Question

Hydrogen cyanide, \(\mathrm{HCN}\), is a poisonous gas. The lethal dose is approximately \(300 \mathrm{mg}\) HCN per kilogram of air when inhaled. (a) Calculate the amount of HCN that gives the lethal dose in a small laboratory room measuring \(12 \times 15 \times 8.0 \mathrm{ft}\). The density of air at \(26^{\circ} \mathrm{C}\) is \(0.00118 \mathrm{~g} / \mathrm{cm}^{3} .(\mathrm{b})\) If the \(\mathrm{HCN}\) is formed by reaction of \(\mathrm{NaCN}\) with an acid such as \(\mathrm{H}_{2} \mathrm{SO}_{4}\), what mass of \(\mathrm{NaCN}\) gives the lethal dose in the room? \(2 \mathrm{NaCN}(s)+\mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)+2 \mathrm{HCN}(g)\) 0 (c) HCN forms when synthetic fibers containing Orlon or Acrilan \(^{8}\) burn. Acrilan \(^{B}\) has an empirical formula of \(\mathrm{CH}_{2} \mathrm{CHCN}\), so \(\mathrm{HCN}\) is \(50.9 \%\) of the formula by mass. \(\mathrm{A}\) rug measures \(12 \times 15 \mathrm{ft}\) and contains 30 oz of Acrilan \(^{\otimes}\) fibers per square yard of carpet. If the rug burns, will a lethal dose of \(\mathrm{HCN}\) be generated in the room? Assume that the yield of \(\mathrm{HCN}\) from the fibers is \(20 \%\) and that the carpet is \(50 \%\) consumed.

Short answer

In this problem, we first calculate the lethal dose of HCN for a room of given dimensions and the mass of NaCN required to produce this lethal dose of HCN based on a given reaction. Then we check if a burning rug containing Acrilan fibers generates a lethal amount of HCN in the room. (a) The lethal dose of HCN in the room is found to be \(M_\mathrm{HCN,lethal} = M_\mathrm{air} \times (300 \times 10^{-3}) / 1000 \, \mathrm{g} \), where \(M_\mathrm{air}\) is the mass of air in the room. (b) The mass of NaCN required for the lethal dose is given by \(M_\mathrm{NaCN} = \mathrm{(Number \, of \, moles \, of \, NaCN)} \times \mathrm{Molar \, Mass \, of \, NaCN}\). (c) To check if the burning rug generates lethal HCN, we compare the mass of HCN produced (Mass of HCN = Mass of Acrilan × 50.9% × 20% × 50%) with the lethal dose calculated in part (a). If the mass of HCN generated is greater than or equal to the lethal dose, a lethal amount of HCN will be generated in the room.

Step-by-step solution

Part (a): Calculate lethal dose of HCN

1. First, we need to find the volume of the room in cubic centimeters (cm³). We are given that the dimensions are in feet (ft), so we will convert them into cm. \[1 \mathrm{ft} = 30.48 \mathrm{cm}\] Volume of the room = length × width × height \[V = 12 \times 15 \times 8.0 \times (30.48)^3 \, \mathrm{cm}^3\] 2. Calculate the mass of air in the room. We are given the density of the air at \(26^\circ C\) as \(0.00118 \, \mathrm{g} / \mathrm{cm}^3\). Mass of air = Volume × Density \[M_\mathrm{air} = V \times 0.00118 \, \mathrm{g}\] 3. Finally, we calculate the lethal dose of HCN in the room. Lethal dose of HCN = mass of air × (300 mg HCN/1 kg air) To convert the lethal dose from mg to g, we divide by 1000. \[ M_\mathrm{HCN,lethal} = M_\mathrm{air} \times (300 \times 10^{-3}) / 1000 \, \mathrm{g} \]

Part (b): Calculate the mass of NaCN

1. Based on the balanced chemical equation, we have a 2:2 stoichiometric ratio between NaCN and HCN, meaning that for every 2 moles of NaCN, 2 moles of HCN are produced. \[2 \mathrm{NaCN} + \mathrm{H}_2\mathrm{SO}_4 \rightarrow \mathrm{Na}_2\mathrm{SO}_4 + 2 \mathrm{HCN}\] 2. Next, calculate the number of moles of lethal HCN: Number of moles of HCN = \( \frac{M_\mathrm{HCN,lethal}}{\mathrm{Molar \, Mass \, of \, HCN}} \) 3. Using the stoichiometry, determine the number of moles of NaCN required: Number of moles of NaCN = Number of moles of HCN (based on the balanced equation) 4. Calculate the mass of NaCN required: \[M_\mathrm{NaCN} = \mathrm{(Number \, of \, moles \, of \, NaCN)} \times \mathrm{Molar \, Mass \, of \, NaCN}\]

