# Random Walk of KMnO4 in Water: Why Do We Observe Non-Probabilistic Behavior?

• Jigyasa
In summary, the conversation discusses the random walk problem in statistical mechanics and the observation that the spread of KMnO4 in a beaker of water does not align with probabilistic assumptions. The probability of all molecules ending up at the same place is computed, but this may not be the desired distribution. The conversation also mentions that the limiting case as N becomes large is an even distribution and that boundary effects cannot be disregarded in a finite beaker.
Jigyasa
I had a question regarding the random walk problem in statistical mechanics. If I drop, say, one molecule of KMnO4 in a beaker of water, what we generally observe (spread of KMnO4 to the ends of the beaker) is different from what we should get from probabilistic assumptions. I must be going wrong somewhere in what I'm thinking but I can't point my finger at it.If I consider one molecule of KMnO4 , then the probability of it taking r steps to the right (or left), out of a total of N steps, is NCr *(1/2)^N (assuming unbiased random walk). For Avogadro's number of molecules, this probability is now raised to the power of Avogadro's number. This is maximum if r = N/2. Physically this means that the probability of the whole solution becoming coloured (KMnO4 traveling to the far ends ) is less than the probability of only a part of the solution becoming coloured (because if KMnO4 moves a total of10 steps, the probability of it moving 5 steps to the right and 5 steps to the left is maximum. In a sense, it oscillating is more probable ) But we almost always see that the whole solution turns purple in due course of time.Is it that the need to overcome concentration gradient dominates so KMNo4 has to reach the ends? If yes, then why do we use probabilities in statistical mechanics when systems may or may not be governed by probabilistic assumptions?

Or maybe I'm wrong in assuming this to be an unbiased random walk

Jigyasa said:
For Avogadro's number of molecules, this probability is now raised to the power of Avogadro's number.
It is not clear what you are trying to do here. What you are actually computing is the probability of all molecules ending up at the same place, r steps away. This would typically not be what you want. What you would typically want is the distribution of the molecules, which will be the same as that of a single molecule.

If I only consider a single molecule, even then the probability of the molecule reaching the far ends of the beaker is coming out to be less than the probability of it oscillating somewhere in between (because NCr is max for r = N/2, this will always be the case assuming unbiased random walk of KMNO4)

Obviously. However, the limiting case as N becomes large is an even distribution.

Unless your beaker is infinite, you also cannot disregard boundary effects.

## 1. What is a random walk?

A random walk is a mathematical concept that describes the movement of an object or particle in a random or unpredictable manner. It is often used to model the behavior of molecules in a liquid or gas, or the movement of stock prices in financial markets.

## 2. How does KMnO4 move in water during a random walk?

In a random walk, KMnO4 molecules will move in a zig-zag pattern, constantly changing direction in a random manner. This is due to the constant collisions and interactions with water molecules, which cause the KMnO4 molecules to change direction and speed.

## 3. What factors can influence the random walk of KMnO4 in water?

The random walk of KMnO4 in water can be influenced by factors such as temperature, pressure, and concentration of KMnO4 and other particles in the water. These factors can affect the speed and direction of the KMnO4 molecules, leading to different patterns of movement.

## 4. What is the significance of studying the random walk of KMnO4 in water?

Studying the random walk of KMnO4 in water can provide valuable insights into the behavior of molecules in liquids, which has important implications in fields such as chemistry, physics, and biology. It can also help in understanding and predicting the diffusion and mixing of substances in water.

## 5. How is the random walk of KMnO4 in water related to Brownian motion?

The random walk of KMnO4 in water is closely related to Brownian motion, which describes the random movement of particles in a fluid due to collisions with other particles. In fact, Brownian motion is often used as a model to explain the random walk of particles, including KMnO4, in liquids.

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