While I'm not an expert on geysers and would need to visit the library to verify my ideas, I believe that they operate the same way a coffee percolator does. Both objects involve a narrow water-filled channel that's heated from below. As the temperature at the bottom of the water column increases, the water's stability as a liquid decreases and its tendency to become gaseous steam increases. What prevents this heated water from converting into gas is the weight of the water and air above it, or more accurately the pressure caused by that weight. But when the water's temperature reaches a certain elevated level, it begins to turn into steam despite the pressure. Since steam is less dense than liquid water, the hot water expands as it turns into steam and it lifts the column of water above it. Water begins to spray out of the top of the channel, decreasing the weight of water in the channel and the pressure at the bottom of the channel. With less pressure keeping the water liquid, the steam forming process accelerates and the column of water rushes up the channel and into the air. Once the steam itself reaches the top of the channel, it escapes freely into the air and the pressure in the channel plummets. Water begins to reenter the channel and the whole process repeats.
To keep soda carbonated, you should minimize the rate at which carbon dioxide molecules leave the soda and maximize the rate at which those molecules return to it. That way, the net flow of molecules out of the soda will be small. To reduce the leaving rate, you should cool the soda—as long as ice crystals don't begin to form, cooling the soda will make it more difficult for carbon dioxide molecules to obtain the energy they need to leave the soda and will slow the rate at which they're lost. To increase the return rate, you should increase the density of gaseous carbon dioxide molecules above the soda—sealing the soda container or pressurizing it with extra carbon dioxide will speed the return of carbon dioxide molecules to the soda. Also, minimizing the volume of empty bottle above the soda will make it easier for the soda to pressurize that volume itself. The soda will lose some of its carbon dioxide while filling that volume, but the loss will quickly cease.
One final issue to consider is surface area: the more surface area there is between the liquid soda and the gas above it, the faster molecules are exchanged between the two phases. Even if you don't keep carbon dioxide gas trapped above soda, you can slow the loss of carbonation by keeping the soda in a narrow-necked bottle with little surface between liquid and gas. But you must also be careful not to introduce liquid-gas surface area inside the liquid. That's what happens when you shake soda or pour it into a glass—you create tiny bubbles inside the soda and these bubbles grow rapidly as carbon dioxide molecules move from the liquid into the bubbles. Cool temperatures, minimal surface area, and plenty of carbon dioxide in the gas phases will keep soda from going flat.
As for pouring the soda over ice causing it to bubble particularly hard, that is partly the result of air stirred into the soda as it tumbles over the ice cubes and partly the result of adding impurities to the soda as the soda washes over the rough and impure surfaces of the ice. The air and impurities both nucleate carbon dioxide bubbles—providing the initial impetus for those bubbles to form and grow. Washing the ice to smooth its surfaces and remove impurities apparently reduces the bubbling when you then pour soda of it.
You are right that adding salt to water raises the water's boiling temperature. Contrary to one's intuition, adding salt to water doesn't make it easier for the water to boil, it makes it harder. As a result, the water must reach a higher temperature before it begins to boil. Any foods you place in this boiling salt water (e.g. eggs or pasta) find themselves in contact with somewhat hotter water and should cook faster as a result. That's because most cooking is limited by the boiling temperature of water in or around food and anything that lowers this boiling temperature, such as high altitude, slows most cooking while anything that raises the boiling temperature of water, such as salt or the use of a pressure cooker, speeds most cooking. However, it takes so much salt to raise the boiling temperature of water enough to affect cooking times that this can't be the main motivation for cooking in salted water. By the time you've salted the water enough to raise its boiling temperature more than a few degrees, you've made the water too salty for cooking. It's pretty clear that salting your cooking water is basically a matter of taste, not temperature.
Dissolving solids in water always lowers the water's freezing temperature by an amount that's proportional to the density of dissolved particles. If you double the density of particles in water, you double the amount by which the freezing temperature is lowered.
