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According to Tiefenbacher,

Egg white is naturally alkaline with an initial pH of 7.6–7.9 but in storage the loss of carbon dioxide by diffusion through the shell increases the pH of the egg white up to about 9.7 with age.

The loss of $\ce{CO2}$ means loss of $\ce{H2CO3}$, an acid with pKa of 6.3. This reduces acid and in a neutral solution pH would approach 7. Since it goes up towards 9-10, what is the base? Albumin is a buffer in biological systems but I don't think it is that strong a base. What is the alkali causing pH of around 9-10?

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    $\begingroup$ This question has attracted some oddly out of character negative attention in the comments; let's keep discussions civil, please. If you want to ask for clarification, do so politely; if you think it's a good question, put your efforts toward an answer instead of a comment. $\endgroup$ – Bryan Krause Jul 2 at 2:53
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    $\begingroup$ I have provided a link for your attribution and corrected it to the author of the book that is listed in Google books. Please note that Google is not the author of material found using its search engine. $\endgroup$ – David Jul 3 at 13:54
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In the terms of the question, this can be regarded as calcium hydroxide, with the carbonate being in the form of calcium carbonate. Calcium hydroxide is a stronger base (pH of aqueous solution at 1mm = 11.27) than carbonic acid is an acid (pH of aqueous solution at 1mm = 4.68), so that 1mm calcium carbonate has a pH of 9.91. (Values from Aquion)

Calcium carbonate is, of course, the constituent of eggshell, which crystallizes out of solution when it is deposited in the uterus. Its presence within the egg white is presumably a consequence of this.

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I would add that one does not generally regard there being a particular base balancing the carbonic acid: when the hydrogen ions are removed there is an increase in the free hydroxyl ions and hence an increase in pH. The way the question is formulated only really makes sense if one assumes that the egg white acquired extra carbonic acid to reduce the alkalinity resulting from the calcium carbonate/hydroxide.

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