability of finding the molecules in the same distribution as before (500 nitrogen on the left and 100 oxygen on the right) is:

(500/600)^500 * (100/600)^100

This is because the probability of finding each molecule in its original location is independent of the other molecules, so we can multiply the individual probabilities together.

I hope this helps to answer your question. Let me know if you have any further questions or need clarification. Happy experimenting!

Yes, that is the correct approach. To find the probability of the molecules being in the same distribution as before, we need to calculate the number of ways the molecules can be arranged in the two halves of the box. This can be done using the formula for permutations, which is n!/r!, where n is the total number of molecules and r is the number of molecules in one half of the box. In this case, n=600 and r=500 for nitrogen and r=100 for oxygen. So the probability would be (500!*100!)/600!, which is approximately 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

## What is the concept of probability in relation to molecules?

The concept of probability in relation to molecules refers to the likelihood or chance of a molecule being in a specific location or state at a given time. It is a measure of uncertainty and is affected by factors such as temperature, pressure, and molecular interactions.

## How do scientists calculate the probabilities of molecules being somewhere?

Scientists use mathematical models and statistical methods to calculate the probabilities of molecules being somewhere. These methods take into account various physical properties of the molecules, such as their mass, size, and energy, to determine their potential locations and states.

## Can the probabilities of molecules being somewhere be predicted with 100% accuracy?

No, the probabilities of molecules being somewhere cannot be predicted with 100% accuracy. This is because there are inherent uncertainties in the behavior of molecules, and the calculations are based on statistical averages rather than exact values.

## How do external factors affect the probabilities of molecules being somewhere?

External factors such as temperature, pressure, and molecular interactions can significantly affect the probabilities of molecules being somewhere. For example, an increase in temperature can lead to higher molecular motion and a wider range of potential locations for the molecules.

## Why is understanding the probabilities of molecules being somewhere important in science?

Understanding the probabilities of molecules being somewhere is crucial in science because it allows us to make predictions and explanations about the behavior of matter. It is also essential in fields such as chemistry, physics, and biology, where the movement and interactions of molecules play a significant role in various processes and phenomena.

Replies
32
Views
1K
Replies
6
Views
2K
Replies
4
Views
1K
Replies
2
Views
3K
Replies
1
Views
3K
Replies
16
Views
2K
Replies
9
Views
3K
Replies
19
Views
971
Replies
2
Views
4K
Replies
23
Views
3K