Question

A gas containing chlorine and oxygen has a density of \(2.875 \mathrm{g} / \mathrm{L}\) at \(756 \mathrm{mmHg}\) and \(11^{\circ} \mathrm{C}\) a. Calculate the molar mass of the gas. b. What is the most likely molecular formula of the gas?

Short answer

Answer: The molar mass of the gas is approximately 85.5 g/mol, and the most likely molecular formula is ClO3.

Step-by-step solution

List the given information and the unknowns

We are given: - Gas density = \(2.875 \frac{g}{L}\) - Pressure, \(P = 756 \ mmHg\) - Temperature, \(T = 11^\circ C\) We need to find: - Molar mass of the gas (in g/mol) - Most likely molecular formula of the gas

Convert the given information to appropriate units

Convert pressure from mmHg to atm and temperature to Kelvin: - Pressure, \(P = 756 \ mmHg\) = \(\frac{756}{760} \ atm \approx 0.995 \ atm\) - Temperature, \(T = 11^\circ C\) = \(11 + 273.15 = 284.15 \ K\)

Calculate the molar mass using the ideal gas equation

Recall the ideal gas equation: \(PV = nRT\), where \(n\) is the amount of gas (in moles), \(R\) is the ideal gas constant (\(= 0.0821 \frac{L \cdot atm}{mol \cdot K}\)). We will first find \(\frac{n}{V}\) (the amount of gas per volume) and then use the density to find the molar mass. Density = \(\frac{mass}{volume}\) = \(\frac{n \cdot molar \ mass}{V}\) So, \(\frac{n}{V} = \frac{density}{molar \ mass}\) From the ideal gas equation, \(\frac{n}{V} = \frac{P}{RT}\) Equating both, \(\frac{density}{molar \ mass} = \frac{P}{RT}\) Rearrange the equation to find the molar mass: \(molar \ mass = \frac{density \cdot RT}{P}\) Substitute the values: \(molar \ mass = \frac{2.875 \ g/L \cdot 0.0821 \ L \cdot atm/mol \cdot K \cdot 284.15 \ K}{0.995 \ atm} \approx 85.5 \ g/mol\)

Find the most likely molecular formula of the gas

We know that the gas contains chlorine and oxygen atoms. Let's determine the most likely molecular formula based on the molar mass of chlorine (35.45 g/mol) and oxygen (16 g/mol). Possible combinations of chlorine (Cl) and oxygen (O) in the gas at different molar masses: - ClO: 35.45 + 16 = 51.45 g/mol - ClO2: 35.45 + (2 * 16) = 67.45 g/mol - ClO3: 35.45 + (3 * 16) = 83.45 g/mol - ClO4: 35.45 + (4 * 16) = 99.45 g/mol Comparing the molar masses, it is evident that the calculated molar mass of \(85.5 \ g/mol\) is closest to that of ClO3 (\(83.45 \ g/mol\)). Therefore, the most likely molecular formula of the gas is ClO3.

Summary

To conclude, we calculated the molar mass of the gas containing chlorine and oxygen to be approximately 85.5 g/mol. Based on this, the most likely molecular formula of the gas is ClO3.

Key concepts

Ideal Gas Law

When working with gases, we often rely on the Ideal Gas Law to link pressure, volume, temperature, and the amount of gas. This fundamental principle is expressed as \(PV = nRT\), where:

• \(P\) denotes pressure (in atm),
• \(V\) is volume (in liters),
• \(n\) represents the number of moles,
• \(R\) is the ideal gas constant, equal to \(0.0821 \, \frac{L\cdot atm}{mol\cdot K}\),
• and \(T\) is temperature in Kelvin.

In practical applications, we rearrange this equation to solve for unknown variables, such as the molar mass of a gas. It allows us to describe how a gas behaves under various conditions. For instance, to find the moles of gas per unit volume, use the formula \(\frac{n}{V} = \frac{P}{RT}\). This equation comes in handy when a gas's density is known, helping us determine its molar mass by intertwining with density-based formulas.

Molecular Formula Determination

Determining a gas's molecular formula requires knowledge of its composition and molar mass. After computing a substance's molar mass, comparison with the sums of known atomic weights is essential. Take chlorine and oxygen, for instance. Knowing their atomic masses, we can calculate potential molecular combinations. If the molar mass of a chlorine-oxygen compound is calculated to be roughly 85.5 g/mol, the task is to find a matching formula. Possible molecular weights could be derived from combining varying numbers of chlorine and oxygen atoms:

• ClO: 51.45 g/mol
• ClO₂: 67.45 g/mol
• ClO₃: 83.45 g/mol
• ClO₄: 99.45 g/mol

The closest match to our target molar mass (85.5 g/mol) is ClO₃ (83.45 g/mol). By aligning calculated figures with the experimental molar mass, we infer that the most plausible molecular formula is ClO₃.

Gas Density Conversion

Gas density links mass to volume. To find density in contexts like this, it's expressed as \(\frac{mass}{volume}\). However, when cross-referencing with the molar mass via the Ideal Gas Law, it requires appropriate unit conversions.Typically, we start with density in grams per liter, while pressure is often in units like mmHg. Our task is to align these units with the Ideal Gas constant's requirements. Hence, converting pressure to atm (using \(1\ atm = 760\ mmHg\)) is crucial. The temperature must also be in Kelvin, converting from Celsius using \(T_{K} = T_{C} + 273.15\).Upon performing these conversions, the molar mass formula \(molar \ mass = \frac{density \cdot RT}{P}\) becomes practical. It transforms given density data seamlessly with critical parameter setups into useful molecular insights.