Question

Hydrogen cyanide, 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 \(\mathrm{HCN}\) 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 \(\mathrm{HCN}\) 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 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. A rug measures \(12 \times 15 \mathrm{ft}\) and contains \(30 \mathrm{oz}\) of Acrilan \({ }^{\circ}\) 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 calculated to be 14.495 grams. To produce this lethal dose, 26.29 grams of NaCN is required through the reaction with H₂SO₄. Burning the rug with given conditions will generate 865 grams of HCN, which is significantly more than the lethal dose; thus, a lethal dose of HCN will be produced in the room.

Step-by-step solution

a) Calculate the lethal dose of HCN in the room

First, we need to find the volume of the room in cubic meters (to make the units consistent), then find the mass of air in the room using the given density. Next, we will find the amount of HCN required for the lethal dose. 1. The room measures 12 × 15 × 8.0 ft. To convert it to meters, use the conversion factor: \(1 \mathrm{ft} = 0.3048 \mathrm{m}\) Room dimensions in meters: Length: \(12 \mathrm{ft} \times 0.3048 = 3.658 \mathrm{m}\) Width: \(15 \mathrm{ft} \times 0.3048 = 4.572 \mathrm{m}\) Height: \(8.0 \mathrm{ft} \times 0.3048 = 2.4384 \mathrm{m}\) 2. Calculate the volume of the room. Volume \(= l \times w \times h = 3.658 \times 4.572 \times 2.4384 = 40.946 \mathrm{m^3}\) 3. Convert the given density of air from g/cm³ to kg/m³: Density = \((0.00118 \frac{g}{cm^3})\times(\frac{1 kg}{1000 g})\times(\frac{100^3 cm^3}{1 m^3}) = 1.18 \frac{kg}{m^3}\) 4. Calculate the mass of air in the room: Mass of air = density × volume = 1.18 \(\frac{kg}{m^3}\) × 40.946 \(\mathrm{m^3}\) = 48.317 kg 5. Calculate the amount of HCN required for the lethal dose. Lethal dose = \((300 \mathrm{\frac{mg}{kg}}) \times (\frac{1 g}{1000 mg}) \times (48.317 kg) = 14.495 g\) The lethal dose of HCN in the room is 14.495 grams.

b) Calculate the mass of NaCN needed to produce the lethal dose of HCN

Now that we know how much HCN is needed for the lethal dose, we need to find the mass of NaCN needed to produce it. For this, we will use the stoichiometry of the reaction: \(2 \mathrm{NaCN(s) + H_2SO_4(aq) \rightarrow Na_2SO_4(aq) + 2 \mathrm{HCN}(g)}\) 1. Calculate the moles of HCN required for the lethal dose: Moles of HCN \(= \frac{mass}{molar\,mass} = \frac{14.495 g}{27.03 g/mol} = 0.5366 mol\) 2. Using the stoichiometry of the reaction, we can find the moles of NaCN needed \(2 \mathrm{NaCN} \longrightarrow 2 \mathrm{HCN}\) Moles of NaCN needed = 0.5366 mol (since in a 2:2 ratio) 3. Calculate the mass of NaCN: Mass of NaCN = moles × molar mass = 0.5366 mol × 49.01 g/mol = 26.29 g The mass of NaCN required to produce the lethal dose of hydrogen cyanide in the room is 26.29 grams.

