A diabasic acid containing C,H and O was found to contain C=26.7% and H=2.2%. The vapour density of its dimethyl ester was found to bc 73. The molecular formula of the acid is

Options

A. `CH_(2)O_(2)`
B. `C_(2)H_(2)O_(4)`
C. `C_(3)H_(3)O_(4)`
D. `C_(2)H_(4)O_(4)`

Answer

To determine the molecular formula of the di-basic acid containing carbon (C), hydrogen (H), and oxygen (O), we will follow these steps:

### Step 1: Calculate the moles of each element from the given percentages.
We are given:
- Carbon (C) = 26.7%
- Hydrogen (H) = 2.2%
- To find the percentage of Oxygen (O), we subtract the sum of the percentages of C and H from 100%:
\[ O = 100 - (26.7 + 2.2) = 100 - 28.9 = 71.1\% \]

Now, we convert these percentages into moles:
- Moles of C = \(\frac{26.7}{12} \approx 2.225\)
- Moles of H = \(\frac{2.2}{1} = 2.2\)
- Moles of O = \(\frac{71.1}{16} \approx 4.44\)

### Step 2: Find the simplest mole ratio.
Next, we divide each mole value by the smallest number of moles calculated:
- For C: \(\frac{2.225}{2.2} \approx 1.01\)
- For H: \(\frac{2.2}{2.2} = 1\)
- For O: \(\frac{4.44}{2.2} \approx 2\)

This gives us a ratio of approximately:
\[ C : H : O \approx 1 : 1 : 2 \]

### Step 3: Write the empirical formula.
From the mole ratio, we can write the empirical formula as:
\[ \text{Empirical formula} = C_1H_1O_2 \text{ or } CHO_2 \]

### Step 4: Calculate the molecular mass of the empirical formula.
The molar mass of the empirical formula (CHO₂) is calculated as follows:
- Molar mass of C = 12 g/mol
- Molar mass of H = 1 g/mol
- Molar mass of O = 16 g/mol

Calculating the total:
\[ \text{Molar mass of CHO}_2 = 12 + 1 + (2 \times 16) = 12 + 1 + 32 = 45 \text{ g/mol} \]

### Step 5: Use the vapor density to find the molecular mass.
The vapor density (VD) of the dimethyl ester is given as 73. The molecular mass (MM) can be calculated using the formula:
\[ \text{Molecular mass} = 2 \times \text{Vapor density} = 2 \times 73 = 146 \text{ g/mol} \]

### Step 6: Determine the ratio of the molecular mass to the empirical formula mass.
Now, we find the ratio of the molecular mass to the empirical formula mass:
\[ \text{Ratio} = \frac{146}{45} \approx 3.24 \]
Since this ratio is approximately 3, we can conclude that the molecular formula is three times the empirical formula.

### Step 7: Write the molecular formula.
Thus, the molecular formula of the di-basic acid is:
\[ \text{Molecular formula} = C_{3}H_{3}O_{6} \]

### Final Answer:
The molecular formula of the acid is \( C_2H_2O_4 \).

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