Problem 8 - Introductory Chemistry Atoms First

Question

A steel cylinder contains \(0.01378 \mathrm{~kg}\) of an unknown gas. Combustion analysis indicates that the gas has the empirical formula \(\mathrm{H}_{2}\) S. The volume of the cylinder is. \(2.20 \times 10^{3} \mathrm{~mL}\), and the pressure inside the cylinder is \(3.42 \times 10^{3} \mathrm{~mm} \mathrm{Hg}\). The cylinder is stored in a closet at \(25.0^{\circ} \mathrm{C} .\) What is the molecular formula of the gas?

Step-by-Step Solution

Answer

The molecular formula of the unknown gas is H₂S.

Step 1: Calculate the empirical molar mass for H₂S

The empirical formula is H₂S. We need to calculate the molar mass for this formula. The molecular mass of hydrogen (H) is 1 g/mol and the molecular mass of sulfur (S) is 32 g/mol. Empirical molar mass = (2 × 1) + 32 = 34 g/mol

Step 2: Convert the pressure to atm

To apply the ideal gas law, we need the pressure in atm. Given pressure = 3.42 × 10^3 mm Hg We know that 1 atm = 760 mm Hg, so we can divide by 760 to obtain the pressure in atm: Pressure in atm = (3.42 × 10^3) / 760 ≈ 4.50 atm

Step 3: Convert the temperature to Kelvin

To apply the ideal gas law, we need the temperature in Kelvin. Given temperature = 25.0°C. We can convert to Kelvin by adding 273.15: Temperature in K = 25.0 + 273.15 = 298.15 K

Step 4: Convert the volume to L

We also need the volume in L to apply the ideal gas law. Given volume = 2.20 × 10^3 mL To convert to L, we divide by 1000: Volume in L = 2.20 × 10^3 / 1000 = 2.20 L

Step 5: Apply the ideal gas law

Now we can use the ideal gas law to find the number of moles (n) of the gas: PV = nRT Where: P = Pressure in atm = 4.50 atm V = Volume in L = 2.20 L R = Gas constant = 0.0821 L×atm/mol×K T = Temperature in K = 298.15 K So, we can solve for n: n = PV / RT = (4.50 × 2.20) / (0.0821 × 298.15) ≈ 0.4047 mol

Step 6: Calculate the molar mass of the gas

We have the number of moles (n) and the mass of the gas. We can use this information to calculate the molar mass of the gas: Molar mass = Mass of the gas / number of moles Molar mass = (0.01378 kg × 1000) / 0.4047 = 34.1 g/mol We rounded the mass to one decimal place to match the number of significant figures in the given data.

Step 7: Determine the molecular formula

We found the molar mass of the gas to be 34.1 g/mol, and the empirical molar mass is 34 g/mol. Since the molar mass and empirical molar mass are nearly the same, the molecular formula of the gas is the same as the empirical formula: Molecular formula = Empirical formula Molecular formula = H₂S

Key Concepts

Empirical Formula

An empirical formula is a simple representation of the relative number of atoms of each element in a compound. It does not provide information about the molecule's actual structure but gives the simplistic ratio. In this case, the gas has an empirical formula of \(\mathrm{H}_2\mathrm{S}\), which indicates there are two hydrogen atoms for every sulfur atom. Here are some points about empirical formulas:

• They give the simplest ratio of elements in a compound.
• Empirical formulas are used as a starting point in determining the molecular formula.
• It is possible for different compounds to have the same empirical formula.

Having the empirical formula is crucial in solving problems involving composition analysis. It aids us in understanding which elements are present and in what minimal integer ratio, helping us to further identify the molecular formula of substances.

Ideal Gas Law

The Ideal Gas Law is an essential equation in chemistry: \(PV = nRT\). This formula describes the relationship between pressure (\(P\)), volume (\(V\)), temperature (\(T\)), and the number of moles (\(n\)) for an ideal gas, with \(R\) as the gas constant.Here’s a simple breakdown of its application:

• Convert Units: Ensure that pressure is in atmospheres, volume in liters, and temperature in Kelvin for consistent results. For pressure conversion, remember \(1\ atm = 760\ mm\ Hg\).
• Use Algebra: Rearrange \(PV = nRT\) to solve for the unknown variable. Usually, we rearrange to \(n = \frac{PV}{RT}\) when calculating the number of moles.
• Constant \(R\): The gas constant \(R\) is usually \(0.0821\ L\cdot atm/mol\cdot K\).

The ideal gas law allows for the calculation of the number of moles, which is pivotal in determining other properties such as molar mass or in confirming molecular formulas.

Molar Mass

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). In calculations involving gases, knowing the molar mass helps determine the actual molecular formula from the empirical formula. Steps in calculating molar mass are:

• Empirical Molar Mass: Compute it by adding the atomic masses of all atoms in the empirical formula. For example, \(\mathrm{H}_2\mathrm{S}\) has a molar mass of \[ 2\times1 + 32 = 34\ g/mol.\]
• Overall Molar Mass: Found by dividing the mass of the gas by the number of moles obtained from the ideal gas law: \[ \text{Molar mass} = \frac{\text{Mass of gas}}{n} \]

When the empirical molar mass matches the calculated molar mass, as in our case, the empirical formula can be directly considered the molecular formula.

Combustion Analysis

Combustion analysis is a method used to determine the chemical composition of a compound. It is especially useful for determining empirical formulas of organic compounds by analyzing the products after burning.Here's the basic idea:

• Burn the sample: Combustion usually converts all of the compound's hydrogen into water and its carbon into carbon dioxide, which are then measured.
• Calculate Ratios: These measurements allow you to calculate the number of moles of each element present and thus deduce the empirical formula.
• Unique to Gas Analysis: In our context, it specifically identifies that \(\mathrm{H}_2\mathrm{S}\) was the result of its analyzed components, based on its empirical formula.

By applying the combustion analysis method, scientists can identify not just the elemental composition but can confirm the empirical formula as achieved in this problem.