While salt and sugar both dissolve in water and thus both lower its freezing temperature, salt is much more effective than sugar. That's because salt produces far more dissolved particles per pound or per cup than sugar. First, table salt (sodium chloride) is almost 40% more dense than cane sugar (sucrose), so that a cup of salt weighs much more than a cup of cane sugar. Second, a salt molecule (NaCl) weighs only about 8.5% as much as a sucrose molecule (C12H22O11), so there are far more salt molecules in a pound of salt than sugar molecules in a pound of sugar. Finally, when salt dissolves in water, it decomposes into ions: Na+ and Cl-. That decomposition doubles the density of dissolved particles produced when salt dissolves. Sugar molecules remain intact when they dissolve, so there is no doubling effect. Thus salt produces a much higher density of dissolved particles than sugar, whether you compare them cup for cup or pound for pound, and thus lowers water's freezing temperature more effectively. That's why the salt water is so slow to freeze.
To make good ice cream, you want to freeze the cream in such a way that the water in the cream forms only very tiny ice crystals. That way the ice cream will taste smooth and creamy. The simplest way to achieve this goal is to stir the cream hard while lowering its temperature far enough to freeze the water in it and to make the fat solidify as well. That's where the ice and salt figure in.
By itself, melting ice has a temperature of 0° C (32° F). When heat flows into ice at that temperature, the ice doesn't get hotter, it just transforms into water at that same temperature. Separating the water molecules in ice to form liquid water takes energy and so heat must flow into the ice to make it melt.
But if you add salt to the ice, you encourage the melting process so much that the ice begins to use its own internal thermal energy to transform into water. The temperature of the ice drops well below 0° C (32° F) and yet it keeps melting. Eventually, the drop in temperature stops and the ice and salt water reach an equilibrium, but the mixture is then quite cold—perhaps -10° C (14° F) or so. To melt more ice, heat must flow into the mixture. When you place liquid cream nearby, heat begins to flow out of the cream and into the ice and salt water. More ice melts and the liquid cream get colder. Eventually, ice cream starts to form. Stirring keeps the ice crystals small and also ensures that the whole creamy liquid freezes uniformly.
A dehumidifier makes use of the fact that water tends to be individual gas molecules in the air at higher temperatures but condensed liquid molecules on surfaces at lower temperatures. At its heart, a dehumidifier is basically a heat pump, one that transfers heat from one surface to another. Its components are almost identical to those in an air conditioner or refrigerator: a compressor, a condenser, and an evaporator. The evaporator acts as the cold surface, the source of heat, and the condenser acts as the hot surface, the destination for that heat.
When the unit is operating and pumping heat, the evaporator becomes cold and the condenser becomes hot. A fan blows warm, moist air from the room through the evaporator coils and that air's temperature drops. This temperature drop changes the behavior of water molecules in the air. When the air and its surroundings were warm, any water molecule that accidentally bumped into a surface could easily return to the air. Thus while water molecules were always landing on surfaces or taking off, the balance was in favor of being in the air. But once the air and its surroundings become cold, any water molecules that bump into a surface tend to stay there. Water molecules are still landing on surfaces and taking off, but the balance is in favor of staying on the surface as either liquid water or solid ice. That's why dew or frost form when warm moist air encounters cold ground. In the dehumidifier, much of the air's water ends up dripping down the coils of the evaporator into a collection basin.
All that remains is for the dehumidifier to rewarm the air. It does this by passing the air through the condenser coils. The thermal energy that was removed from the air by the evaporator is returned to it by the condenser. In fact, the air emerges slightly hotter than before, in part because it now contains all of the energy used to operate the dehumidifier and in part because condensing moisture into water releases energy. So the dehumidifier is using temperature changes to separate water and air.
The amount of hot water that's cooling doesn't necessarily determine which bowl of water will cool fastest. That depends on how quickly each gram of the hot water loses heat, a rate that depends both on how much hotter the water is than its surroundings and on how that water is exposed to those surroundings. In general, hot water loses heat through its surface so the more surface that's exposed, the faster it will lose heat. But surface that's exposed to air will lose heat via evaporation and will be particularly important in cooling the water.
In answer to your question, my guess is that the larger bowl of water also exposes much more of that water to the air. Although the larger bowl had more water in it, it allowed that water to exchange heat faster with its environment. If the larger bowl contained twice as much water but let that water lose heat twice as fast, the two bowls would maintain equal temperatures. If you want to see the effect of thermal mass in slowing the loss of temperature, you'll need to control heat loss. Try letting equal amounts of hot water cool in two identical containers—one wrapped in insulation and covered with clear plastic wrap (to prevent evaporation) and one open to the air. You'll see a dramatic change in cooling rate. And if you want to compare unequal amounts of water, use two indentical containers that are only exposed to the cooler environment through a controlled amount of surface area. For example, try two identical insulated cups, one full of water and one only half full. If both lose heat only through their open tops, the full cup should cool more slowly than the half full cup.
