Reaction of hydrogen peroxide with sodium carbonate In my book the following acid base reaction is written:

$$\ce{H2O2 + Na2CO3 -&gt; Na2O2 + H2O + CO2}$$

But I found that the $\mathrm{p}K_\mathrm{a}$ value of $\ce{H2CO3}$ is less than the one for $\ce{H2O2},$ which means $\ce{H2CO3}$ is more acidic than $\ce{H2O2}.$

As $\ce{H2CO3}$ is getting formed as the product (and then decomposing to give $\ce{CO2}$), shouldn't the reverse reaction be more favourable due to greater acidity of $\ce{H2CO3}$ and the above reaction not happen significantly? 

Is $\ce{H2CO3}$ maybe getting decomposed as soon as it forms allowing this reaction to occur?
The impossibility of a reaction producing simultaneously $\ce{Na_2O_2, H_2O_2}$ and $\ce{CO_2}$ can be proved by the following arguments, drawn from the Merck Index, under the reference "$8728$. Sodium Peroxide". 

1) $\ce{Na_2O_2}$ is produced by burning metallic sodium in a current of air, from which carbon dioxide has been removed. 

2) $\ce{Na_2O_2}$ reacts so easily with $\ce{CO_2}$ that it is used for rendering air charged with $\ce{CO_2}$ respirable as in submarines and diving bells.

3) $\ce{Na_2O_2}$ absorbs water from the air forming sodium hydroxyde and hydrogen peroxide, the latter quickly decomposed into Oxygen and water. 

Ref.: Maryadele O'Neil, Ann Smith, et al., The Merck Index, An Encyclopedia of Chemicals, 13th Ed. 2001, Merck &amp; Co Inc., Whitehouse Station, NJ, USA.