A 100ml mixture of $N a_{2} C O_{3}$ and $N a H C O_{3}$ is titrated against $1 M H C l .$ if $V_{1}$ and $V_{2} L$ are consumed when Phenolphthalein and methyl orange are used as indicators respectively in two separate titrations. Which of the following is true for molarities in the original solutions

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Molarity of $N a_{2} C O_{3} = 10 (V_{2} + V_{1})$

Molarity of $N a_{2} C O_{3} = 10 (V_{2} - V_{1})$

Molarity of $N a_{} H C O_{3} = 10 (V_{2} - 2 V_{1})$

Molarity of $N a_{2} C O_{3} = 20 V_{1}$

answer is B.

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Detailed Solution

Let $M_{N a_{2} C O_{3}} = x M, M_{N a H C O_{3}} = y M$
$\begin{array}{l} N o w, V_{1} \times 1 \times 1 = \frac{100 x}{1000} \times 1 (P h e n o l p h t h a l e i n) \\ a n d V_{2} \times 1 \times 1 = \frac{100 x}{1000} \times 2 + \frac{100 y}{1000} \times 1 (m e t h y l o r a n g e) \\ x = 10 V_{1}; y = 10 (V_{2} - 2 V_{1}) \end{array}$