Re: How does an egg shell works like a cell membrane when osmosis happens?

You didn't say if you were using a whole egg, or a boiled one with the shell off. The shell itself has no part in osmosis, it's the membrane between the shell and the white that's important. You can see this membrane when peeling a boiled egg - it's the white, alomst transparent film around the egg-white itself.

The membrane will let water pass, but not many other molecules such as proteins and sugars.

If you put an egg in pure water, or any dilute solution like your vinegar, then this is what happens:
The egg is full of water and protein, but outside the egg it's mostly water. Water flows into the egg. This is simply because there is more water outside the egg flowing in than there is water inside the egg flowing out. This process makes the egg become bloated with water and the solution outside will lose water.

The flowing process is based on the random movement of each water molecule and is properly called diffusion.

Another effect of the vinegar would be to corrode away the eggshell if you use an unshelled egg. This happens with any carbonate and any acid. It won't affect the osmosis but it might make things bubble a bit.

If you put the egg in a strong solution like your syrup what happens is this:
The strong solution is mostly sugar with very little water. There is more water in the egg even with the egg's protein. So there is more water in the egg, flowing out, than there is water in the syrup, flowing in.

This means that your egg in syrup will dry up and the syrup will be diluted. Here are a couple of web-sites with background info and diagrams: http://tidepool.st.usm.edu/crswr/osmosis.html http://biology.arizona.edu/sciconn/lessons/mccandless/reading.html