To determine the molecular formula of the carbohydrate with the empirical formula CH₂O, we can follow these steps:
### Step 1: Calculate the amount of hydrogen in 0.1 moles of the carbohydrate.
The empirical formula CH₂O indicates that there are 2 moles of hydrogen (H) for every mole of the compound. Therefore, in 0.1 moles of CH₂O, the amount of hydrogen can be calculated as follows:
\[
\text{Hydrogen in 0.1 moles} = 0.1 \, \text{moles} \times 2 \, \text{moles of H/mole of CH}_2\text{O} = 0.2 \, \text{moles of H}
\]
### Step 2: Convert moles of hydrogen to grams.
The molar mass of hydrogen (H) is approximately 1 g/mol. Therefore, the mass of hydrogen in 0.2 moles is:
\[
\text{Mass of H} = 0.2 \, \text{moles} \times 1 \, \text{g/mol} = 0.2 \, \text{g}
\]
### Step 3: Relate the given mass of hydrogen to find the number of moles.
The problem states that 0.1 moles of the carbohydrate contains 1 g of hydrogen. We can find out how many moles of hydrogen correspond to 1 g:
\[
\text{Moles of H} = \frac{\text{Mass}}{\text{Molar mass}} = \frac{1 \, \text{g}}{1 \, \text{g/mol}} = 1 \, \text{mole of H}
\]
### Step 4: Determine the moles of the carbohydrate that would contain 1 g of hydrogen.
Since we have established that 0.1 moles of the carbohydrate contains 0.2 g of hydrogen, we can set up a proportion to find out how many moles would contain 1 g of hydrogen:
\[
\frac{0.1 \, \text{moles}}{0.2 \, \text{g}} = \frac{x \, \text{moles}}{1 \, \text{g}}
\]
Cross-multiplying gives:
\[
0.2x = 0.1 \implies x = \frac{0.1}{0.2} = 0.5 \, \text{moles}
\]
### Step 5: Calculate the total mass of the carbohydrate.
Now, we know that 0.5 moles of the carbohydrate corresponds to 1 g of hydrogen. The empirical formula CH₂O has a molar mass of:
\[
\text{Molar mass of CH}_2\text{O} = 12 + 2(1) + 16 = 30 \, \text{g/mol}
\]
Thus, the mass of 0.5 moles of the carbohydrate is:
\[
\text{Mass} = 0.5 \, \text{moles} \times 30 \, \text{g/mol} = 15 \, \text{g}
\]
### Step 6: Determine the molecular formula.
To find the molecular formula, we need to determine the ratio of the empirical formula mass to the molar mass of the compound. The empirical formula mass of CH₂O is 30 g/mol. Since we have 15 g of the empirical formula, we can find the number of empirical units in the molecular formula:
\[
\text{n} = \frac{\text{Molar mass}}{\text{Empirical formula mass}} = \frac{15 \, \text{g}}{30 \, \text{g/mol}} = 0.5
\]
Since this doesn't seem to fit, we need to check our calculations again.
### Final Calculation:
We have established that the empirical formula CH₂O contains 2 moles of hydrogen. Given that we have 1 g of hydrogen, we can calculate the total moles of hydrogen in the molecular formula:
1 g of hydrogen corresponds to 1 mole. Therefore, if we have 10 grams of hydrogen, we can relate this back to the empirical formula:
\[
\text{Molecular formula} = \text{Empirical formula} \times n
\]
Where n = 5 (since 10 g of hydrogen corresponds to 5 moles in the empirical formula).
Thus, the molecular formula is:
\[
\text{C}_5\text{H}_{10}\text{O}_5
\]
### Answer:
The molecular formula of the carbohydrate is **C₅H₁₀O₅**.
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