Question

Hydrogen cyanide, 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 \(\mathrm{g} / \mathrm{cm}^{3} .\) (b) If the HCN is formed by reaction of \(\mathrm{NaCN}\) with an acid such as \(\mathrm{H}_{2} \mathrm{SO}_{4}\) what mass of NaCN gives the lethal dose in the room? $$ 2 \mathrm{NaCN}(s)+\mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)+2 \mathrm{HCN}(g) $$ (c) HCN forms when synthetic fibers containing Orlon or Acrilan burn. Acrilan has an empirical formula of \(\mathrm{CH}_{2} \mathrm{CHCN},\) so HCN is 50.9\(\%\) of the formula by mass. A rug measures \(12 \times 15 \mathrm{ft}\) and contains 30 oz of Acrilan fibers per square yard of carpet. If the rug burns, will a lethal dose of HCN be generated in the room? Assume that the yield of HCN from the fibers is 20\(\%\) and that the carpet is 50\(\%\) consumed.

Short answer

The lethal dose of HCN in the small laboratory room is approximately 1377 mg. To generate this lethal dose, we require 2.5 grams of NaCN reacting with H2SO4. When burning the rug containing Acrilan fibers, a significantly higher amount of HCN (865.8 grams) is generated compared to the lethal dose of 1.377 grams, thus confirming that a lethal dose of HCN will indeed be created in the room.

Step-by-step solution

Calculate the lethal dose of HCN in the laboratory room

To do this, we first need to determine the volume of the room in cubic centimeters. The room dimensions are given in feet; we'll convert them to centimeters using the conversion factor 1 ft = 30.48 cm. Room volume in \(ft^3\) = 12 ft × 15 ft × 8 ft = 1440 \(ft^3\) Now, let's convert this into \(cm^3\): Room volume in \(cm^3\) = 1440 \(ft^3\) × (30.48 cm/ft) × (30.48 cm/ft) × (30.48 cm/ft) = 3889,108.8 \(cm^3\) Next, we'll find the mass of air in the room using the given density of air: Mass of air = Density × Volume = 0.00118 g/\(cm^3\) × 3,889,108.8 \(cm^3\) = 4590.05 g Now, we can find the lethal dose of HCN in the room: Lethal dose of HCN = 300 mg/kg × mass of air / 1000 mg/g = (300 mg/kg × 4590.05 g) / 1000 mg/g = 1377.015 mg

Calculate the mass of NaCN required to produce the lethal dose of HCN

First, we will covert lethal dose of HCN from mg to grams: 1g = 1000 mg Lethal dose of HCN = 1377.015 mg × (1 g / 1000 mg) = 1.377 g HCN Using the stoichiometry of the given reaction, we can determine the mass of NaCN needed: 2 NaCN + H2SO4 → Na2SO4 + 2 HCN From the reaction, we see that two moles of NaCN are required to produce two moles of HCN. Therefore, the ratio is 1:1. We'll now convert the lethal dose of HCN to moles: 1 mole HCN = 27.03 g/mol (12.01 g/mol + 14.01 g/mol + 1 g/mol) Moles of HCN = mass / molar mass = 1.377 g / 27.03 g/mol = 0.051 mol HCN Since the ratio is 1:1, we need 0.051 mol NaCN. Now we can calculate the mass of NaCN required: 1 mole NaCN = 49.01 g/mol Mass of NaCN = 0.051 mol × 49.01 g/mol = 2.500 g NaCN

