Lesson Explainer: Brownian Motion | Nagwa Lesson Explainer: Brownian Motion | Nagwa

# Lesson Explainer: Brownian Motion Physics

In this explainer, we will learn how to describe the Brownian motion of particles and how this explains the diffusion of gases.

At a small scale (at the molecular level) particles in fluids are numerous and can be considered identical.

Individually, particles in a gas, for example, move in a predictable way based on their collisions with other particles.

Consider a situation in which just very few particles are confined in a container, and imagine further that all the particles except one are fixed in place. The particle that is free to move may collide with the stationary particles as shown.

Next, imagine that we “release” one of the previously fixed particles so that it, too, is free to move.

Continuing this trend, all of the particles will eventually be in motion, colliding off one another and the container walls.

With all the particles moving, together, the particles constitute a gas. A gas consists of freely moving particles.

If more particles are added to the container, any given particle will experience collisions more frequently, as shown below.

This will cause the highlighted particle to change directions more often and follow a more seemingly random path, but again, the particle’s motion is determined by collisions with other matter. If we knew the position and velocity of every particle in the container at some initial moment, we could, in theory, calculate the resulting motion of any of the particles precisely.

For large systems of particles, however, such calculations may be beyond available computing power. In such cases, particle motion may appear random, while actually being predictable given enough information about the system.

In such an environment, particle motion is known as Brownian motion.

### Definition: Brownian Motion

It is the seemingly random motion of particles in a system of many identical particles such as a liquid or a gas.

Particles in such a system are equally likely to be pushed in any direction by a collision. Over time, a particle experiencing Brownian motion will tend toward a displacement of zero from its original position.

### Example 1: Identifying Brownian Motion of an Individual Particle

Which of the following objects’ motion best represents Brownian motion?

Brownian motion is the seemingly random movement of small particles in a gas or liquid. This movement is caused by collisions between particles, so a particle experiencing Brownian motion will rapidly change direction. These collisions are equally likely to redirect a given particle in any direction; therefore, Brownian motion appears random and irregular.

Choice A shows a stationary object. Since objects experiencing Brownian motion do move, we can eliminate this choice from consideration.

Choice B depicts an object rotating about an axis through its center. Brownian motion, however, involves particles moving like billiard balls: frequently changing direction due to collisions. An object that stays in one place while rotating does not demonstrate this type of motion.

Choices C and D show objects in motion, but the motion is either in a straight line (C) or a perfect circle (D). This is not the sort of motion an object would display as it experiences many collisions with identical moving objects.

The last option, choice E, shows movement that we would struggle to predict, reflecting the apparently random nature of Brownian motion. For our answer, we select this option.

### Example 2: Choosing the Most Correct Description of an Object in Brownian Motion

Which of the following most correctly explains the Brownian motion of objects?

1. An object in Brownian motion has no net force acting on it.
2. An object in Brownian motion collides repeatedly with other objects, each of which moves independently of the other.
3. An object in Brownian motion has a constant mechanical energy, so its net displacement over time is zero.
4. An object in Brownian motion is attracted to its initial position, so its net displacement over time is zero.
5. An object in Brownian motion collides perfectly elastically with anything that it contacts, so it always tends to return to its initial position.

Rather than applying to objects of everyday size, Brownian motion describes how particles move at a small scale. Specifically, we can imagine molecules of a gas or liquid colliding with many other identical particles contained in a volume.

Under these conditions, the only significant factor determining a given particle’s motion is the collisions it experiences. The force between any pair of particles in the volume is otherwise effectively zero.

Considering our choices, only option D, claiming a force of attraction between an object experiencing Brownian motion and its initial position, is explicitly incorrect. There is no object to cause this force, so this force does not exist.

Choice A is correct that, in general, an object in Brownian motion experiences no net force. However, there are brief instants in time when the object collides and does experience a net force; this is how its direction of motion changes.

Choice C says an object in Brownian motion has a constant mechanical energy, so its net displacement over time is zero. It is true that if we add together the kinetic energy and potential energy of a particle in this motion, the (mechanical) energy will be constant over time. It is also true that the average displacement over time of an object in Brownian motion is zero; it will tend to move equally in all directions from its initial position.