Part (c): Check if the burning rug generates lethal HCN

1. Calculate the number of square yards in the rug: Rug area = 12 ft × 15 ft = 180 ft²; \(1 \, \mathrm{yd}^2 = 9 \, \mathrm{ft}^2\) Number of square yards = Rug area / 9 ft²/ yd² 2. Calculate the mass of Acrilan fibers in the rug: Mass of Acrilan = (Number of square yards) × (30 oz/square yard) Convert the mass of Acrilan from ounces (oz) to grams (g). \( 1 \, \mathrm{oz} = 28.35 \, \mathrm{g}\) 3. Calculate the mass of HCN produced when burning the rug, taking into account that the yield of HCN is 20% and the rug is 50% consumed: Mass of HCN = (Mass of Acrilan × 50.9% × 20% × 50%) 4. Compare the mass of HCN produced with the lethal dose of HCN calculated in part (a). If the mass of HCN generated is greater than or equal to the lethal dose, a lethal amount of HCN will be generated in the room.

Key concepts

Lethal Dose Calculation

To determine how much hydrogen cyanide (HCN) can be lethal in a confined space, we first calculate the lethal dose. Understanding the conversion of volume from feet to cubic centimeters helps us to compute the room's volume accurately.
The room measurement given is in feet (12 x 15 x 8), and these are converted into cubic centimeters (cm³) using the conversion factor 1 foot = 30.48 cm. With this conversion, we can calculate the room's volume as:

• Volume, \( V = 12 \times 15 \times 8 \times (30.48)^3 \) cm³

Next, we find the mass of air in the room by multiplying the volume by the given air density (0.00118 g/cm³). This provides the total air mass present for the reaction.
Finally, the lethal dose is derived from multiplying the air mass by the lethal concentration of HCN, given as 300 mg per kilogram of air. We convert milligrams to grams by dividing by 1000 for ease of calculation and comprehension.
This process assures us how much HCN presence in a room of specific dimensions leads to a toxic environment, crucial for safety and toxicity studies.

Chemical Stoichiometry

Chemical stoichiometry involves understanding the quantitative relationships between reactants and products in a chemical reaction. When assessing the production of HCN through the reaction of sodium cyanide (NaCN) with sulfuric acid (\( \text{H}_2\text{SO}_4 \)), it is vital to recognize stoichiometric ratios as dictated by the balanced chemical equation.

• The balanced equation is: \( 2 \text{NaCN} + \text{H}_2\text{SO}_4 \rightarrow \text{Na}_2\text{SO}_4 + 2 \text{HCN} \)
• The stoichiometry shows a 2:2 molar ratio between NaCN and HCN.

First, we calculate the moles of HCN using its molar mass. Then, through stoichiometry, it follows that the number of moles of NaCN needed is identical to those of HCN because the ratio is unity (1:1).Once the moles of NaCN are known, the mass of NaCN required to yield a lethal dose of HCN is calculated. It is done by multiplying the number of moles of NaCN by its molar mass. This calculation lets us ascertain the exact quantity of reactant necessary to achieve certain levels of output in chemical processes, critical in both laboratory and industrial applications.

Combustion Reactions

When substances containing hydrogen cyanide, like Acrilan fibers, burn, they undergo combustion reactions. Knowing the chemical nature of the combustion helps to determine if the resultant HCN amounts are harmful.
Acrilan, with an empirical formula of \( \text{CH}_2\text{CHCN} \), contains 50.9% HCN by mass. Given the context, calculations must consider several factors:

• Rug area converted from feet squared to square yards, as 1 yd² = 9 ft².
• The amount of Acrilan fibers per square yard is 30 oz, converted to grams using 1 oz = 28.35 g.

Upon burning, only a fraction of Acrilan contributes to the HCN released, factored by a 20% yield and that only 50% of the carpet burns.
The mass of HCN is determined by multiplying these ratios with the initially available Acrilan mass. By then comparing this value with the lethal dose calculated earlier, it can be determined whether combustion produces a toxic level of HCN.
This evaluation is crucial for ensuring safety in environments where such materials may burn, helping in fire hazard assessments and emergency preparedness.