A better term than "coagulation" is "denaturation". To denature a protein is to unfold it, and this can be done by thermal energy. Initially the egg white proteins are each folded up into little ball-like bundles which limits their interactions, but when they denature the become strands. The charges on these strands then attract to each other and form a tangled mess. Of course, this changes the entropy, but not as much as for simple molecules. That is, we need to look up the "heat of denaturization" for albumin, which is 12.2 J/g (compared to water melting, 334 J/g, so it's fairly small).
That is, denaturing a 60g all albumin egg, should take 60g * 12.2 J/g or 732 J. Given the heat capacity of water as 4200 J/kg K, or 168 J/K for 40 mL of water, this should cause a temperate change of about 4.3C in 40 mL.
So the specific answers are:
The heat of denaturation is small but not completely negligible. Since you start with 100C water, and want to end around 65C, for the small amount of water you mention (40 mL), it's around a 10% error.
Yes, if you're trying to hit it right at 63C, you'll need to account for the heat going into denaturation.
For a temperature controlled system like a sous vide cooker, the amount of energy going into denaturation is negligible compared to energy lost to steam from the bath, etc. (Overall though, this question doesn't quite make sense to me and I wonder whether you're confusing "temperature" with "heat".)
Also, keep in mind that denaturing the albumin does not necessarily mean the egg is cooked sufficiently to be safe to eat.
Finally, I wouldn't trust the 63C number, especially since you're taking the low end of a given range. If you really want to be exact, you'll probably need to go to the source of those numbers. (And don't forget the heat capacity of your container, and also I suspect heat loss will, in fact, be significant here.)
From the original assumptions:
So far, my understanding of water evaporating is the following:
The higher the temperature, the higher the vapor pressure, therefore the faster water vaporizes.
The rate of water boiling, assuming a constant boiling temperature, is dependent on the rate of heat transfer to the water, not the vapor pressure.
At sea level, water boils at 100°c. Boiling temperature decreases as athmospheric pressure decreases.
It is somewhat more correct to say that the boiling temperature decreases as ambient pressure decreases, no matter what the source of the ambient pressure.
It takes a fixed amount of heat / energy to evaporate water (once it reaches boiling point?)
The heat of vaporization of water decreases as the boiling temperature (and boiling pressure) increases, up to the critical temperature (and critical pressure) of water. This means that if pressure changes substantially during the boiling process, the heat of vaporization of the water will also change substantially during the boiling process.
To answer the question directly, there are a few industrial situations where water is given enough heat to instantly evaporate if. For the case of hot oil systems in industry, a furnace heats up the oil to 450 deg F (maybe hotter), and sends the oil to process equipment for the purpose of boiling the process in a heat exchanger. Very infrequently (approximately 5 year cycles), the hot oil system is shut down for maintenance, and water is used to hydroblast equipment in order to clean it. This means that water collects in low points in the associated piping. After maintenance is complete, the proper start-up procedure calls for slowly heating the oil, checking all low points in the piping for water, and draining all water before the hot oil gets above the boiling point of water. On very rare occasions, there have been cases where a small slug of water was trapped in piping during startup without anyone knowing it. This slug of water was isolated from the hot oil by closed valves, so its temperature was too low to cause boiling. Unfortunately, in the event of a process operator opening those valves (for whatever reason), the trapped slug of water immediately contacts 450 deg F hot oil, causing an extremely rapid evaporation rate, and resulting in an explosion. And yes, this has indeed happened. So the answer is: if the water has access to enough heat, at a high enough temperature, it will indeed instantly vaporize.
Best Answer
Okay, first we have the phenomenon: Yes. adding salt increases the boiling point of water, which means that you have to input more energy to get the water to boil, but your egg or pasta will cook faster once you do, because the water will be hotter.
Then there's the why. The boiling point of a liquid is the temperature at which the vapor pressure of the liquid is the same as the atmospheric pressure above the liquid. If we can artificially increase the vapor pressure of the liquid, we decrease the boiling temperature. If we can artificially decrease the vapor pressure of the liquid, we increase the boiling temperature. So the question has now become: why does the vapor pressure of water decrease when we add salt to it?
So imagine a pot of water. At any given temperature there will be some water molecules in the gas phase above the pot (that's the origin of the vapor pressure), and some in the liquid phase in the pot. The proportion in the two phases is determined by the interplay of lowering potential energy (by decreasing elevation in gravity, by forming hydrogen bonds, by lining up the polar ends of the molecules, etc.) and increasing the entropy (there's more accessible states in the gas phase, most liquids are incompressible, etc.). The potential energy part favors the liquid phase, while the entropy part favors the gas phase. The real requirement here is to minimize the free energy, F = U - TS, with F the free energy, U the potential, T the temperature, and S the entropy. Since S is paired with the temperature, increasing the temperature increases the impact of the entropy part, which is why the vapor pressure increases as we increase the temperature.
So now we toss in some salt, while keeping the temperature fixed. The volume fraction of the water decreases, and suddenly there are new accessible states for the water molecules in the liquid phase -- so the vapor pressure decreases. We keep adding salt and the vapor pressure keeps decreasing. If we keep going, eventually there's no vapor pressure.
Raoult's law says that the vapor pressure of a solution is proportional to the vapor pressure of the pure solvent (basically that there is a straight line between the pure vapor pressure and zero, when we've buried it in salt). That's taken as the definition of an ideal solution. Real solutions have a curved functional form between the two boundary conditions, with the deviations from linearity coming from interactions between the solute (the salt) and the solvent (the water). Those interactions might be things like breaking up the network of hydrogen bonds in the water, disrupting the polarization arrangement (both of which will favor gas phase), or bonding/pairing up with water molecules (which will favor liquid phase). At relatively low concentrations of solute the interaction effects are pretty small, so the dependence of vapor pressure on solute concentration remains roughly linear. The cool observation though is that at most temperatures and for most solvents, it doesn't matter what solute you use (as long as the solute itself doesn't have a vapor pressure), the vapor pressure of the solvent is still decreased by adding solute (which indicates that the entropic contribution is the most important part, and the interactions don't play a big role).
Now to sum up: for a given concentration of salt dissolved in water, there are more states accessible to the water molecules in the liquid phase than there are in pure water. So at every water temperature as we pour in energy to make it boil, there will be a lower vapor pressure than there would have been without the salt, and thus we won't get to the boiling point until the water has reached a higher temperature (until we've poured in more energy than we would have had to). Salt does disrupt the network of hydrogen bonds in the water molecules, but the effect isn't very big at reasonable concentrations of salt, and it's never big enough to counteract the entropic effect.