Question

Cumene, a hydrocarbon, is a compound composed only of \(\mathrm{C}\) and \(\mathrm{H}\). It is \(89.94 \%\) carbon, and its molar mass is \(120.2 \mathrm{g} / \mathrm{mol} .\) What are the empirical and molecular formulas of cumene?

Short answer

Cumene's empirical formula is \(\text{C}_3\text{H}_4\) and its molecular formula is \(\text{C}_9\text{H}_{12}\).

Step-by-step solution

Determine Moles of Carbon in 100g of Cumene

Since cumene is 89.94% carbon, in 100g of cumene, there are 89.94 grams of carbon. Calculate the moles of carbon as follows: \[ \text{Moles of carbon} = \frac{89.94\,\text{g}}{12.01\,\text{g/mol}} \approx 7.49\,\text{mol} \]

Determine Moles of Hydrogen in 100g of Cumene

The remaining percentage of cumene is hydrogen, which is \(100\% - 89.94\% = 10.06\%\). Thus, in 100g of cumene, there are 10.06 grams of hydrogen. Calculate the moles of hydrogen: \[ \text{Moles of hydrogen} = \frac{10.06\,\text{g}}{1.008\,\text{g/mol}} \approx 9.98\,\text{mol} \]

Calculate Empirical Formula

Determine the simplest whole number ratio of moles of carbon to moles of hydrogen. Divide by the smallest number of moles: \[ \frac{7.49}{7.49} = 1 \quad \text{for carbon, and} \quad \frac{9.98}{7.49} \approx 1.33 \quad \text{for hydrogen} \]To express a whole number ratio, multiply by 3 to clear the fraction: \(1 \times 3 = 3\) for carbon, and approximately \(1.33 \times 3 \approx 4\) for hydrogen. The empirical formula is \(\text{C}_3\text{H}_4\).

Determine Molar Mass of Empirical Formula

The empirical formula \(\text{C}_3\text{H}_4\) has a molar mass of: \[ 3(12.01\,\text{g/mol}) + 4(1.008\,\text{g/mol}) = 36.03\,\text{g/mol} + 4.032\,\text{g/mol} = 40.062\,\text{g/mol} \]

Calculate Molecular Formula

Use the given molar mass of cumene to find how many times the empirical formula fits into the molecular formula: \[ \frac{120.2\,\text{g/mol}}{40.062\,\text{g/mol}} \approx 3 \]Thus, the molecular formula is \(\text{C}_{3 \times 3}\text{H}_{4 \times 3} = \text{C}_9\text{H}_{12}\).

Key concepts

Empirical Formula

When analyzing chemical compounds, understanding the empirical formula is crucial as it represents the simplest ratio of elements within a compound. The empirical formula does not indicate the exact number of atoms, but it provides the reduced form, which is very helpful in identifying substances. To determine the empirical formula of cumene:

• First, examine the percentage composition of each element. Cumene has a composition of 89.94% carbon and 10.06% hydrogen.
• Convert these percentages into masses, assuming you have 100 grams of the compound. Therefore, this results in 89.94 grams of carbon and 10.06 grams of hydrogen.
• Calculate the moles of each element to find their ratio. You can do this by using their atomic masses: for carbon (12.01 g/mol) and for hydrogen (1.008 g/mol).
• The ratio determined (in simplest whole numbers) gives the empirical formula as C₃H₄.

Breaking this down into smaller steps helps in understanding how the empirical formula relates to the composition of the compound.

Moles of Carbon

In chemistry, the concept of moles is foundational for understanding quantities of substances. It allows chemists to work at the macro level using atomic mass units. Here’s how to calculate the moles of carbon in cumene:

• We know that 89.94% of cumene is carbon, which translates to 89.94 grams if we consider a 100g sample.
• To find the number of moles, divide the mass of carbon (89.94 g) by its molar mass (12.01 g/mol). This calculation yields approximately 7.49 moles of carbon.

This calculation is an essential step in determining the empirical formula, as it establishes the number of moles of carbon present, which is then compared to the number of moles of other elements.

Moles of Hydrogen

Just like with carbon, understanding the number of moles of hydrogen in a compound like cumene is vital for determining its proportions. Here is how you determine the moles of hydrogen:

• Since the rest of cumene's mass is hydrogen, you have 10.06 grams of it (100% - 89.94% carbon = 10.06% hydrogen).
• Using the molar mass of hydrogen, which is 1.008 g/mol, divide the mass of hydrogen (10.06 g) by this number. The result is approximately 9.98 moles of hydrogen.

Calculating the moles of each component allows chemists to find their relative quantities, a critical step in establishing the empirical formula.

Molar Mass Calculation

Calculating the molar mass of the empirical formula and using it to find the molecular formula is an important part of analytical chemistry. Here's how it is done for cumene:

• First, calculate the molar mass of the empirical formula C₃H₄. This involves adding the atomic weights of each element present: 3 × (12.01 g/mol for carbon) and 4 × (1.008 g/mol for hydrogen).
• The sum for the empirical formula is 40.062 g/mol.
• To determine the molecular formula, compare this to the known molar mass of cumene, 120.2 g/mol.
• Dividing the compound's molar mass by the empirical formula's molar mass (120.2/40.062) shows a factor of 3, indicating the empirical unit occurs three times.

Therefore, the molecular formula can be determined as C₉H₁₂, which shows the entire molecule's structure based on its empirical representation.