Entropy and Ice

There are many more ways to be messy than organized. Things tend toward disorder because, statistically, there's just so many more ways to be disordered than ordered. This is the central idea behind the Second Law of Thermodynamics. Exhale and the carbon dioxide that comes from your mouth is not likely to float there in front of your face, bound up like a little ball of gas. It's not likely to stay so organized. The molecules are going to randomly move about and distance themselves from one another. There's no force pushing them apart; it's random motion. Divide the room you're in into a billion equal-sized cubes. The cubes right in front of your mouth have a large number of carbon dioxide (CO₂) molecules in them, right after you exhale. The other cubes around the room have some CO₂ molecules scattered about them. With all these molecules whizzing about at large velocities, what are the odds that as many CO₂ molecules will enter the cubes (the space; the volume) in front of your mouth as will leave that volume? Practically zero. The universe naturally tends towards disorder. In fact, nothing will happen spontaneously unless it (i.e. the process, the reaction) increases the disorder of the universe. Entropy is a property of a system that measures the system's disorder. When a system becomes more disordered, we can say its entropy has increased. While we can talk of entropy in a qualitative sense, it's a number. (Its units are joules per Kelvin; energy is measured in joules, while temperature is measured in Kelvin, so we have amount of energy per unit (think degree) of temperature.) We see some process occur spontaneously and we think, entropy must have increased.

Go outside on a winter day and you may see ice -- solid water. And solids are more organized than liquids. In a piece of ice the water molecules are arranged in a certain pattern, and the ice itself takes up a small amount of space and doesn't spread out like liquid water. How could water suddenly become more organized in its structure? The ice you see formed spontaneously. Does this violate the Second Law of Thermodynamics, which states that the entropy of an isolated system must increase for any spontaneous process?

It's true that the entropy of the water decreased when the ice froze, but what about the entropy of the "isolated system"? Does the water/ice constitute an isolated system? No. It's in direct contact with the ground and with the air. We realize the Second Law holds only when we realize that the entropy of something else increased when the entropy of the water decreased. And not only did the entropy of something else increase, but it increased by more than the entropy of the water decreased. What became more disordered in this process? The air around the water-turned-ice. When ice freezes, it releases heat, and that heat goes into the air. The heat warms the air and gets some air molecules to speed up a bit. It increases the air's local temperature ever so slightly.

I need to mention that our intuitive notion of temperature is a result of atoms and molecules in motion (i.e. thermal energy). When something is heated up, its atoms/molecules move and vibrate more rapidly. When something is cooled down, its atoms/molecules move and vibrate less rapidly. Our skin can sense these motions, in the aggregate, and our brains interpret these sensations as temperature. With increased thermal energy comes increased entropy (or disorder). So hot air is more disordered than cold air. And, in our example, the air around the water-turned-ice is slightly more disordered than before the water froze, because the air has just been heated up a bit.

What dictates when the increase in the entropy of the air around a pool of water is enough to allow the water to become more organized (i.e. to turn to ice)? We now know that the increase in the entropy of the air must be greater than the decrease in the entropy of the water for such a process to occur spontaneously. (Thanks to the Second Law of Thermodynamics.) When is this the case? Why doesn't a pool of water, out on the sidewalk, turn to ice on a summer day? Again, what determines when the entropy of the air will increase by a greater amount than the entropy of the water will decrease? ... The initial temperature of the air, of course. But why?

Scenario A: I'm in the mood to dance. I go to a night club and make my way to the dance floor. Some hundred people are already shaking their bodies to the beat. I join in.
Scenario B: I'm in the mood to dance. I head over to the campus library and walk directly to the large reading room. Some hundred students are spread about the room, at tables, quietly reading. I start dancing in the middle of the room. They stare.

In which scenario have I contributed most to the energy in the room? Definitely scenario B. At the night club, there's already a scene. A hundred people dancing. I go relatively unnoticed. At the library, however, I'm really stirring things up. I'm definitely noticed. With it so calm and quiet prior to my arrival, I significantly contribute to the energy in the room once I start dancing. Likewise, adding heat to a cool system differs from adding heat to a hot system. "[T]he arriving [thermal] energy more noticeably stirs up the molecules of a cool system, which have little thermal motion, than those of a hot system, in which the molecules are already moving vigorously." (Chemical Principles, 3rd edition, Atkins & Jones, p. 247)

We can now reason that the heat released by freezing water really stirs things up on a cold day, when the air molecules have little thermal motion. And the heat that would be released by the water on a warm day would be an insignificant contribution to the thermal motion of the warm air. It would add little to the entropy of the air. So for a given amount of heat, introduced into a system, entropy will increase more when the system is at a low temperature than when it is at a high temperature. What's the magic temperature below which a bit of released heat (from water) will increase the entropy of the air more than the entropy of the water will decrease? Zero degrees centigrade, of course. Water, we know, freezes at 0 degrees C. When the outside air temperature is greater than 0 degrees C, the heat that would come from the freezing of water is not capable of increasing the entropy of the air enough to warrant the freezing. Only when the temperature is 0 degrees or less is the entropy of the entire system increased by the freezing of water, so only then is freezing spontaneous.

Sources: Chemical Principles, 3rd edition, by Atkins & Jones;
Principles of Chemistry, by Michael Munowitz