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As you can see in this demonstration and read in this Wikipedia article if you were to place a cup of liquid nitrogen in a low pressure environment then it would reach a triple point and begin to solidify. However I do not understand why.

As we know lowering the atmospheric pressure reduces the boiling point of a liquid. This is because there are less, in this case, nitrogen particles above the liquid nitrogen, allowing more particles from the liquid to escape into a gas state. However I do not understand what effect this has on it's solid state. Yes, I've seen the phase diagram but a liquid turns to a solid when the energy of its particles lowers to the point they form an ordered structure. The only thing I can think of that would lower the freezing point is that at lower pressure since atoms more easily escape from the liquid into the vapor phase the liquid is left with all the low energy atoms, which turns it to a solid. I couldn't find any relevant information on this.

Another thing I don't understand is why is the temperature of the nitrogen falling? It starts at ~ -196 C but according to the diagram and Wikipedia then falls to ~ -210 C causing it to form solid layers. Considering how cold it is it should only be absorbing temperature from the environment and thus increasing its temperature. Is this linked to my hypothesis above?

To summarize:
1. Considering the linked video and article - what is the mechanism that causes the nitrogen to become solid at lower pressures? Is it because high energy atoms leave the liquid, thus leaving only their low energy brethren behind?
2. Why is the temperature of the nitrogen falling? Is it linked to the hypothesis from 1. where the leaving high energy atoms, by leaving only low energy particles, lower the temperature of the system?

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A couple different phenomena are at work here - first, the presenter in the video is right that the higher energy molecules in the liquid are more likely to escape, leaving lower temperature ones behind, but that's only part of the picture. Here's another part:

When a gas expands, its temperature drops. Think of the Ideal Gas Law, which says that pV = nR*T or pressure x volume = number of particles x ideal gas constant x temperature. This is the way your air conditioner works - compress a gas, it gets hot. Let it cool for a bit toward ambient temp, then expand it again, and it will be colder than ambient.

Now think about what is happening in the vacuum chamber as it is pumped down. You have two regions in the chamber - a high density region of liquid nitrogen with a limited volume and very low temperature, and the air, which is room temp, low density, and larger volume. The liquid nitrogen is boiling as its temperature increases, which will increase the pressure in the bell jar and lower the temperature in the jar toward an equilibrium (before the pump is turned on). If you just leave it alone, the nitrogen boiling will slow when the internal pressure reaches a higher level (raising the boiling point) and the internal temperature reaches a lower level, until eventually the heat flux through the walls of the bell jar is equal to the heat flux required to boil a specific amount of nitrogen per second, and the boil rate remains steady.

Now you turn on the vacuum pump. Suddenly two things are true - first, the pressure increase from boil off no longer works - the pump can remove gas faster than the boil can generate it. So, the rate of boil will get faster. Second, the transmission of heat into the interior of the bell jar gets slower - vacuum is an excellent insulator; essentially the media that transmitted the heat outside the jar to the liquid is gone, so the main heat flux now is conduction through the bottom of the table rather than convection through the gas in the flask. In the video, you saw the top of the liquid freeze before the bottom - that's because the bottom has more heat input in the lowered pressure.

Now, think back to the ideal gas law. Within the jar, volume is constant, but pressure is dropping, so the left side of pV=nR*T is decreasing. So, the right side is decreasing too. R is a constant, so it can't decrease - that leaves n and T. In the gas portion of the jar, n is definitely decreasing - the pump is pulling particles out, so there are fewer of them. But in the liquid portion, the decrease in n is rate limited by the boil off rate, and we've already said that the pump is reducing pressure faster than the nitrogen can boil off to refill it (otherwise pressure would not change in the jar), so that leaves temperature as the only variable that can decrease to keep the equation balanced.

Additionally, in the video you saw the frozen nitrogen repeatedly pop off the glass and vaporize. The solid phase basically forms a cap that prevents boiling, so the pressure drops even further - you'd expect that to cause the solid to stay, given the logic above. However, now the two volumes aren't actually interacting - the cap isolates the liquid system from the rest of the jar, and keeps internal pressure in the liquid constant for a few seconds. Meanwhile, the main heat flux into the liquid is coming from the bottom of the cup where it touches the table, adding heat to the system that can no longer escape by phase changing some of the liquid into gas. Pressure in the liquid increases, and eventually the pressure difference between the bell jar vacuum and the liquid in the cup pops the solid nitrogen cap out of the cup. The burst of pressure into the jar brings things back above the triple point, and the solid cap liquifies and then boils off the warmer table surface very rapidly - this outpaces the pump for a few seconds, and then the oscillation repeats.

It really is an interesting demo!

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