Physics and Astronomy

# The Science of Boiling an Egg

## Introduction

This document is an introduction to of some of the science relevant to preparing boiled eggs. It has evolved from a letter published in the Last Word section of New Scientist magazine (04-April-98) which answered a question by Chris Finn, who asked 'Does anybody have a formula to calculate the boiling time for a soft-boiled egg, given its weight and initial temperature?'.

Much of the information presented here is well-known and compiled from secondary sources (e.g. nutritional information printed on egg boxes!). As far as I know, the formula for the time required to boil an egg given its size and initial temperature is novel. However, it's very easy to derive so I would be surprised if it hadn't been previously discovered independently by others. Please let me know by email C.D.H.Williams@exeter.ac.uk if there are points in this document that need clarifying or correcting.

## A Formula for Soft-Boiling Eggs

### The Derivation

To obtain a simple formula the problem must be idealised somewhat, so the egg will be treated as a spherical homogeneous object of mass M and initial temperature Tegg. If the egg is placed straight into a pan of boiling water at Twater, it will be ready when the temperature at the boundary of the yolk has risen to Tyolk~63°C. With these assumptions, the cooking time t can be deduced by solving a heat diffusion equation.

### The Result

The full derivation (PDF, 19kB), which can be viewed with the free Adobe Acrobat Reader, is quite complicated but the final result is relatively simple:

$$t_{\rm cooked} = \frac{M^{2/3} c \rho^{1/3}}{K \pi^2 ( 4 \pi / 3 )^{2/3}}\log_e \left [ 0.76 \times \frac{( T_{\rm egg}-T_{\rm water} ) }{ ( T_{\rm yolk}-T_{\rm water}) } \right ]$$

where ρ is density, c the specific heat capacity, and K thermal conductivity of 'egg'. According to this formula, a medium egg (M~57 g) straight from the fridge (Tegg=4°C) takes four and a half minutes to cook, but the same egg would take three and a half minutes if it had been stored at room temperature (Tegg=21°C). If all the eggs are stored in the fridge, then a small (size 6, 47 g) egg will require four minutes to cook, and a large egg (size 2, 67 g) will take five minutes.

### Illustrative Graph

Here is a graph of the values given by the equation for the cases when the egg is initially either at room temperature or has been in the 'fridge.

This graph is plotted using a value Tyolk=63°C, which gives a slightly less-than-firm white. The values used were (see full derivation (PDF, 19kB)) ρ = 1.038 g cm−3, K = 5.4×10−3 W cm−1 K−1, and c = 3.7 J g−1 K−1 The upper x-axis gives the European standard size numbers corresponding to the range of masses between the tick marks.

### Accuracy

The model of an egg used to derive the above formula involves several approximations. For a more accurate treatment one would need to note that the thermal properties of the white, yolk and shell are all different, and treat the egg as three concentric ellipsoids with Dirichlet boundary-conditions at the water-shell interface and Neuman boundary-conditions for the shell-white and white-yolk parts. Changes in its thermal properties when the white changes state from sol to gel, and the latent heat associated with this change would also need to be accounted for.

## Boiling Eggs on a Mountain

I was recently told that it takes noticeably longer to boil an egg on a mountain than at sea-level. This must be because the boiling point of water falls with decreasing atmospheric pressure Patm. The graph below uses data derived from the Manual of ICAO Standard Atmosphere (1964) to plot how ambient temperature and Twater decrease with height in a temperate climate.

The next graph shows how the reduction in water temperature increases the time needed to boil an egg with increasing altitude.

Two cases are shown. The 'egg inside' curve is for an egg taken from a refrigerator at 4°C. The 'egg outside' curve is for an egg that is initially at the ambient temperature (plotted above), e.g. during a camping trip. Of course, if circumstances were such that the egg has become frozen, then the cooking time would increase dramatically due to the latent heat required to melt it again.

Dr Edward Brooks is reported to have said "It took me five minutes to cook a 'three minute egg'" at Mount Washington Observatory (1917m). It seems that the additional cooking time predicted by this application of the formula is too small.

Assuming that room temperature at the observatory was similar to that sea level, the discrepancy probably arises because the single 'coagulation temperature' Tyolk used to derive the formula is an approximation. Coagulation is a complicated process and does not occur instaneously at a single temperature, but via chemical reactions that get increasingly rapid with increasing temperature. The result is that a higher 'coagulation temperature' is needed to model the lower temperature cooking that occurs at high altitude.

## Structure and Composition of Eggs

There is a considerable amount of detailed structure to a hen's egg, but the gross features and properties are as follows:

### Shell

The shell accounts for about 9 to 12% of its total weight depending on egg size. It comprises about 94% calcium carbonate with small amounts of magnesium carbonate, calcium phosphate and other organic matter including protein.

