Problem 49 - Chemistry

Question

A certain anesthetic contains 64.9 percent \(\mathrm{C}, 13.5\) percent \(\mathrm{H},\) and 21.6 percent \(\mathrm{O}\) by mass. At \(120^{\circ} \mathrm{C}\) and \(750 \mathrm{mmHg}, 1.00 \mathrm{~L}\) of the gaseous compound weighs \(2.30 \mathrm{~g}\). What is the molecular formula of the compound?

Step-by-Step Solution

Answer

The molecular formula of the compound is \(C_{9}H_{8}O\). The empirical formula is \(C_{7}H_{8}O\).

Step 1: Convert mass percentages to grams

Assume you have a 100g sample. Then, by the percentages given, it contains 64.9g of C, 13.5g of H, and 21.6g of O.

Step 2: Convert mass to moles

Divide the masses by the atomic masses of the elements to determine the number of moles of Carbon \(C\), Hydrogen \(H\), and Oxygen \(O\). Thus, \(C\) has \(\frac{64.9}{12.01}\) moles, \(H\) has \(\frac{13.5}{1.01}\) moles, and \(O\) has \(\frac{21.6}{16.00}\) moles.

Step 3: Normalize the mole ratios

Divide each mole ratio by the smallest one to normalize the ratio to its lowest whole numbers. Round to the nearest whole number.

Step 4: Calculate the empirical formula mass

Calculate the empirical formula mass by adding the masses corresponding to the empirical formula. This is obtained by multiplying the number of atoms of each element by the atomic weight of the element and adding them together.

Step 5: Obtain the molar mass

From the ideal gas law, the molar mass is calculated using \(PV = nRT\). Where we are given: \(P = 750mmHg = 0.9869atm, V = 1.00L, R = 0.0821L.atm/mol.K, T = 120^\circ C = 393.15K\). Rearranging the equation gives us \(n = PV/RT = (0.9869.1.00)/(0.0821.393.15)\). Thus, the molar mass of unknown compound is mass/moles = \(2.30g/n\).

Step 6: Determine the molecular formula

Divide the given molar mass by the empirical formula mass to find the number of empirical formula units per molecule. The molecular formula of the compound is obtained by multiplying the entire empirical formula by this number.

Key Concepts

Empirical Formula Calculation

When chemists analyze a compound, one of the first things they determine is the empirical formula. This formula represents the simplest whole-number ratio of the atoms within the compound. To calculate the empirical formula, you first need to convert the percentage composition of each element to mass, assuming a standard sample size, typically 100 grams for simplicity.

Here's how you would proceed with our example: The anesthetic compound consists of 64.9% Carbon (C), 13.5% Hydrogen (H), and 21.6% Oxygen (O). In a 100g sample, you'd have 64.9g of C, 13.5g of H, and 21.6g of O. Next, these masses are converted to moles by dividing by each element's atomic mass, since moles give us a way to compare amounts of substances based on the number of particles, not mass. Once you find the moles, you can derive the mole ratio by dividing all mole amounts by the smallest one. This gives you the lowest whole number ratio, which is the empirical formula.

Mole Concept

The mole concept is utterly essential in chemistry as it allows chemists to count atoms, molecules, and other particles using mass. One mole is defined as the amount of substance that contains as many particles as there are atoms in exactly 12 grams of carbon-12, which is approximately 6.022 x 10²³ particles - a number known as Avogadro's number.

For substances with different elements, the number of moles is the mass of the substance divided by the molar mass (the sum of the atomic masses of all the atoms in the molecule). In the case of our anesthetic, we calculate the moles of each element by dividing their mass (assuming a sample of 100g) by their respective atomic masses. This helps us arrive at molar ratios, a crucial step for deducing the empirical formula and, eventually, the molecular formula.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry and physics, describing the behavior of gases and providing a relationship between pressure (P), volume (V), number of moles (n), the ideal gas constant (R), and temperature (T). The law is usually stated as PV = nRT. It's useful for calculating one of these variables if the others are known, or in the case of our problem, determining the molar mass of a gaseous compound.

For the anesthetic gas at 120°C and 750 mmHg in a volume of 1.00 L, we can determine the number of moles using the ideal gas law. By first converting the temperature to Kelvin and the pressure to atmospheres, we apply the ideal gas equation to find the number of moles of the gas present. This calculated number of moles, when compared to the given mass of the gas, allows us to solve for the molar mass. This crucial step is instrumental in determining the molecular formula from the empirical formula.