How much whipped egg white can we make from one egg white?

Transcription

1 How much whipped egg white can we make from one egg white? Let's say (always) the question again : how much whipped egg white can we make from one egg Indeed, one simple mind would answer : from one egg white, one can make one whipped egg white. But if the question is asked, it means that such a simple answer is not the right one. We have to put intelligence in what we do, in particular answering questions. How much? This means that we need a unit. A whipped egg white has a masse, for sure, but also a volume. So that we can rephrase the question as : which volume of whipped egg white can we make from one egg white? Now, of course, we could whip an egg white, or try to remember last whipping, and we would conclude that from one egg white, we get about one third of a liter. Tfhis answer has very few interest, because it is again too simple. Let's consider the question : - we start from an "egg white" - we whip - we get a whipped egg white - there is a volume. An egg white? We know that an egg is about 60 g, with 30 g of egg white. This egg white is made of 10 % proteins and 90 % water. When we whipp in a traditional way, the whisk pushes air bubbles in the liquid. And these bubbles are also divided when the whisks goes through them after they are in the liquid. And of course, if there are air bubbles in the liquid, the whole system is getting larger, has a larger volume. Finally, the final system is a foam, with air bubbles in the liquid. Let's make a sketch for this : We now see that there are large air bubbles, and small bubbles. But remember that in science, adjectives and adverbs are forbidden: they have to be replaced by the answer to the question "how much".

2 And now, we see how imprecise the question is: would a single bubble already make a "whipped egg white"? And which would the volume, then? Before calculating, we need a strategy, a method (general way, of reaching the solution), and then a tactif (implementation of the strategy, various steps...). By the way, I did not had time to know if this was strategy or tactic (can you tell me?), but there is a Rule : always use formal symbols for calculation, or make orders of magnitude, but newer mix litteral and numerical calculus. Certainly the strategy asks that we begin by being clear with the question, make a model, and characterize this model quantitatively the model. In general, this is enough for making the solution obvious. Let's ask again the question : we start from an egg white, i.e. a solution of various proteins in water. Using this solution, we shall make a sphere (air bubble) with the solution around : The proteins are at the interface : The relative stability (compared to pure water) of the liquid film around the sphere is due to proteins, which come at the interface, because when whipping, they are partly denaturated, so that the hydrophobic part comes in contact with air, whereas the hydrophilic remains in water. Mind that the former picture is not right, as there are probably two layers of proteins at the interface []. Now, some tactic : making the model simpler. For example, we see on the picture proteins with complex shapes (and remember that the system is dynamic: do you remember how fast do water

3 molecules move in liquid water at room temperature? answer: read it only after you tried to do the calculation by yourself! First simplification (S1): we shall begin by connsidering that any protein has an area s, without going into more details. Another tactic: characterize quantitatively the model, i.e. simply introduce symubols in order to describe what you see on the picture that you did. Here, for example, one can see a bubble of radius R (external radius), and thickness of the liquid film equal to e. Another piece of tactic: without waiting, let's ask again the question, let's "ruminate" it, and transform it in a formal question. Here, for example, we are looking for the volume of the bubble : V. This volume is linked to the radius R by: Looking for V is also looking for R. Another piece of tactic: put down the initial date, the a priori knowledge... but put it down formally. Here, for example, the film is egg white, an aqueous solution of proteins. proteins. We simplify (let's keep track of the simplification : S2) by assuming that there is only one kind of proteins. We assume that this protein is ovalbumine, because it is the most abundant (about 45 %)[]. An egg white has a mass M, and it contains a mass M protéines of proteins. This mass corresponds to a number of molecules, using a simple relationship : 1 mol of proteins (a mass MM protéines, MM describing the molar mass) contains N A molecules, where N A stands for the Avogadro number. In M protéines g of proteins, there are moles, i.e. molecules of proteins.

4 But we saw that each protein has an area s. So that the total area covered by all proteins is. Here, we can see that considering the volume of the radius of the sphere was not enough. We need the area. We know that the area of the external surface of the sphere is : For the inner area: Which means taht the total area has to be covered by proteins: Here a problem appears: we have two unknowns parameters R and e for only one equation. However we see that the second unknown value, the thickness of the liquid film, corresponds to a part of the egg white that we did not discuss, i.e. the proportion of water in the egg white. One should add that there is no reasonn that the quantity of water is exactly the one for which the radius is such that it is entirely covered with proteins. So that we should consider two cases: - what would be the radius if water were the limiting factor? - what is water is limiting? For the first case, we simply write that the volume of the film is equal to the volume of the egg white: But the question of two unknowns remains. Indeed we did not considered another data: the limit is a film of one water molecule thick. Which means that the volume is ggiven when the volume of the film is equal to the volume of the egg white : Here, it is interesting to get a numerical application, without waiting. We use the second solution, as it is the only positive one: - e will be chosen as m - the volume of egg white will be like for 30 g of water: kg, soit m 3. (1.1) (1.2) More than 100 meters for the radius! You see, there are ample reasons to ask if the proteins can cover such a big area!let us now look for the area covered by proteins. We could see that it was:

