Why does diffusion always go down the concentration gradient?

[TVI_1405]

I have been wondering if there is an explanation to why diffusion always goes down the concentration gradient? If it is a random molecular movement than why do we always end up with a uniform distribution of molecules?

 

[mfb]

The mixed state has the highest entropy.
Put 100 blue balls in a container, add 100 red balls on top of them, mix it. Remove the upper 100 balls. What do you expect?

While there is the option to have 100 red balls, or 100 blue balls, most options to draw "100 out of 200" are close to 50 blue and 50 red balls.

 

[AlephZero]

Diffusion goes at the same rate in both directions, but the amount of material that moves from any point is proportional to the amount of material that was there to start with.

Supposed there is 50% concentration at A and 20% at B, and in a given time one tenth the material diffuses the distance between A and B.

The material diffusing from A reduces the concentration at A by 5% (one tenth of 50%) and increases the contentration at B by 5%.
The material diffusing from B reduces B by 2% and increases A by 2%.
So overall, the concentration at A reduces by 3% and at B increases 3%.

If all the material is the identical, you can't measure the fact that some of it moved UP the concentration gradient, and the only interesting numbers are the overall 3% change.

 

[HallsofIvy]

You can also think of it in terms of "random motion". There are many possible motions that will mix "red balls" with "blue balls", very few that will result in separating them. And that is simply because there are far more possible arrangements for the balls mixed together than for the same balls separated by color.

For a very simple example, if you have n₁ blue balls and n₂ red balls, arranged in a line, there are a total of (n₁+ n₂)! possible arrangements. Of those only 2(n₁!n₂!) have "all red" on one side and "all blue" on the other. For large n₁ and n₂ the first will be far larger than the latter.
2000! is on the order of 10³⁷⁴ while 2(100!100!) is of the order 10³²¹⁶, not particularly different, but the difference increases as the numbers increase.

 

[Edgardo]

As mfb and the others have already mentioned, it has something to do with entropy. Entropy basically means that you count the number of possibilities that a certain state can occur.

Imagine that you sprayed perfume in your room. And imagine further that you divided your room in 1m³ cubes. Then consider the two states:
State 1: All perfume molecules are in a 1m³ cube, e.g. in the corner of your room.
State 2: All perfume molecules are "evenly" distributed, i.e. every 1m³ cube has the same number of perfume molecules.

It turns out that there are much more possibilities to "realize" state 2.

I recommend reading the websites below. They explain the concept of macrostates and microstates.

An Introduction to Entropy by Gary Felder

Lecture 5 : Statistical Interpretation of Entropy

Entropy and a box of air

Checkerboard example on page 3 of the PDF
This explains how to calculate the number of possibilities using the binomial coefficient.

 

[sophiecentaur]

It is a bit hard to swallow the idea that 'statistics' can cause a 'force', though. In the end, you just need to accept that sort of thing and see the statements fro what they really mean.

 

[TVI_1405]

Diffusion goes at the same rate in both directions, but the amount of material that moves from any point is proportional to the amount of material that was there to start with.

Supposed there is 50% concentration at A and 20% at B, and in a given time one tenth the material diffuses the distance between A and B.

The material diffusing from A reduces the concentration at A by 5% (one tenth of 50%) and increases the contentration at B by 5%.
The material diffusing from B reduces B by 2% and increases A by 2%.
So overall, the concentration at A reduces by 3% and at B increases 3%.

If all the material is the identical, you can't measure the fact that some of it moved UP the concentration gradient, and the only interesting numbers are the overall 3% change.

Thank you, this is a great explanation. So, I guess if the material is not the same we can trace the exact mechanism of diffusion. That's helpful.

 

[TVI_1405]

It is a bit hard to swallow the idea that 'statistics' can cause a 'force', though. In the end, you just need to accept that sort of thing and see the statements fro what they really mean.

True, but sometimes it is worthwhile reviewing the existing axioms too...

 

[TVI_1405]

As mfb and the others have already mentioned, it has something to do with entropy. Entropy basically means that you count the number of possibilities that a certain state can occur.

Imagine that you sprayed perfume in your room. And imagine further that you divided your room in 1m³ cubes. Then consider the two states:
State 1: All perfume molecules are in a 1m³ cube, e.g. in the corner of your room.
State 2: All perfume molecules are "evenly" distributed, i.e. every 1m³ cube has the same number of perfume molecules.

It turns out that there are much more possibilities to "realize" state 2.

I recommend reading the websites below. They explain the concept of macrostates and microstates.

An Introduction to Entropy by Gary Felder

Lecture 5 : Statistical Interpretation of Entropy

Entropy and a box of air

Checkerboard example on page 3 of the PDF
This explains how to calculate the number of possibilities using the binomial coefficient.

Thank you for providing so many useful links. Highly appreciated.

