When an egg is kept in vinegar for one week, its hard calcium carbonate shell changes into a soft rubbery membrane. As vinegar is weak acetic acid, how does vinegar change calcium carbonate into a soft rubbery membrane? What is the reason for this?
It doesn't make calcium carbonate rubbery, it removes the calcium carbonate. Egg shells are not purely calcium carbonate they are more like a composite with a continuous matrix of calcium carbonate and a smaller continuous matrix of protien. When you put the egg in vinegar, you etch away this calcium carbonate matrix leaving the formerly less noticeable protein matrix as a gel filled with vinegar and creates a gel.
As far as calcium carbonate is considered, it must be getting dissolved in the vinegar solution and left over is the protein matrix.
The concept of denaturation of proteins may be helpful to explain your so called phenomenon of formation of soft rubbery membrane.
A process that changes the physical and biological properties of proteins i.e., native state of proteins without effecting the chemical composition of the protein is called denaturation.
It can be caused by change in pH, temperature, presence of salts or certain other chemical substances.
Change in pH has the greatest disruptive effect on the hydrogen bonding and salt bridges in proteins. This denaturation results from the rearrangement of the secondary and tertiary structure of the proteins. The globular proteins present in egg shell change to a rubber like insoluble mass.
This is what happens to the egg albumin when kept in a vinegar solution.