Video Transcript
A gas consists of a very large
number of particles that are in Brownian motion. Which of the following most
correctly relates the most likely speed of a particle in a gas to the direction of
its motion? (A) The more particles in the gas
moving in the same direction as the particle, the greater its speed is likely to
be. (B) The more particles in the gas
moving in the same direction as the particle, the smaller its speed is likely to
be. (C) The more particles in the gas
moving in the opposite direction to the particle, the greater its speed is likely to
be. (D) The more particles in the gas
moving in the opposite direction to the particle, the smaller its speed is likely to
be. (E) The speed of a particle is
independent of the direction of its motion.
To begin, let’s recall that
Brownian motion describes the seemingly random movement of many identical fluid
particles. To help us think about this, we can
make some room on screen and draw a diagram. A particle experiencing Brownian
motion changes direction rapidly due to repeated collisions with other
particles. Since the many particles in a
system are colliding with each other over and over again, their motion seems
random. And over time, any given particle
tends to have a net zero displacement since it’s equally likely to be redirected in
any direction at any time.
Now, let’s think about the question
at hand. We want to relate the speed of a
particle in a gas to the direction of its motion. Answer options (A) and (B) talk
about many particles in the gas moving in the same direction, so let’s picture this
scenario. Imagine that somehow we have a
group of particles with most of them moving in the same direction. Let’s focus on the motion of one
single particle. To help us keep track, we can color
this one particle magenta, but note that all the particles are identical in every
other way.
At first, with lots of particles
moving along nicely together, the magenta particle will likely carry on without as
many collisions as usual. But all it takes is one straight
particle to start a chain reaction of collisions in different directions. And we should note that the
identical particles collide elastically. So the bulk of the particles won’t
experience an overall change in momentum. The magenta particle would likely
still experience several changes to its own momentum. So as time passes and the particles
interact, it’s not reasonable to predict that generally the magenta particle would
either be moving faster or slower. So answer options (A) and (B)
aren’t looking great.
Next, let’s think about options (C)
and (D), which talk about many particles moving in the opposite direction to one
particle of interest. Again, let’s sketch out this
scenario and color one particle magenta to help us think about its motion.
Say we have lots of particles
moving to the right and the magenta particle is initially moving left. At first, this one particle is
still not likely to experience as many collisions as it would if all the particles
had random motion in all sorts of directions. But it would eventually collide
with another particle. And in that case, the two
particles’ incoming speeds will determine the magenta particle’s speed after the
collision. Likely, though, it would simply be
accelerated to the right. And then, it would be moving in the
same direction as the bulk of the particles, like we saw before in answer options
(A) and (B).
So none of these four options are
particularly convincing. Considering what we know about the
chaotic nature of fluid particle motion, it would be naive to think that we could
predict a single particle’s speed by noting the direction it moves at a certain
moment. And the thing is, Brownian motion
doesn’t work like this. Options (A) through (D) don’t even
describe Brownian motion, which is what this question is supposed to be about.
In real life, we’re not going to
find a very large number of particles all synchronized and moving together. Even if a smaller group of
particles were somehow sent in the same direction, this configuration wouldn’t be
stable as collisions would likely soon decay any sense of order into the randomness
of Brownian motion. Realistically, the motion of each
gas particle is independent of the others. And the speed of a particle is
independent of its direction of motion, which is why the motion seems so random.
Thus, we know that option (E) is
the best answer. The speed of a particle is
independent of the direction of its motion.
