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I'm a complete novice in biology. This is just a question that arose during Saturday night warmup and there is no better place to get answers :)

So, as probably most of you know, if you want to decide whether an egg is rotten/gone bad/addled you just put it in a bowl filled up with cold water and some salt.

If the egg stays on the bottom of the bowl then it is good. If it floats to the top of the water then it is rotten/addled.

I imagine the egg is floating because there is an air bubble in the rotten egg.

But how would that air bubble get in the egg?

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Egg shells look as if they're solid, but in fact they're filled with tiny pores through which air can and does diffuse.

...up to the time when internal pipping takes place, when pulmonary ventilation is initiated, about 20 liters (O2 + CO2 + water vapor) have passed through 10,000 pores of an 80 gm egg.

--Pores and gas exchange of avian eggs: a review.

A typical 60-g chicken egg has about 10,000 pores, each with a functional cross section of 150–200 μm^2; therefore, the total pore area is about 1.5–2 mm^2.

--Gas exchange in avian embryos and hatchlings

You only asked about chicken eggs, but chickens have it pretty easy compared to many other avian species. For example, ostrich eggs have a much larger volume-to-surface area, meaning that exchange is less effective. One way ostriches may compensate is by incubating eggs at lower temperatures, reducing oxygen requirements; more significantly and related, incubation time is relatively slower for birds with large eggs. Birds at high altitudes have larger total pore size, adapting to the reduced O2 pressure.

Actually, a more serious problem than oxygen exchange is water exchange. The egg needs to manage water loss (not too much, not too little), and the more gas exchange the more water loss. Accordingly, birds adapted to different humidities lay eggs with different gas exchange capacities (changing pore densities for the most part); birds in very arid environments, like gulls in Chile's Atacama Desert, have very low gas and water exchange, to the point where their developing chicks are relatively hypoxic:

--Reduced oxygen diffusion across the shell of Gray gull (Larus modestus) eggs.

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As an egg ages, it becomes progressively less dense and this is a natural consequence of the development of the embryo. If an egg becomes contaminated by bacteria, then it will also become less dense. The reason for this is that as the contents of the egg develop (or decompose, as the case may be) gases are produced. These gases dissolve into fluids of the egg and then migrate out of the egg into the surrounding environment.

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I would not use the terms "rotten/addled". The test you describe is a simple kitchen test to determine if eggs are "old" vs "fresh" (before the days of refrigeration). Fresh eggs will sink while old eggs will tend to float. This is because all eggs have a "bubble of air" at the blunt end (presumably for the chick to breath while hatching). As the egg gets older, this bubble will increase in size due to simple desiccation (moisture loss) over time.

You can try the test using fresh vs. expired eggs from the market. You will find that many more of them will float when they are weeks past their expiration date, but they are definitely not rotten/addled and some people will still use them. It is more a test for the cautious person to avoid possible health risks from older eggs. A rotten/addled egg will give off the characteristic "rotten egg" smell due to bacterial growth.

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