c) Determine if burning the rug will produce a lethal dose of HCN

Given the size of the rug, the amount of Acrilan fibers in it, and the conditions of burning, we must determine if a lethal dose of HCN will be generated. 1. Calculate the total mass of Acrilan fibers in the rug: We are given that the rug contains 30 oz of Acrilan fibers per square yard. Convert 30 oz to grams: \(30\, \mathrm{oz}\, \times(\frac{28.35 \mathrm{g}}{1 \mathrm{oz}})= 850.5 g\) 2. Determine the square feet of the rug: Total area of the rug = Length × Width = 12 ft × 15 ft = 180 ft² 3. Convert the area of the rug to square yards: 180 ft² × \((\frac{1\, \mathrm{yd}}{3\, \mathrm{ft}})^2 = 20\, \mathrm{yd^2}\) 4. Determine the total mass of Acrilan fibers in the rug: Total mass = mass of Acrilan fibers/square yard × total square yards = 850.5 g/yd² × 20 yd² = 17,010 g 5. Account for the burning conditions when calculating the mass of Acrilan fibers burned: Mass of fibers burned = total mass × yield × burn amount = 17010 g × 0.2 × 0.5 = 1701 g 6. Calculate the mass of HCN produced from the burned Acrilan fibers: Mass of HCN = Mass of burned fibers × mass fraction of HCN = 1701 g × 0.509 = 865 g We found that burning the rug will produce 865 g of HCN, which is significantly more than the lethal dose (14.495 g) we calculated for the room. Therefore, burning the rug will generate a lethal dose of HCN in the room.

Key concepts

Chemistry Stoichiometry

In understanding chemistry, one of the fundamental concepts is stoichiometry, which involves calculations of the quantities of reactants and products in chemical reactions. It's based on the conservation of mass where the total mass of reactants equals the total mass of products. For instance, when calculating the lethal dose of hydrogen cyanide (HCN), we employ stoichiometry to relate the mass of a reactant (like NaCN) to the mass of the desired product (HCN).

Stoichiometric calculations start with a balanced chemical equation. In our exercise, the equation is:
\[ 2 \mathrm{NaCN} + \mathrm{H_2SO_4} \longrightarrow \mathrm{Na_2SO_4} + 2 \mathrm{HCN} \]
This equation tells us that two moles of sodium cyanide (NaCN) react to produce two moles of hydrogen cyanide gas. Hence, there is a one-to-one molar relationship between NaCN and HCN. By calculating the number of moles of HCN required for a lethal dose, you can directly ascertain the equivalent moles and, thus, the mass of NaCN using stoichiometry.

Gas Density Calculations

Gas density calculations are used to determine the mass of a gas within a certain volume, which is essential when dealing with gaseous substances like HCN. The density is typically given in grams per cubic centimeter (g/cm³) and can be converted to kilograms per cubic meter (kg/m³) to match SI units. To find the mass of a gas, we multiply the density by the volume it occupies.

In our problem, the density of air is given at specific conditions. We used the density formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
Inverting this relationship allows us to find mass if we know the density and volume of the air in the room. It is crucial to comprehend this concept to initially determine the mass of air in the room, as it helps in assessing the necessary quantity of HCN to reach a lethal concentration.

Molar Mass

Molar mass is a pivotal physical property defined as the mass of one mole of a substance. It is typically expressed in grams per mole (g/mol) and can be found by summing the atomic masses of the atoms in a molecule. For instance, HCN has a molar mass of 27.03 g/mol, which means one mole of HCN weighs 27.03 grams.

In our context of calculating lethal doses, knowledge of molar mass permits us to transition from mass to moles, which is foundational to stoichiometry. Molar mass serves as a conversion factor between the weight of a substance and the number of moles, which then allows us to relate the amount of different chemicals in a reaction using their molar ratios as illustrated in the balanced chemical equation.

Chemical Reaction

A chemical reaction is a process through which one or more substances, the reactants, are converted into one or more different substances, the products. In the given exercise, the reaction involves the formation of HCN from the reactant NaCN when it reacts with an acid like sulfuric acid (H₂SO₄). Understanding the nature of chemical reactions is crucial for predicting the quantities of products formed from given reactants.

When examining a reaction like:
\[ 2 \mathrm{NaCN} + \mathrm{H_2SO_4} \longrightarrow \mathrm{Na_2SO_4} + 2 \mathrm{HCN} \]
we look at the stoichiometry to deduce that two moles of NaCN yield two moles of HCN. Knowing the empirical formula of substances like Acrilan helps us to understand how much HCN will be produced under combustion. In a chemical reaction, it's also important to consider practical aspects such as yield, which refers to the efficiency of a reaction, and in the case of a burning rug, the percentage of material that actually burns. These factors affect the theoretical and actual outcome of the chemical reaction involved in producing a toxic gas like hydrogen cyanide.