Calculate the amount of HCN generated during the burning of Acrilan fibers

First, we find the total mass of Acrilan fibers in the 12 ft × 15 ft rug: 1 yard = 3 ft Size of the rug = (12 ft / 3 ft) × (15 ft / 3 ft) = 4 yd × 5 yd = 20 yd² Mass of Acrilan fibers in the rug = 20 yd² × 30 oz/yd² = 600 oz The rug burns and is 50% consumed; thus, only half of the Acrilan fibers are burned: Mass of burned Acrilan fibers = 600 oz × 0.5 = 300 oz We'll convert this to grams: 1 oz = 28.35 g Mass of burned Acrilan fibers = 300 oz × 28.35 g/oz = 8505 g HCN is 50.9% of the empirical formula by mass. After accounting for the yield of 20%, we calculate the amount of HCN generated: Mass of HCN generated = 8505 g × 0.509 (50.9% HCN by mass) × 0.2 (20% yield) = 865.8 g Now we'll check whether the generated HCN is greater than the lethal dose calculated in Step 1: 865.8 g HCN > 1.377 g HCN Since the generated amount of HCN is significantly greater than the lethal dose of 1.377 g, burning the rug will indeed generate a lethal dose of HCN in the room.

Key concepts

Lethal Dose Calculation

Understanding how to calculate lethal doses of substances is an important aspect of toxicology. A lethal dose is the amount of a substance that is sufficient to cause death. For chemicals such as hydrogen cyanide (HCN), lethal dose calculations are vital for ensuring safety in various environments, such as laboratories.

The calculation of a lethal dose usually involves identifying the concentration that can lead to fatality and then using the volume of the space to determine the total fatal quantity. For instance, HCN's lethal dose is given per kilogram of air; we then translate that into the volume of the room by providing the density of air. This process involves converting units appropriately and utilizing dimensional analysis to ensure accuracy. A step often recommended to increase students’ understanding is visualizing the problem by sketching out the room and labeling the known values to create a clear picture of the scenario before tackling the calculations.

When teaching, remind students that lethal doses can differ based on the method of exposure and the physical characteristics of the individual or environment exposed. This informs them why specific contextual details, such as the weight of the air in a room, are necessary for calculating the lethal quantity of a substance.

Chemical Reaction

Chemical reactions form the foundation of stoichiometry, which is the study of the quantitative relationships between the substances involved in a chemical reaction. The balanced chemical equation provided in the hydrogen cyanide problem, for example, illustrates the reaction between sodium cyanide (NaCN) and sulfuric acid (H2SO4) to produce sodium sulfate (Na2SO4) and hydrogen cyanide (HCN).

To effectively teach chemical reaction concepts, emphasize the importance of a balanced equation, as it ensures the law of conservation of mass is satisfied. The ratio of reactants to products, known as stoichiometric coefficients, dictates how much of each substance is needed or produced. For our problem, the stoichiometry of the reaction shows a 1:1 molar relationship between NaCN and HCN, which is crucial for determining how much NaCN is needed to produce a lethal dose of HCN.

Incorporating real-life examples, such as the reaction that produces a toxic substance, allows students to grasp the significance of correctly calculating and predicting the amounts of reactants and products, emphasizing the real-world applications of stoichiometry. It's also important to point out that while equations give a theoretical yield, the actual yield in practical situations is often less, due to factors such as incomplete reactions and side reactions.

Empirical Formula

The empirical formula of a compound represents the simplest whole-number ratio of elements in that compound. For substances like Acrilan, the empirical formula, which is CH2CHCN, indicates the ratio of carbon (C), hydrogen (H), and nitrogen (N) in its structure. The mass percentage of each element within the empirical formula is a crucial piece of information when it comes to determining the actual amount of a specific compound, such as HCN, generated in a reaction or a scenario like the combustion of materials.

To enhance students’ understanding of empirical formulas, you can use visual aids, such as molecular models, to represent the simplified ratios. Point out that while the empirical formula gives us the ratio of elements, it does not convey the actual number of atoms in a more complex molecular structure. It's also valuable to discuss how the empirical formula relates to the molecular formula, especially if the molecular formula is a whole-number multiple of the empirical formula.

For the exercise at hand, students must apply their knowledge of the empirical formula to calculate the potential release of HCN when Acrilan burns. This involves using the percentage composition by mass to extract the amount of HCN that can be generated, considering the yield, which can be lower than 100% due to incomplete combustion or other side reactions.