In chemistry we did an analysis of a mixture of Na2CO3 and NaOH using double indicators and a standard HCl solution. Can you explain this to me?

This type of reaction is called a titration reaction and can be used to determine the concentrations of the components of a solution. In your reaction, your unknown solution was a mixture of Na2CO3 and NaOH. You titrated with HCl. Phenolphthalein and methyl orange were your indicators.

When you added the HCl (in the burette that you used) to your flask containing your mixture, the acid reacted with the bases in solution. It reacted with the strongest base first (-OH from NaOH). This was not detected by either of your indicators.  

NaOH + HCl --> NaCl + H2O

The next reaction was between the HCl and the Na2CO3:

Na2CO3 + HCl --> NaHCO3 + NaCl

This reaction involved half neutralization of Na2CO3 and you observed it as the pink color disappeared from the solution in the flask. It occurred as the pH reached 8.3. We will call the volume of HCl added to reach this point V1.

You probably then added a few drops of methyl orange to your flask and continued titrating HCl. The next reaction (around a pH of 4) was:

NaHCO3 + HCl --> NaCl + H20 + CO2

Notice that the remaining carbonate was neutralized by the completion of this step. The color of your solution turned red as this reaction came to completion. We will call the total volume of HCl added to reach this point V2.

So we have 2 volumes at this point, V1 and V2. V1 was the volume of HCl added to neutralize all of the NaOH and half of the carbonate (CO3). V2 was the volume of HCl added to neutralize all of the NaOH and all of the carbonate.  

To calculate how much of the HCl was just used only to neutralize only half of the carbonate, you subtract V1 from V2 (V2-V1). This is doubled...2(V2-V1) to get the volume of HCl to neutralize all of the carbonate in the solution.

The equation NaVa=NbVb can be used to find the concentrations of our mixture components (Na2CO3 and NaOH). N is the normality of the component.

To find the concentration of Na2CO3:

-assume that 25 mL of Na2CO3/NaOH was initially added to the flask. This may be different for the experiment that you performed.

-assume that the normality of your HCl was 0.99. This may be different for the experiment that you performed.

-assume that the volume of HCl added to make the pink color disappear was 19.5 mL. This may be different for the experiment that you performed.

-assume that the volume of HCl to make the red color appear after adding methyl orange was 25 mL. This may be different for the experiment that you performed.

.99*(2(V2-V1))=NNa2CO3*25mL

.99*2(5.5mL)=NNa2CO3*25mL

.99*11=NNa2CO3*25mL

10.89=NNa2CO3*25mL

NNa2CO3=0.4356

Because a 1M solution of Na2CO3 = 2N (remember the 2 steps needed to neutralize it), 1N Na2CO3=53g/L (1mol=106g).

So, if the assumptions were correct, 0.4356*53 g/L or a 23.09g/L solution of Na2CO3 was added initially to the flask.

Keep the same assumptions to determine the concentration of the NaOH also in the flask.  

To calculate the volume of HCl needed to neutralize the NaOH only you would have to subtract the volume needed to neutralize all carbonate (see above) from V2 (again, see above). This volume is equal to:  V2-(2(V2-V1)) or V2-2V2+2V1. In this case (given the assumptions made above), the volume would be equal to 14mL.

Again, NaVa=NbVb.

0.99*14mL=NNaOH*25mL

13.86mL=NNaOH*25mL

NNaOH=0.5544

1 M NaOH = 1 N NaOH (dissociates in 1 step) and 1mol NaOH=40g

0.5544*40g/L = 22.18 g/L NaOH added to the flask.

The link below is to an image showing the titration curve. Remember that you cannot clearly see the point on the curve where NaOH is neutralized, but it comes before the point (at a higher pH) where phenolphthalein is indicated on the graph.