Problem 67 - Chemistry and Chemical Reactivity

Question

There are five compounds in the family of sulfur-fluorine compounds with the general formula \(\mathrm{S}_{x} \mathrm{F}_{y}\). One of these compounds is \(25.23 \%\) S. If you place \(0.0955 \mathrm{g}\) of the compound in a \(89-\mathrm{mL}\). flask at \(45^{\circ} \mathrm{C},\) the pressure of the gas is 83.8 mm Hg. What is the molecular formula of \(\mathrm{S}_{x} \mathrm{F}_{y} ?\)

Step-by-Step Solution

Answer

The molecular formula is \( \mathrm{S}_2 \mathrm{F}_{10} \).

Step 1: Calculate Molar Mass Using the Ideal Gas Law

To find the molar mass, use the ideal gas law \( PV = nRT \), solving for \( n \) (moles): \[ n = \frac{PV}{RT} \]. Convert 83.8 mm Hg to atm: \( \frac{83.8}{760} \) atm = 0.11026 atm. Convert 45°C to Kelvin: 45 + 273.15 = 318.15 K. Use \( R = 0.0821 \) L·atm/mol·K. Volume (\( V \)) = 0.089 L. Substitute values: \[ n = \frac{0.11026 \times 0.089}{0.0821 \times 318.15} = 3.79 \times 10^{-4} \text{ mol} \].

Step 2: Calculate Molar Mass of the Compound

The molar mass (M) is the mass of the compound divided by the number of moles: \[ M = \frac{0.0955 \text{ g}}{3.79 \times 10^{-4} \text{ mol}} \approx 251.2 \text{ g/mol} \].

Step 3: Calculate the Molar Mass of Sulfur in Compound

The mass percentage of sulfur is given as 25.23%. If the molar mass of the compound is 251.2 g/mol, the mass of sulfur can be calculated as: \[ \text{Mass of Sulfur} = 0.2523 \times 251.2 \approx 63.4 \text{ g/mol} \]. The molar mass of sulfur (S) is 32.07 g/mol, hence number of sulfur atoms: \[ x = \frac{63.4}{32.07} \approx 2 \].

Step 4: Determine the Formula

With \( x = 2 \) for sulfur, calculate the remainder of the molar mass for fluorine: \[ 251.2 \text{ g/mol} - 63.4 \text{ g/mol} = 187.8 \text{ g/mol} \]. The molar mass of fluorine (F) is 19 g/mol. Number of fluorine atoms: \[ y = \frac{187.8}{19} \approx 9.9 \approx 10 \].

Step 5: Deduce the Molecular Formula

The approximate values for \( x \) and \( y \) lead to the molecular formula \( \mathrm{S}_2 \mathrm{F}_{10} \), which matches the known compound sulfur hexafluoride \( \mathrm{SF}_6 \).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a crucial concept in understanding the behavior of gases. It is expressed by the formula \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. This equation allows us to calculate the amount of gas in moles when other properties like volume, pressure, and temperature are known.

The pressure in the formula needs to be converted to atmospheres (atm) if it's not initially provided in this unit. Similarly, temperature must be in Kelvin, requiring a conversion from degrees Celsius by adding 273.15. These conversions are essential for correct calculation.

The Ideal Gas Law is versatile and often used for calculating molar mass as it links the physical properties of a gas with the number of moles. This connection allows us to derive the molar mass when the mass of the compound and the conditions of its gaseous state are known.

Molar Mass

Molar Mass is the mass of one mole of a substance, usually expressed in g/mol. It is critical for converting the weight of a sample to moles, which provides a way to relate macroscopic quantities to microscopic properties.

Calculating molar mass using the Ideal Gas Law involves dividing the mass of the gas sample by the number of moles obtained from \( n = \frac{PV}{RT} \). This results in the molar mass, offering insights into the substance's identity.

Understanding molar mass helps in determining how many atoms or molecules are present in a given weight of the compound, ensuring precise chemical calculations. In the context of sulfur-fluorine compounds, knowing the molar mass aids in discovering the exact arrangement of atoms in the compound.

Sulfur-Fluorine Compounds

Sulfur-fluorine compounds represent a family with various chemical structures, marked by unique interactions between sulfur and fluorine atoms. They typically follow a general formula of \( \mathrm{S}_x \mathrm{F}_y \).

The molecular arrangement of these compounds is significant due to their stability and interesting chemical properties. An example is sulfur hexafluoride \( \mathrm{SF}_6 \), known for its strong covalent bonds and being an inert gas. Characteristics like these make investigating their molecular compositions a fascinating task.

Determining the number of sulfur and fluorine atoms in such compounds involves chemical analysis, revealing the specific arrangement that dictates the properties of the compound. These insights are essential for applications in various fields, from chemical engineering to product manufacturing.

Chemical Composition Analysis

Chemical Composition Analysis is the process of determining what elements and how much of each is present in a compound. This analysis often includes calculating mass percentages of elements within the compound, aiding in the determination of molecular formulas.

From there, the ratio of atomic masses provides a way to deduce the number of particular atoms from a known sample mass. For instance, in sulfur-fluorine compounds like \( \mathrm{S}_2 \mathrm{F}_{10} \), the known mass percentage of sulfur coupled with the total molar mass indicates the number of atoms present.

This method enables precise formulation of the compound’s molecular structure through careful calculation and consideration of atomic masses, enhancing our understanding of its chemical nature. Such analysis is a vital tool for chemists in identifying compounds and understanding their potential reactivity and uses.