On strong heating, one gram of the silver salt of an organic dibasic acid yields 0.5934g of silver. If the mass percentage of carbon in it 8 times the mass percentage of hydrogen and one-half the mass percentage of oxygen, determine the molecular formula of the acid.

Options

A. `C_(4)H_(6)O_(4)`
B. `C_(4)H_(6)O_(6)`
C. `C_(4)H_(6)O_(2)`
D. `C_(5)H_(10)O_(5)`

Answer

To determine the molecular formula of the organic dibasic acid based on the information provided, we can follow these steps:

### Step 1: Calculate the moles of silver produced
Given that 0.5934 g of silver (Ag) is produced, we can calculate the number of moles of silver.

\[ \text{Molar mass of Ag} = 108 \, \text{g/mol} \]

\[ \text{Moles of Ag} = \frac{\text{mass of Ag}}{\text{molar mass of Ag}} = \frac{0.5934 \, \text{g}}{108 \, \text{g/mol}} \approx 0.00549 \, \text{mol} \]

### Step 2: Relate moles of Ag to moles of the acid
Since the silver salt of the acid is Ag2A (where A represents the acid), the stoichiometry tells us that 2 moles of Ag correspond to 1 mole of the acid.

\[ \text{Moles of acid (A)} = \frac{0.00549 \, \text{mol Ag}}{2} \approx 0.002745 \, \text{mol} \]

### Step 3: Calculate the molar mass of the acid
The mass of the silver salt is given as 1 g. The mass of the acid can be calculated using the mass of silver produced.

\[ \text{Mass of the acid} = \text{mass of silver salt} - \text{mass of silver} = 1 \, \text{g} - 0.5934 \, \text{g} = 0.4066 \, \text{g} \]

Now, we can calculate the molar mass of the acid:

\[ \text{Molar mass of acid} = \frac{\text{mass of acid}}{\text{moles of acid}} = \frac{0.4066 \, \text{g}}{0.002745 \, \text{mol}} \approx 148.5 \, \text{g/mol} \]

### Step 4: Set up mass percentages
Let the mass percentage of hydrogen be \( x \). Then, according to the problem:

- Mass percentage of carbon = \( 8x \)
- Mass percentage of oxygen = \( 2x \)

The total mass percentages must equal 100%:

\[ x + 8x + 2x = 100 \]

\[ 11x = 100 \implies x = \frac{100}{11} \approx 9.09\% \]

### Step 5: Calculate individual mass percentages
Now we can calculate the mass percentages:

- Mass percentage of hydrogen = \( x \approx 9.09\% \)
- Mass percentage of carbon = \( 8x \approx 72.73\% \)
- Mass percentage of oxygen = \( 2x \approx 18.18\% \)

### Step 6: Calculate the number of moles of each element
Assuming we have 100 g of the acid:

- Moles of C = \( \frac{72.73 \, \text{g}}{12 \, \text{g/mol}} \approx 6.06 \, \text{mol} \)
- Moles of H = \( \frac{9.09 \, \text{g}}{1 \, \text{g/mol}} \approx 9.09 \, \text{mol} \)
- Moles of O = \( \frac{18.18 \, \text{g}}{16 \, \text{g/mol}} \approx 1.14 \, \text{mol} \)

### Step 7: Determine the simplest ratio
To find the simplest ratio, divide each by the smallest number of moles (1.14):

- C: \( \frac{6.06}{1.14} \approx 5.32 \)
- H: \( \frac{9.09}{1.14} \approx 7.96 \)
- O: \( \frac{1.14}{1.14} = 1 \)

### Step 8: Empirical formula
The approximate ratio is C5H8O. To find the molecular formula, we need to find the factor \( n \) such that:

\[ n \times \text{empirical mass} = \text{molar mass} \]

Calculating the empirical mass:

\[ \text{Empirical mass} = 5(12) + 8(1) + 1(16) = 60 + 8 + 16 = 84 \, \text{g/mol} \]

Now, calculate \( n \):

\[ n = \frac{148.5}{84} \approx 1.77 \approx 2 \]

### Step 9: Molecular formula
Thus, the molecular formula is:

\[ \text{Molecular formula} = C_{5 \times 2}H_{8 \times 2}O_{1 \times 2} = C_{10}H_{16}O_{2} \]

### Final Answer
The molecular formula of the organic dibasic acid is \( C_{10}H_{16}O_{2} \).

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