Question

The most common constituent of gasoline is iso-octane. It is a hydrocarbon, composed by weight of \(84.12 \%\) carbon, and \(15.88 \%\) hydrogen. Given that it contains \(5.27 \times 10^{21}\) molecules per gram, what is its molecular formula?

Short answer

The molecular formula of iso-octane is \(C_9H_{18}\), which was found by calculating the empirical formula (\(CH_2\)) and then determining the molecular formula using the given number of molecules per gram.

Step-by-step solution

Determine the Empirical Formula

To find the empirical formula, we'll first assume that we have 100 grams of iso-octane. This means that we have 84.12 grams of carbon and 15.88 grams of hydrogen. Now, we'll want to know the number of moles of carbon and hydrogen. The molar mass of carbon is 12.01 g/mol, and the molar mass of hydrogen is 1.01 g/mol. Number of moles of carbon = \(\frac{84.12 \text{ g}}{12.01 \text{ g/mol}} = 7.005\) mol Number of moles of hydrogen = \(\frac{15.88 \text{ g}}{1.01 \text{ g/mol}} = 15.723\) mol To find the empirical formula, we'll divide the number of moles of each element by the smallest number of moles. \(\frac{7.005}{7.005} = 1\) for carbon \(\frac{15.723}{7.005} = 2.24\) for hydrogen Since the ratio is close to 1:2, the empirical formula for iso-octane would be \(CH_2\).

Determine the Molecular Formula

Now we'll use the number of molecules per gram to find the molecular formula. First, Convert the empirical formula mass to grams per mole: Empirical formula mass of \(CH_2\) = 12.01 g/mol (carbon) + 2.01 g/mol (hydrogen) = 14.02 g/mol Given that there are \(5.27 \times 10^{21}\) molecules of iso-octane per gram, we can calculate the molar mass of iso-octane by multiplying the molecules by the mass of a single carbon plus two hydrogen. Molar mass of iso-octane = \((5.27 \times 10^{21} \text{ molecules}) \times (14.02 \frac{\text{g/mol}}{\text{molecule}}) \times (\text{1 mol} / 6.022\times 10^{23}\text{ molecules}) = 122.22 \text{ g/mol}\) Now, we'll find the ratio of the molar mass of iso-octane to the empirical formula mass. \(\frac{122.22 \text{ g/mol}}{14.02 \text{ g/mol}} = 8.725 \approx 9\) The ratio is approximately 9, so the molecular formula of iso-octane would be 9 times the empirical formula, which is \(C_9H_{18}\).

Key concepts

Empirical Formula

The empirical formula is the simplest representation of a compound's composition. It shows the smallest whole-number ratio of the elements present. In the case of iso-octane, we begin by assuming a 100-gram sample for ease of calculation.- Iso-octane is composed of 84.12% carbon and 15.88% hydrogen.- To determine the amount of each element, we convert these percentages into grams: - 84.12 grams of carbon - 15.88 grams of hydrogen- Next, we calculate the number of moles of each element since this is necessary for finding the empirical formula: - For carbon: the moles are given by dividing the mass by the molar mass of carbon (12.01 g/mol). \[ \text{Moles of Carbon} = \frac{84.12}{12.01} = 7.005 \text{ moles} \] - For hydrogen: the moles are determined similarly, using the molar mass of hydrogen (1.01 g/mol). \[ \text{Moles of Hydrogen} = \frac{15.88}{1.01} = 15.723 \text{ moles} \]- To form the empirical formula, the ratio of the number of moles of each element is determined by dividing each by the smallest number of moles calculated: - The smallest is 7.005, hence: \[ \frac{7.005}{7.005} = 1 \text{ for carbon} \] \[ \frac{15.723}{7.005} \approx 2.24 \text{ for hydrogen} \]- This results in an elemental ratio close to 1:2, giving the empirical formula \(CH_2\).

Moles Calculation

Calculating moles is a fundamental concept in chemistry. It measures the amount of a substance. In converting grams to moles, one uses the molar mass specific to each element. - Molar mass is the mass of one mole of an element and is generally expressed in grams per mole (g/mol). - For iso-octane, the calculated moles derived from the masses are: - Carbon: 84.12 grams converted to moles using its molar mass (12.01 g/mol) resulting in 7.005 moles. - Hydrogen: 15.88 grams converted using a molar mass of 1.01 g/mol resulting in 15.723 moles. - By determining the smallest whole number ratio of these moles, we can identify the empirical formula of the compound. The conversion from grams to moles aids in finding the simplest elemental ratio. This step is crucial as it lays the groundwork for understanding the compound's basic composition.

Iso-octane

Iso-octane is an important component in gasoline, characterized by its structural formula and role in determining octane ratings for fuels.- Iso-octane is a hydrocarbon, primarily made up of carbon and hydrogen atoms.- It plays a crucial role in fuels because it has a high resistance to knocking in engines.- The structure of iso-octane permits it to interact smoothly within engines, providing efficiency.- Understanding its empirical formula helps in comprehending how many molecules make up specific amounts of carbon and hydrogen. The empirical formula of \(CH_2\) initially points to the simplest representation before determining the actual molecular composition.

Hydrocarbon

Hydrocarbons are organic compounds consisting solely of carbon and hydrogen. They come in various forms, from simple structures to complex arrangements. - Iso-octane, a type of hydrocarbon, is particularly noteworthy in the energy industry. - The molecular structure of hydrocarbons like iso-octane impacts their chemical properties and usability in different applications. - As hydrocarbons serve primarily in energy provision, their molecular and empirical formulas offer insight into fuel efficiency and environmental impact. - Comprehension of hydrocarbons is critical for appreciating their roles as fuels, including combustion efficiency, pollution formation, and energy content.