There is no reason though for these two true claims to be logically connected. For example, a particle moving in a straight line at constant speed and elevation would have constant mechanical energy, but this would not imply that its displacement over time is zero. Choice C does not offer a sound inference between the facts it reports.

Choice E is similar, reporting two correct facts about Brownian motion yet not connecting them in a way that makes physical sense.

Compared to all these choices, choice B gives the most correct description of an object in Brownian motion. A system giving rise to this motion consists of many independently moving objects that interact through collisions. The irregular path an object follows as a result of these many collisions is an indicator of Brownian motion.

### Example 3: Determining the Direction of an Object in Brownian Motion

The object shown moves with Brownian motion between the points shown at the instants , , and . Which of the directions shown is the object most likely to move in after ?

1. I
2. II
3. III
4. IV
5. All directions are equally likely.

We can see that between and , as well as between and , the object shown moves to the right. If we knew nothing else about this object, we could reasonably predict that its next movement would also be to the right.

We are told, however, that this object is in Brownian motion. This means the object moves due to collisions with other particles moving independently around it. These other particles can move in any direction and do so with equal likelihood.

The object we are considering then is just as likely to be bumped up, down, left, right, or in any direction at all. We choose as our answer choice E: all directions are equally likely.

One effect of Brownian motion is for particles in a container to distribute themselves evenly throughout the container. For example, if more particles are initially on a given container’s right side, over time, the particles will tend to move left and become uniformly distributed.

This happens because any particles that happen to be moving into a less-dense region of the container will face fewer barriers to motion. Fewer particles exist for them to collide with, making this expansion easier than movement through more-dense regions.

The process of particles spreading out evenly over a volume is called diffusion, and it occurs generally for liquids and gases.

### Example 4: Predicting the Diffusion of Two Different Gases

The container shown in the diagram consists of two equally sized chambers. The chambers hold two different gases, both at the same temperature. The particles of both gases undergo Brownian motion. The separating wall between the chambers is removed. Which of the following most correctly represents the distribution of the particles of the gases a few minutes after the separating wall has been removed?

Considering this scenario, we want to think through what will happen to each gas, if anything, when the barrier separating the gases is removed.

The gases on either side of the barrier are different. We can say then that initially, the concentration of the gray-colored gas particles on the right side of the barrier is zero and so is the initial concentration of yellow-colored gas particles on the left side of the barrier.

We can understand how both of these gases will respond to the removal of the separating wall by thinking of them as completely independent. That is, ignoring collisions with the other type of gas, the gray-colored particles will move as though the starting conditions were as shown below.

And the yellow-colored particles will behave as though the starting conditions were as shown below.

When the barrier is removed, both gases will diffuse throughout the larger chamber. This involves spreading out evenly through all available space. When the gases have diffused completely, the concentration of each one will be nearly equal throughout the larger volume.

After several minutes of the gases mixing freely, we expect roughly the same number of yellow particles on the right and on the left, and the same goes for the gray particles. Reviewing the available choices, only one shows this equal mixing of gases: choice D.

### Example 5: Determining Particle Distribution for Given Brownian Motion

A very small object moves with Brownian motion as it passes through a region containing gas particles. The path of the object is shown in the diagram. Which of the following most correctly represents the distribution of gas particles in the region?

Studying the path followed by our object in Brownian motion, we see from the lengths of the arrows how far the object moved before it collided with a particle of the gas in that region. This tells us roughly how far apart the particles of the gas are.

We want to choose which diagram most correctly shows the distribution of gas particles. Since we know these particles will diffuse and, therefore, spread out nearly evenly, we can eliminate choices B and D from consideration, as both show uneven particle distributions.

To choose between A, C, and E, we want to figure out which of these diagrams shows the average distance between particles to be approximately equal to the average length of the arrows in the diagram in the question.

Choice A shows particles separated by less than this average distance, while choice E shows particles separated by too great a distance.

Choice C shows a diffuse gas of appropriate density for the Brownian motion of the object shown.

### Key Points

• For large systems of identical particles, the motion of those particles is considered predictable or random depending on whether it can be calculated or not.
• Brownian motion describes object motion at a small scale in liquids and gases. This motion is caused by collisions with surrounding particles that make the object of interest move in an irregular, seemingly random path.
• Over time, an object in Brownian motion will tend toward a net displacement of zero.
• Due to collisions between particles, fluids spread out evenly over any available volume, a process called diffusion.