Hydration number: Carbonic acid in water Carbonic acid ($\ce{H2CO3}$) is known to be soluble in water, and that this particular compound is unstable, so much so that according to the 2014 article from Berkeley Lab: 
New Insights on Carbonic Acid in Water, it only exists for a fraction of a second (300 nanoseconds according the article).

Newly reported experiments and observations from the Berkeley Lab found that:


  “Our results support an average hydration number of 3.17 with the acid’s two protons each donating a strong hydrogen bond to solvating waters, the carbonyl oxygen accepting a strong hydrogen bond from solvating water, and the hydroxyl oxygen molecules accepting weak hydrogen bonds from the water”


What is not clear is how is the hydration number calculated and how does this relate to the short existence of carbonic acid in water?
You may wish to refer to the publication of Saykally et al., The hydration structure of aqueous carbonic acid from X-ray
absorption spectroscopy

Basically, to determine hydration number, X-ray spectroscopy was used. This relates to the short existence of carbonic acid as, since it is so short lived, it is necessary to analyze it in such a time frame. As such, this was "the first measurement of X-ray absorption spectra for aqueous carbonic acid, enabled by the use of rapid-flow liquid microjet technology" which allowed it to be captured.

Then, by analyzing the spectra, it is possible to calculate the approximate hydration number, as "the increasing H-bond strength is proportional to the hydration
number (measured for distances up to $2.5\mathrm{\overset{\circ}A}$): $\ce{Na2CO3}$ (5.55), $\ce{HCO3}$
(4.26), $\ce{H2CO3}$ (3.17)".