Entropy and Freezing Point Depression

Hi, I am a middle school teacher, teaching general science. My students asked me the same question on why salt lowers the melting point of ice. I have read your explanation. However I have some lingering questions:[1] Can you elaborate about the part on "entropy"?[2] I look up "entropy". It seems so complicated as it is under the topic of thermodynamics. Any recommendation on a good website that introduces thermodynamic entropy conceptual. The complex equations are giving me a headache.[3] Is this possible to relate this to the kinetic model of matter?
- Jerry (age 35)
Singapore

I'm not sure where to find a good website to introduce entropy, but Reif's book in the Berkeley series is very good: . (The Wikipedia entropy article is currently (6/15) hopelessly complex and confusing for somebody first learning the topic.)

The basic idea is this. Take some set of gross fixed conditions, e.g.
{number of molecules, temperature of environment, pressure}. How likely are you to find the molecules in some type of arrangement (liquid, solid)? Nature turns out to be extremely indifferent to which state things are in. In equilibrium the probability of finding something (e.g. liquid) is just proportional to the number of quantum states that look like that. Will the water be liquid or solid? It just depends on which type is consistent with more quantum states. The total number of quantum states that look like the liquid is the product of the number of states of the water that are liquid times the number of states of the environment that go along with the liquid. It's a product for the same reason that if you roll two dice there are 6*6=36 possible results. The same goes for the solid. The number of environment states depends on whether the water is liquid or solid, because the solid has lower energy, so that leaves more energy (and thus more states reachable) for the environment. 

So finding the most likely form means finding the form with the biggest product of system state number times environment state number. It's a nuisance to maximize a product. So we convert it to a sum by taking the logarithm. The most likely form has the biggest log of the total number of states, meaning the biggest sum of the  log of the number of states of the system plus the log of the number of states of the environment.

These words are getting awkward. Let's make up a new one for "natural log of number of states". The word is "entropy".  So the most likely result is the one that maximizes total entropy. 

You can  see why whether the water ends up liquid or solid depends on how much the environment's entropy goes up when it gets the energy released by conversion to solid. It would be nice to have a name for that too. We call the energy required to increase the entropy by one unit the "temperature". At high temperature the environment entropy doesn't change much so the water find the form that gives itself a lot of entropy: liquid. At low temperature, the environment entropy is very sensitive to an energy it can pick up, so the water takes the form that releases the most energy: solid.

Now you can also see why solutes in water favor the liquid state. If some of the water freezes to ice, the solutes (left in the liquid) have less room to run around and thus have lower entropy. That means you have to lower the environment temperature a little more to make the solid the form that maximizes total entropy.

Unfortunately some of this argument is obscured by traditional definitions of entropy, which use weird historical units and don't bring out the relation to number of states. To get more precise, there are some subtle adjustments when some of the quantum states are more likely than others, but you don't need that for starters.

Yes, it is possible to relate all this to the classical kinetic model, especially for something simple like a gas. For example, on another question () we derive pV=NkT for an ideal gas from this picture, which you can connect to the common derivation from a classical kinetic picture.

Please follow up if more explanation is needed.

Mike W.

p.s. Googling around turned up this  good discussion:   .

 

(published on 06/16/2015)

What is the scientific reason salt water lowers the freezing point when added to water? What properties of the salt force this action?
- Jane (age 14)
New York City, U.S

Hi Jane- We discuss the reason in some of the answers above. The key properties needed are

1. The salt dissolves in the water.

2. The salt almost completely does not dissolve in ice.

That means that when ice forms, it reduces the room for the salt ions to run around in the liquid. That makes it harder to freeze. 

Exactly the same reasoning applies to anything that dissolves in water but not in ice. So sugar, alcohol, other salts, and so forth all lower the melting point.

Mike W.

 

(published on 11/08/2015)