When he established his temperature scale, Daniel Gabriel Fahrenheit defined 32 degrees "Fahrenheit" (32 F) as the melting temperature of ice—the temperature at which ice and water can coexist. When you assemble a mixture of ice and water and allow them to reach equilibrium (by waiting, say, 3 minutes) in a reasonably insulated container (something that does not allow much heat to flow either into or out of the ice bath), the mixture will reach and maintain a temperature of 32 F. At that temperature and at atmospheric pressure, ice and water are both stable and can coexist indefinitely.
To see why this arrangement is stable, consider what would happen if something tried to upset it. For example, what would happen if this mixture were to begin losing heat to its surroundings? Its temperature would begin to drop but then the water would begin to freeze and release thermal energy: when water molecules stick together, they release chemical potential energy as thermal energy. This thermal energy release would raise the temperature back to 32 F. The bath thus resists attempts at lowering its temperature.
Similarly, what would happen if the mixture were to begin gaining heat from its surroundings? Its temperature would begin to rise but then the ice would begin to melt and absorb thermal energy: separating water molecules increases their chemical potential energy and requires an input of thermal energy. This lost thermal energy would lower the temperature back to 32 F. The bath thus resists attempts at raising its temperature.
So an ice/water bath self-regulates its temperature at 32 F. The only other quantities affecting this temperature are the air pressure (the bath temperature could shift upward by about 0.003 degrees F during the low pressure of a hurricane) and dissolved chemicals (half an ounce of table salt per liter of bath water will shift the bath temperature downward by about 1 degree F).
The relative stabilities of liquid and gaseous water depend on both temperature and pressure. To understand this, consider what is going on at the surface of a glass of water. Water molecules in the liquid water are leaving the water's surface to become gas above it and water molecules in the gas are landing and joining the liquid water below. It's like a busy airport, with lots of take-offs and landings. If the glass of water is sitting in an enclosed space, the arrangement will eventually reach equilibrium—the point at which there is no net transfer of molecules between the liquid in the glass and the gas above it. In that case, there will be enough water molecules in the gas to ensure that they land as often as they leave.
The leaving rate (the rate at which molecules break free from the liquid water) depends on the temperature. The hotter the water is, the more frequently water molecules will be able to break away from their buddies and float off into the gas. The landing rate (the rate at which molecules land on the water's surface and stick) depends on the density of molecules in the gas. The more dense the water vapor, the more frequently water molecules will bump into the liquid's surface and land.
As you raise the temperature of the water in your glass, the leaving rate increases and the equilibrium shifts toward higher vapor density and less liquid water. By the time you reach 100° Celsius, the equilibrium vapor pressure is atmospheric pressure, which is why water tends to boil at this temperature (it can form and sustain steam bubbles). Above this temperature the equilibrium vapor pressure exceeds atmospheric pressure. The liquid water and the gas above it can reach equilibrium, but only if you allow the pressure in your enclosed system to exceed atmospheric pressure. However, if you open up your enclosed system, the water vapor will spread out into the atmosphere as a whole and there will be a never-ending stream of gaseous water molecules leaving the glass. Above 100° C, liquid water can't exist in equilibrium with atmospheric pressure gas, even if that gas is pure water vapor.
So how can you superheat water? Don't wait for equilibrium! The road to equilibrium may be slow; it may take minutes or hours for the liquid water to evaporate away to nothing. In the meantime, the system will be out of equilibrium, but that's ok. It happens all the time: a snowman can't exist in equilibrium on a hot summer day, but that doesn't mean that you can't have a snowman at the beach... for a while. Superheated water isn't in equilibrium and, if you're patient, something will change. But in the short run, you can have strange arrangements like this without any problem.
Ice will melt fastest in whatever delivers heat to it fastest. In general that will be water because water conducts heat and carries heat better than air. But extremely hot air, such as that from a torch, will beat out very cold water, such as ice water, in melting the ice.
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