Lemon juice and carbonic acid reaction

I was doing a chemistry project to see if the pH of a carbonated solution effects the time taken for it to settle after being disrupted, and I am trying to map out the reaction when the components are combined. 


I was able to figure out a few, but I got stuck on carbonic acid and lemon juice. For some reason, I was getting ethane as a by product of the reaction, and I'm sure that's not right. Could someone explain what I'm doing wrong? 


the formula is $\ce{H2CO3 + C6H8O7}$




These two weak acids have similar disassociation constants in water. A low Ka indicates a relatively small number of acid molecules will form an anion stabilized by water. Two weak acids with similar disassociation constants will coexist in a solution without interacting. If two weak acids have very different Ka then there can be a reaction. In this case nothing would happen. For more generalized information search for mixture of weak acids.



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> Carbonic Acid
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$$\ce{H2CO3 + H2O <=> HCO3- + H3O+}$$
$$K\_a = 4.3 \times 10^{-7} = \frac{[\ce{H3O+}][\ce{HCO3^-}]}{[\ce{H2CO3}]}$$



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> Citric Acid
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$$\ce{H3C6H5O7 + H2O <=> H2C6H5O7- + H3O+}$$
$$K\_a = 1.8 \times 10^{-6} = \frac{[\ce{H3O+}][\ce{H2C6H5O7^-}]}{[\ce{H3C6H5O7}]}$$


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By Le Chatelier's principle ,a new equilibrium will be formed for the H3O+ concentration , since H2CO3 has the lower dissociation constant.H3O+ concentration and therefore the pH will lie somewhere between the two acids 


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