Question

Hydrogen cyanide, \(\mathrm{HCN}\), is a poisonous gas. The lethal dose is approximately \(300 \mathrm{mg} \mathrm{HCN}\) per kilogram of air when inhaled. (a) Calculate the amount of HCN that gives the lethal dose in a small laboratory room measuring \(12 \times 15 \times 80 \mathrm{ft}\). The density of air at \(26{ }^{\circ} \mathrm{C}\) is \(0.00118 \mathrm{~g} / \mathrm{cm}^{3}\), (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? \(2 \mathrm{NaCN}(s)+\mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)+2 \mathrm{HCN}(g)\) (c) HCN forms when synthetic fibers containing Orlon \({ }^{8}\) or Acrilan burn. Acrilan 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 \(30 \mathrm{oz}\) of Acrilan \(^{8}\) 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 summary, the lethal dose of HCN in the small laboratory room is calculated by finding the volume of the room, mass of the air, and using the given lethal dose concentration. The required mass of NaCN is calculated from the moles of HCN needed for the lethal dose. Finally, the mass of HCN produced from the rug is found, and the result is compared to the lethal dose to determine if a lethal amount of HCN is generated by the burning rug.

Step-by-step solution

Calculate the volume of the room

To calculate the volume of the room, we will need to multiply the three dimensions: length, width, and height. The room measures \(12 \times 15 \times 80~ft\). First, we need to convert feet to centimeters, as we have the density of air in g/cm³. \(1~ft = 30.48~cm\) Then calculate the volume in cm³: \(V = 12~ft \times 15~ft \times 80~ft \times (30.48~cm/ft)^3 = 12 \times 15 \times 80 \times 30.48^3~cm^3 \)

Calculate the mass of the air in the room

Using the density given, we can calculate the mass of the air inside the room: \(mass = density \times volume\) \(mass = 0.00118~g/cm^3 \times V\)

Calculate the lethal dose HCN amount

The lethal dose is given by: \(lethal~dose = 300~mg~HCN/kg~air\) We have the mass of air in grams, so we can calculate the lethal amount of HCN: \(lethal~amount~HCN = mass~air \times (lethal~dose / 1000)\)

Calculate the required mass of NaCN

From the provided reaction, 2 moles of NaCN produce 2 moles of HCN. Calculate the amount (mol) of HCN required for the lethal dose from its mass and molecular weight (27.03 g/mol): \(n_{HCN} = mass~HCN / 27.03~g/mol\) The moles of NaCN required are the same as the moles of HCN. Now we can calculate the mass of NaCN needed (using the molecular weight of NaCN, 49.01 g/mol): \(mass~NaCN = n_{HCN} \times 49.01~g/mol\)

Calculate the mass of HCN produced from the rug

First, calculate the mass of Acrilan fibers in the rug: - Total area of rug = 12 ft x 15 ft - Acrilan fiber mass per area = 30 oz/square yard - Percent of Acrilan fiber consumed = 50% - Acrilan in the rug = Area × (Acrilan mass per area) × (Percent consumed) Next, calculate the mass of HCN generated from Acrilan fibers: - Percent HCN in Acrilan formula = 50.9% - HCN yield from fibers = 20% - HCN mass = Acrilan in the rug × (Percent HCN in Acrilan formula) x (HCN yield from fibers)

Compare the mass of HCN produced with the lethal dose

Determine if the mass of HCN produced from burning the rug is greater than or equal to the lethal amount of HCN calculated in step 3. If it is, then the burning of the rug will generate a lethal dose of HCN in the room.

Key concepts

Lethal Dose Calculation

Hydrogen cyanide (HCN) is highly toxic, and calculating its lethal dose is crucial for understanding its danger. The lethal dose is defined as the amount needed to potentially cause death in a given situation. For HCN, the lethal dose is given as 300 mg per kilogram of air inhaled. This calculation helps us determine how much HCN needs to be present in an enclosed space to reach this critical level.
To find the lethal dose in a specific room, you first need to know the mass of air present. By converting the room's volume into a usable format and multiplying by the known density of air, we get the mass of air. The lethal dose of HCN can then be calculated by multiplying the air mass by the lethal dose factor (300 mg/kg) and adjusting the units correctly. It's a vital step in assessing the risk in various environments.

Volume and Density

Understanding volume and density is essential when dealing with any gas, as both factors greatly influence calculations involving air or other gases. Volume refers to the space occupied by an object or substance, and it is usually measured in cubic centimeters (cm³), meters (m³), or liters.
In this exercise, we calculated the room's volume by multiplying its three dimensions: length, width, and height. However, because the density is provided in g/cm³, we first converted these dimensions from feet to centimeters using the conversion factor: 1 ft = 30.48 cm.
Density, on the other hand, is a measure of mass per unit volume, typically expressed in g/cm³ or kg/m³. The density of air at 26°C is 0.00118 g/cm³, which allows us to find the mass by multiplying density by the calculated volume. Once we have the mass of air, we can proceed with our calculations for the lethal dose of hydrogen cyanide.

Chemical Reactions

Chemical reactions play a key role in understanding how hydrogen cyanide can be produced and released. One such reaction involves sodium cyanide (\(\mathrm{NaCN}\)) reacting with sulfuric acid (\(\mathrm{H_2SO_4}\)). The balanced chemical equation for this reaction is:
\(2 \mathrm{NaCN}(s) + \mathrm{H_2SO_4}(aq) \longrightarrow \mathrm{Na_2SO_4}(aq) + 2 \mathrm{HCN}(g)\).
This reaction shows that two moles of sodium cyanide react with sulfuric acid to produce two moles of hydrogen cyanide gas. Each mole of NaCN results in an equivalent amount of HCN, making the stoichiometry a simple 1:1 conversion per mole.
Understanding this reaction is important for determining how much of a chemical is needed to produce a lethal dose of hydrogen cyanide, which in this case depends directly on the amount of sodium cyanide used.

Mass Calculation

Mass calculation is crucial when considering how much of a substance is needed or produced in chemical reactions. For this problem, we calculated the mass of NaCN required to produce a lethal dose of HCN. To do this, we first determined the amount (in moles) of HCN needed using its molar mass: 27.03 g/mol.
The moles of HCN required can be directly converted into equivalent moles of NaCN due to the chemical reaction stoichiometry. We then use the molar mass of NaCN, which is 49.01 g/mol, to find the required mass of sodium cyanide by multiplying the moles of NaCN by its molar mass.
When considering the burning of synthetic fibers (like Acrilan), which also release HCN, similar calculations are necessary. Here, the mass of the fibers, the percentage of HCN in the material, and the yield from combustion all factor into determining whether a lethal dose is produced during burning.