Question

The maximum allowable concentration of \(\mathrm{H}_{2} \mathrm{S}(g)\) in air is 20 mg per kilogram of air \((20\) ppm by mass). How many grams of FeS would be required to react with hydrochloric acid to produce this concentration at 1.00 atm and \(25^{\circ} \mathrm{C}\) in an average room measuring 12 \(\mathrm{ft} \times 20 \mathrm{ft} \times 8 \mathrm{ft}\) ? (Under these conditions, the average molar mass of air is 29.0 \(\mathrm{g} / \mathrm{mol.} )\)

Short answer

About 3.34 grams of FeS is required to react with hydrochloric acid to produce the given concentration of H₂S in the room.

Step-by-step solution

Convert room dimensions to meters and find the volume

First, we need to convert the room dimensions from feet to meters. The conversion factor is: 1 ft = 0.3048 m. So, \(12\:\mathrm{ft} \times 20\:\mathrm{"]=> 3.66\:\mathrm{m} \times 6.10\:\mathrm{m}\) \(8\:\mathrm{ft} \Rightarrow 2.44\:\mathrm{m}\) Now, we calculate the volume of the room: \(V = l \times w \times h = 3.66\:\mathrm{m} \times 6.10\:\mathrm{m} \times 2.44\:\mathrm{m} = 54.13\:\mathrm{m}^3\)

Calculate the mass of air in the room

We know the conditions are 1.00 atm and \(25^{\circ} \mathrm{C}\). Using the ideal gas law formula, \(PV = nRT\), we can calculate the moles of air in the room. First, convert the temperature to Kelvin, \(25^{\circ} \mathrm{C}\) = \((25+273.15) \mathrm{K} = 298.15\:\mathrm{K}\). Rearranging the formula to find n: \( n = \frac{PV}{RT} \) Now, plug in the values (R = 0.0821 L·atm/(K·mol)): \( n = \frac{1.00\: \mathrm{atm} \cdot 54.13\:\mathrm{m}^3 \cdot 1000\: \mathrm{L/m}^3}{0.0821\: \mathrm{L\:atm/(K\:mol)} \cdot 298.15\:\mathrm{K}} = 2227.3\: \mathrm{mol}\) Calculating the mass of air in the room: mass of air = moles of air (\(n)\) × molar mass of air (\(29.0\:\mathrm{g/mol}\)): mass of air = \(2227.3\: \mathrm{mol} \times 29.0\:\mathrm{g/mol} = 64691.7\: \mathrm{g}\)

Calculate the mass of H₂S

Now, we'll determine the mass of H₂S present using the given concentration (20 ppm by mass): mass of H₂S = \(\frac{20\:\mathrm{mg\: H_{2}S}}{10^6\:\mathrm{g\: air}} \times 64691.7\:\mathrm{g\: air} = 1.2938\:\mathrm{g\: of\: H_{2}S}\)

Stoichiometry to find moles of FeS

The balanced reaction of FeS with hydrochloric acid is: FeS (s) + 2 HCl (aq) → FeCl₂ (aq) + H₂S (g) Using stoichiometry, determine the moles of FeS needed to react with HCl to produce the calculated mass of H₂S: \(1\: \mathrm{mol\ of\: FeS} \Rightarrow 1\: \mathrm{mol\ of\: H_{2}S}\) Calculate the number of moles of H₂S: moles of H₂S = \(\frac{1.2938\:\mathrm{g}}{34.08\:\mathrm{g/mol} }\) //molar mass of H₂S = 34.08 g/mol moles of H₂S = 0.03797 mol H₂S Therefore, the moles of FeS needed = 0.03797 mol FeS.

Calculate the mass of FeS

Finally, convert the moles of FeS into grams using its molar mass (FeS = 87.91 g/mol): mass of FeS = moles of FeS × molar mass of FeS mass of FeS = 0.03797 mol × 87.91 g/mol = 3.3383 g So, approximately 3.34 grams of FeS is required to react with hydrochloric acid to produce the given concentration of H₂S in the room.

Key concepts

Ideal Gas Law

The ideal gas law is a cornerstone in understanding gas behaviors under various conditions, and it is represented by the formula: \( PV = nRT \). In this formula, \( P \) represents the pressure of the gas, \( V \) is the volume it occupies, \( n \) is the number of moles of the gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

By manipulating the ideal gas law, we can solve for any one of the variables if the others are known. This principle is crucial when calculating the amount of a gas, like in our scenario, where we calculate the moles of air in the room. We converted the given temperature to Kelvin and used the known pressure, volume, and the ideal gas constant to find the number of moles.

Molar Mass

Molar mass, often confused with molecular weight, is fundamental in stoichiometry for converting between the quantity of matter and the number of particles. It is defined as the mass of one mole of a substance and is usually expressed in grams per mole (g/mol).

Every chemical element has its unique molar mass, which can be found on the periodic table, and the molar mass of a compound is the sum of the molar masses of its constituent elements. In our exercise, we used the molar mass of air (29.0 g/mol) to find the mass of air in the room and the molar masses of H₂S and FeS to connect the weight of these compounds to the moles required for the chemical reaction.

Chemical Reaction Calculations

When dealing with chemical reactions, calculations often focus on determining the quantities of reactants or products. Essentials in these calculations include understanding the balanced chemical equation, the mole concept, and the law of conservation of mass.

From the balanced equation, we can see the mole ratio between reactants and products, which helps to find out how much of one substance will react or be produced from a certain amount of another. As we saw in our textbook problem, given the balanced equation for the reaction of FeS and HCl, we used stoichiometry to calculate the exact amount of FeS needed to produce a specified amount of H₂S.

Gas Volume Conversion

Gas volume conversion is a technique used to toggle between different units of volume. Since gases can fill their containers, their volume is an important property. Commonly, gas volumes are measured in liters or cubic meters, but in practical applications, they may be given in other units, such as cubic feet, and need to be converted.

In our problem, we converted the room's dimensions from feet to meters and then calculated the volume in cubic meters because we needed the volume in meters cubed to be compatible with other units in the ideal gas law equation. This step is essential to maintain consistency in units throughout our calculations, ensuring accuracy in the outcome.