Question

what is the molecular formula of a compound given the molar mass of the compound is \\(186.5\text{ gram}\\) and the empirical formula is \\(\text{c}_2\text{h}_7\\)?

\\(\bigcirc\text{ c}_4\text{h}_{14}\\)
\\(\bigcirc\text{ c}_2\text{h}_{14}\\)
\\(\bigcirc\text{ c}_3\text{h}_{21}\\)
\\(\bigcirc\text{ c}_2\text{h}_7\\)
\\(\bigcirc\text{ none of these}\\)

Explanation:

🆕 New Concept Discovered: Empirical and Molecular Formulas
Scaling up the simplest ratio to find the real molecule

Step 1: Calculate the empirical formula mass

The empirical formula is \(\text{C}_2\text{H}_7\). We find its mass by adding the average atomic masses of its constituent atoms:

• Carbon (\(\text{C}\)): \(\approx 12.011\text{ g/mol}\)
• Hydrogen (\(\text{H}\)): \(\approx 1.008\text{ g/mol}\)

$$ \text{Empirical Mass} = (2 \times 12.011\text{ g/mol}) + (7 \times 1.008\text{ g/mol}) $$

$$ \text{Empirical Mass} = 24.022 + 7.056 = 31.078\text{ g/mol} $$

Step 2: Determine the scaling factor

The molecular formula is always a whole-number multiple (\(n\)) of the empirical formula. To find this multiplier, divide the given molar mass of the compound by the empirical formula mass:

$$ n = \frac{\text{Molar Mass of Compound}}{\text{Empirical Formula Mass}} $$

$$ n = \frac{186.5\text{ g/mol}}{31.078\text{ g/mol}} \approx 6.00 $$

This means the actual molecule is \(6\) times heavier than the empirical unit.

Step 3: Determine the molecular formula

Multiply the subscripts in the empirical formula \(\text{C}_2\text{H}_7\) by the scaling factor \(n = 6\):

• Carbon: \(2 \times 6 = 12\)
• Hydrogen: \(7 \times 6 = 42\)

The molecular formula is \(\text{C}_{12}\text{H}_{42}\).

Looking at the given choices:

• \(\text{C}_4\text{H}_{14}\) (this corresponds to \(n = 2\))
• \(\text{C}_2\text{H}_{14}\)
• \(\text{C}_3\text{H}_{21}\) (this corresponds to \(n = 1.5\))
• \(\text{C}_2\text{H}_7\) (this corresponds to \(n = 1\))
• none of these

Since \(\text{C}_{12}\text{H}_{42}\) is not listed among the specific molecular formulas, the correct choice is "none of these".

Answer:

none of these