Question

A 2.85 -g sample of an unknown chlorofluorocarbon decomposes and produces 564 \(\mathrm{mL}\) of chlorine gas at a pressure of 752 \(\mathrm{mm} \mathrm{Hg}\) and a temperature of 298 \(\mathrm{K}\) . What is the percent chlorine (by mass) in the unknown chlorofluorocarbon?

Short answer

The percent chlorine by mass in the chlorofluorocarbon is calculated by determining the moles of chlorine gas produced using the ideal gas law, converting that to mass using the molar mass of chlorine, and then using the mass of chlorine and mass of the sample to find the percentage.

Step-by-step solution

Convert the Volume of Chlorine Gas to Moles

To determine the moles of chlorine gas, we use the ideal gas law, which is given by PV = nRT. We have a volume (V) of 564 mL, which we need to convert to liters by dividing by 1000, a pressure (P) of 752 mmHg, which we convert to atmospheres by dividing by 760, the temperature (T) of 298 K, and the ideal gas constant (R) of 0.0821 L·atm/mol·K. The moles of chlorine gas (n) can be calculated by rearranging the ideal gas law to n = PV / RT.

Calculate Moles of Chlorine Gas

Using the converted values, calculate the moles of chlorine gas (n) as follows: n = (P * V) / (R * T) n = (752 mmHg / 760 mmHg/atm) * (564 mL / 1000 mL/L) / (0.0821 L·atm/mol·K * 298 K)

Calculate Mass of Chlorine Gas

Once you have the moles of chlorine gas, multiply it by the molar mass of chlorine gas (Cl2) to get the mass. The molar mass of Cl2 is approximately 70.90 g/mol.

Calculate Percent Chlorine by Mass

With the mass of chlorine gas, you can find the percent chlorine in the sample by using the formula: percent chlorine = (mass of chlorine / mass of sample) * 100%

Key concepts

Understanding the Ideal Gas Law

One of the foundational principles in chemistry is the ideal gas law, represented by the formula PV = nRT. This equation illustrates the relationship between pressure (P), volume (V), temperature (T), and the number of moles of a gas (n), with R being the ideal gas constant.

When working with gases, it's essential to measure and control conditions to make precise predictions or determinations about a gas's behavior. In the given exercise, the student was asked to use the ideal gas law to find the moles of chlorine gas produced by the decomposition of an unknown chlorofluorocarbon.

To use the ideal gas law effectively, we must ensure all the variables are in the correct units. Pressure must be in atmospheres (atm), volume in liters (L), temperature in Kelvin (K), and the gas constant R in consistent units of L·atm/mol·K. By carrying out the conversion of units, as demonstrated in the first steps of the solution, and applying the law, the problem becomes manageable.

Importance of Standard Units

Ensuring your measurements are in the standard units before using the ideal gas law is crucial. This will prevent errors and provide an accurate value for the moles of gas. When students overlook this step, it can lead to calculation errors that affect the rest of the stoichiometric analysis.

Molar Mass Calculation Explained

Molar mass is a vital constant that enables chemists to convert between moles, the standard unit of quantity in chemistry, and grams, a more familiar unit of mass. The molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol).

Understanding molar mass is essential for a multitude of calculations in chemistry, including those involving the ideal gas law and stoichiometry. When the exercise asks for the mass of chlorine gas, the student uses the previously calculated number of moles and multiplies it by the molar mass of chlorine gas, Cl₂, which is roughly 70.90 g/mol.

How to Find Molar Mass

The molar mass is often found using the periodic table, where the atomic mass of each element, listed in atomic mass units (amu), is effectively the same as the molar mass in grams per mole. For molecules like Cl₂, simply add the atomic masses of the constituent atoms.

• Identify the elements in the compound and their respective quantities in a single molecule.
• Look up each element's atomic mass on the periodic table.
• Multiply the atomic mass of each element by the quantity in the molecule.
• Sum the total mass to find the compound's molar mass.

The Role of Stoichiometry in Chemistry

Stoichiometry is the study of the quantitative relationships, or ratios, between substances involved in chemical reactions. It's the core of chemical equation balancing and allows chemists to predict the amounts of products and reactants involved in a reaction.

In the context of the exercise, stoichiometry is the bridge between the moles of chlorine gas calculated from the ideal gas law and the final percent composition by mass of chlorine in the chlorofluorocarbon. After using the ideal gas law and molar mass calculations to determine the mass of chlorine gas, the student is required to apply stoichiometric principles to express this mass as a percentage of the total mass of the decomposed sample.

Stoichiometric Calculations

To complete a stoichiometric calculation:

• Determine the molar ratio between reactants and products from a balanced chemical equation.
• Convert given quantities into moles if they are not already in that form.
• Use the molar ratio to convert moles of one substance to moles of another.
• If necessary, convert moles back into grams or other units to obtain the required information.

The ability to perform stoichiometric calculations is critical and students should practice various scenarios to solidify their understanding.