5 In this expression, the quantity s is missing. Let's look for it. Previously, we saw that we can consider two cases: - proteins like coils - proteins fully extended For the first case, we can calculate that the surface would be about squares of 5 covalent bonds (try to calculate, using the mass of ovalbumin, finding the number of residues of amino acids, the length of an amino acid, and a folding on a cubic grid); for the second case, we can identify a protein to a rectangl of 150 lengths of covalent bonds, by 5 covalent bonds. Here the total area is: If e is small compared to R (assumption S3, to be tester later), we have: And: Again, we keep only the first solution, in order to calculate: (1.3) (1.4) Here the result is in m, for both cases. We see that: - even with entirely folded proteins, one can make a large bubble, of half of a meter for the radius - with unfolded proteins, the volume is much bigger, and it even much bigger than for ordinary whipped egg whites - our assumption of a radius much larger than thickness holds. Let's now put ourselves one step behind ourself. Why do we usually get so few whipped egg white, from one egg? The answer holds for whipped egg white, but more generally for all

6 "disperse systems", and in particular colloidal systems. Let's have a cube of a material, of edge A. The area is 6A2. If you divide this cube in n3 small cubes of edge a, the total area of a small cube is 6a2, and the number of small cubes is (A/a)³, so that the total area is (A/a)3.6a2, soit 6A³/a. One order of magnitude to fix ideas: if A = 1 cm, and a = 0,0001 cm, then the total area moves from 6 cm² to cm², i.e. 6 m²! The tremendous increase of the area of divided solids goes along with a huge energy being spent: the cohesion of the solid is due to forces (think of electrostatic forces holding together chloride ions Cland sodium ions Na+ in a crystal of salt). These forces have to be broken so that we can divide the solid. The energy to give (a "surface energy") is proportional to the area! For example, if the surface energy were 70 mj.m-², the total energy to give for the division would be 0,42 J! Coming back to the whipped egg white, the bubbles have a radius mm, is it can be easily seen on microscopic picture (don't put a cover slip, because it would change the appearance) : The aire bubbles are in the liquid. So you see that the bubbles are "packed" in water. And this is why the questions of "packing" can be now considered. Close packing The question of packing is yours is you try to make a pile of oranges: how to make a pile, narrow and high? How to minimize the volume of a pile of a certain number of objects? A good way to answer to a complex question is to make first a simplification. All physicists begin the the simplest first, which is the "principle of simplicity", or the Ockham razor, from the name of the monk Guillaume of Ockham (or Occam), who lived in the XIV century : Essentia non sunt multiplicanda praeter necessitam (entities should not be multiplied without necessity). Faraday, but also Einstein, were choosing the simplest laws first. For example, Einstein wrote The Lord is

7 subtle but not malicious. Let' us follow them, keeping in mind that sometimes it can be more complex than wanted. For example, what is the shortest way joining the four corners of a square? Contrary to what For this particular question, I invite you to look for the solution of the problem, and you will find an example of a broken symetry... for which the book How to solve it, by George Polya is very important. For a problem such as the one with oranges, a method is needed: first reduce the number of dimensions. with one dimension is a segment, annd the packing question is solved by putting segments one after the other, which means that the "packing coefficient" (the proportion of space occupied by the spheres) is 1 ; between then? You should be able to calculate that the proportion of space filled by disks of radius r in a square filling is equal to the proportion occupied in a simple square of edge 2r: this means that the paccking coefficient is r²/4r², close to 3/4. However, for this filling, it is easy to see that if you shift a row of disks on the side, we can move them up, so that another piling is obtained. Now the coefficient is... how much? I invite you to calculate by yourself. is 0.740? Let's come back to whipped egg. Knowing the filling coefficient of the hexagonal compact filling, we see that a mass m V such as V (1-0,74) =. For one egg whit of 30 g, and a density of the egg white equal to the density of water (indeed the egg white is densier than water), on can find L. It is less than what we can get in practice, but the picture of the whipped egg white is showing that the real whipped egg white is more packd, because small bubbles can come between the big one, so that the coefficient can reach about 0.95, i.e. a volume of 0,6 L, higher than what we generally get in practice. Such calculation give ideas in order to increase the possible volumes with a traditional whipping: you have to increase the volume of the liquid phase... for example by adding water! And finally, the quantity of proteins will be the limiting factor. The maximum volume assuming that proteins are limiting Let us use an egg white of masse M, with a proportion of proteins p. The mass of proteins is p.m. The molar mass of proteins being MM, the number n of moles of proteins is: Let's a number of proteins N equal to

8 As previously, if we assume that proteins cover an area s, the the total surface covered by proteins is :. Let's assume now that the bubbles have a radius r number N b is given by : r², so that their Here we know S, r (microscopic picture), so that we calculate: The volume of one bubble is: :, so that the total volume of gas in the whipped egg white is Let's now do the numerical applicationwith the same data as before, but first with bubbles of radius 0.1 mm : (1.3.1) The result is between 0,3 and 9 L! Yes, you can get easily 9 L of whipped egg white by the addditioon of water. Of courses, the liquide phase can have a flavour, such as in orange juice, tea, coffee, wine, stock, bisque... Such a system was called a "vauquelin". You can cook it like a meringue, in order to get wind crystals.