Shell strength is influenced by two factors. Firstly, the hen's diet, particularly its calcium, phosphorus, manganese and vitamin D intake. Secondly, egg size, which increases as the hen ages while the mass of shell material that covers cover it stays fixed. Hence the shell is thinner on larger eggs.

Between seven- and seventeen-thousand tiny pores are distributed over the shell surface. As the egg ages, moisture and carbon dioxide diffuses out, and air diffuses in through these causing the air cell to grow and the net mass to decrease. The shell is covered with a protective coating called the cuticle. By blocking the pores, the cuticle helps to preserve freshness and prevent microbial contamination of the contents. This is why good quality eggs (e.g. EC class A) should not be washed, which removes the cuticle, until immediately before they are to be used.

The colour of the shell is determined by the breed of hen. Brown-shelled eggs tend to be more expensive because they come from larger birds, and these are more costly to feed.

In the U.K. mass-market eggs seem to crack more easily these days than they used to; this is apparently because their shells are rather thinner than they were twenty or thirty years ago. I believe that this is mainly because the proportion of 'grit' (a calcium supplement) in chickens' diet has been reduced (it's expensive to transport). Eggs need careful handling to avoid damage, which can be invisible to the naked eye.

Pricking the blunt end of an egg does not, as is commonly thought, prevent the shell from cracking; it relieves pressure and reduces the extent to which the egg is forced out through any crack that does develop. Adding vinegar and/or salt to the water also helps by causing the emerging liquid to coagulate, and thereby plug the gap, more rapidly than it would do in simple hot water.

### White (Albumen)

Albumen accounts for most of an egg's liquid weight, about 67%. It consists of four opalescent layers of alternately thick and thin consistencies. The white of a freshly laid egg has a pH between 7.6 and 7.9 and an opalescent (cloudy) appearance due to the presence of carbon dioxide. As the egg ages the CO2 escapes which increases the pH. Egg white also becomes thinner as an egg ages because its protein changes in character. That's why fresh eggs broken onto a plate sit up tall and firm while older ones tend to spread out. The albumen of older eggs is more transparent than that of fresher eggs. Fresh egg whites coagulate in the range 62° to 65°C, the temperatures decrease with increasing pH and hence age. This is why very fresh eggs require more time to cook than older eggs.

### Yolk

The yolk (yellow portion) makes up about 33% of the liquid weight of the egg. It contains all of the fat in the egg and slightly less than half of the protein. With the exception of riboflavin and niacin, the yolk contains a higher proportion of the egg's vitamins than the white. All of the egg's vitamins A, D and E are in the yolk. Egg yolks are one of the few foods naturally containing vitamin D. The yolk also contains more phosphorus, manganese, iron, iodine, copper, and calcium than the white, and it contains all of the zinc. The yolk of a large egg yields about 250 kJ of energy. Egg yolks have a pH of about 6.0 which stays relatively constant as the egg ages as there is no CO2 loss. Coagulation occurs in the range 65° to 70°C.

## Storing Eggs

According to the American Egg Board, eggs stored at room temperature age more in one day than in one week stored in a refrigerator. Provided they are kept refrigerated, eggs can be stored for 4-5 weeks after the packing date without loss of quality.

## How to Boil an Egg

If you need every detail of the process spelled out consult Learn2.com.

### Soft-Boiling

There are two common approaches: put the eggs into a pan of boiling water, e.g. Learn2 Method One; or put the eggs into a pan of cold water and it bring to the boil.

### Hard-Boiling

Again, there are two common approaches, both are described by Learn2.com. The first is to put the eggs into a pan of boiling water, cook for twice the time required to soft-boil them then plunge into cold water. Alternatively, put the eggs into a pan of cold water and bring it to the boil, then remove the heat and let the pan stand with its lid on for about seventeen minutes, finally cool the eggs with cold water.

These procedures are intended to prevent the yolk temperature getting too high (about 70°C) when hydrogen sulphide generated by the decomposition of sulphur-containing amino acids in the white will react with iron in the yolk causing a (harmless) grey-green film of ferrous sulphide to form on the surface of the yolk. Close inspection of a 'greened' yolk sometimes reveals several concentric rings; the yolk develops within the hen in spherical layers and the rings reflect variation in the iron content of the hen's feed or water.

### Steaming

My preferred method of boiling eggs is to steam them. Use about 1 cm depth of boiling water in a pan with a close-fitting lid. The eggs rest on the bottom of the pan which reduces bumping and presumably reduces the chance of cracking. Since only a small amount of water needs to be boiled the method also saves time and energy.