 

[TVI_1405]

The mixed state has the highest entropy.
Put 100 blue balls in a container, add 100 red balls on top of them, mix it. Remove the upper 100 balls. What do you expect?

While there is the option to have 100 red balls, or 100 blue balls, most options to draw "100 out of 200" are close to 50 blue and 50 red balls.

Thank you. I will try that out of curiosity.

 

[TVI_1405]

You can also think of it in terms of "random motion". There are many possible motions that will mix "red balls" with "blue balls", very few that will result in separating them. And that is simply because there are far more possible arrangements for the balls mixed together than for the same balls separated by color.

For a very simple example, if you have n₁ blue balls and n₂ red balls, arranged in a line, there are a total of (n₁+ n₂)! possible arrangements. Of those only 2(n₁!n₂!) have "all red" on one side and "all blue" on the other. For large n₁ and n₂ the first will be far larger than the latter.
2000! is on the order of 10³⁷⁴ while 2(100!100!) is of the order 10³²¹⁶, not particularly different, but the difference increases as the numbers increase.

I guess it would be interesting to see the distribution of only several molecules of some material in the medium. Although they would have a very high probability to be separated but it would be interesting to find out how large and stable that separation would be (would they ever come up close to one another or would they mostly stay apart)...

 

[mfb]

(would they ever come up close to one another or would they mostly stay apart)...

Both. Sometimes, they are close to each other, but this is really rare (depending on the number of molecules). However, if you wait long enough, you can be sure (with probability 1) that it will happen sometimes.

 

[Edgardo]

The following example makes it clearer what I meant with possibilities for a certain state:

Consider a room with four persons, their names are Alice, Bob, Charly and Daisy.
Each person has a coin. They throw the coin and if it shows heads, they go to the left side of the room,
and if it shows tails they go to the right side.

They throw their coin and the result may be:
[ AB | CD ]
which means that Alice and Bob are on the left side whereas Charly and Daisy are on the right side.

Let's call [ AB | CD ] a microstate. Other examples for microstates are:
[ C | ABD ]
[ - | ABCD].

--

Now we can ask the questions:
(i) How many possible ways are there for 4 persons being on the left side?
(ii) How many ways are there for 2 persons being on the left side and 2 on the right side?
In other words, we are asking for the distribution of the persons.

We can describe this distribution with the macrostate (#left, #right),
e.g. (3,1) means 3 persons on the left and 1 person on the right.Now, back to our questions:
(i) How many ways are there for the macrostate (4,0)?
(ii) How many ways are there for the macrostate (2,2)?

Let's write down a table:

microstates     macrostate
[ ABCD | - ]    (4,0)

Obviously, [ ABCD | -] is the only microstate that fulfills the macrostate (4,0). Therefore, there is only 1 possible way to realize the macrostate (4,0).

microstates      macrostate
[ AB | CD ]       (2,2)
[ AC | BD ]
[ AD | BC ] 
[ BC | AD ]
[ BD | AC ]
[ CD | AB ]

Here, you can see that there are 6 possible ways to realize the macrostate (2,2).To stress the meaning of this imagine that Alice, Bob, Charly and Daisy will throw their coin
and form a macrostate.
You have a friend with whom you make a bet. What macrostate would you place your money on?

Note:
- Can you determine how many ways there are for the other macrostates?
- Do you see what this has to do with the binomial coefficient?

 

[the_emi_guy]

If you want an impressive demonstration of rates of diffusion do the following:

1 - Put a small balloon into a large plastic soda bottle, inflate the balloon with argon and tie it off.

2 - Hold soda bottle with neck downward and fill it with helium (the lighter helium will rise and displace all of the air).

3 -Cap the bottle and wait. Over several hours you will see balloon expand inside the bottle.

Diffusion is acting to equalize concentration of both gasses on both sides of the balloon,
but the rate of diffusion of He is higher due to its smaller molecular mass and thus higher average molecular velocity.

I did this years ago when playing around with gasses. I have never seen this experiment described anywhere but I'm sure I'm not the first to have done it.

 

[Edgardo]

Here are more links on the statistical interpretation of entropy:

1) The Statistical Interpretation of Entropy by Todd Timberlake
with a nice paper.

2) A video on Boltzmann's entropy equation

 

[the_emi_guy]

It is useful to visualize diffusion in terms of kinetic motion.

Consider two containers with a door between.

Left container holds 100 gas molecules, right container holds 2.

Door is closed: The door feels a force from the left container because it is struck by random walking gas molecules more often.

Door is open: gas molecule drift from left to right because doorway is encountered by a random walking molecule from left container more often (because there are more molecules there).

Once mixing is complete and each container has 51 molecules, probability of left and right random walking molecules encountering the doorway are equal. Close door and there is no force because both sides of door are struck by molecules equally.

 

[TVI_1405]

Thanks a lot, everyone. It is really helpful.