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 \(3.5 \times 4.5 \times 2.5 \mathrm{~m}\). The density of air at \(26^{\circ} \mathrm{C}\) is $0.00118 \mathrm{~g} / \mathrm{cm}^{3} .(\mathbf{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 \(^{\text {- }}\) or Acrilan \(^{\circledast}\) burn. Acrilan \(^{\circledast}\) has an empirical formula of \(\mathrm{CH}_{2} \mathrm{CHCN},\) so HCN is \(50.9 \%\) of the formula by mass. A rug measures \(3.5 \times 4.5 \mathrm{~m}\) and contains $850 \mathrm{~g}\( of Acrilan \)^{\circledast}$ 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.

Step by step solution

01

1. Calculate the volume of the room

First, let's find the volume of the laboratory room by multiplying its dimensions: \(V = l \cdot w \cdot h = 3.5 \mathrm{~m} \cdot 4.5 \mathrm{~m} \cdot 2.5 \mathrm{~m}\).

02

2. Convert the volume to cubic centimeters

Next, we need to convert the volume of the room to cubic centimeters so it matches the density of air provided: \(V = V \cdot (100 \mathrm{~cm / m})^3\).

03

3. Calculate the mass of air in the room

Now we can find the mass of air in the room using the density of air formula: \(m_\text{air} = V \cdot \rho_\text{air}\), where \(\rho_\text{air} = 0.00118 \mathrm{~g} / \mathrm{cm}^3\).

04

4. Calculate the lethal dose of HCN in the room

Using the lethal dose of HCN per kilogram of air, we can find the lethal dose of HCN in the room: \(\mathrm{Lethal\:dose\:of\:HCN} = m_\text{air} \cdot \frac{300 \mathrm{mg\:HCN}}{1\mathrm{kg\:air}} \). Convert the lethal dose to grams: \(300\mathrm{mg}\) = \(0.3 \mathrm{g}\). #Part b#

05

1. Determine the stoichiometry of the reaction

From the balanced chemical equation, we can see that 2 moles of NaCN react to form 2 moles of HCN: \(\mathrm{2\:NaCN\:to\:2\:HCN}\).

06

2. Calculate the moles of HCN for the lethal dose

Using the lethal dose of HCN found in part (a), we can calculate the moles of HCN needed for the lethal dose: \(n_\text{HCN} = \frac{m_\text{HCN}}{M_\text{HCN}}\), where \(M_\text{HCN} = 27.03\,\mathrm{g/mol}\).

07

3. Calculate the moles and mass of NaCN for the lethal dose

Using the stoichiometry of the reaction, we can calculate the moles of NaCN needed for the lethal dose: \(n_\text{NaCN} = n_\text{HCN}\). Then, we can find the mass of NaCN required using the molar mass of NaCN, \(M_\text{NaCN} = 49.01\,\mathrm{g/mol} \): \(m_\text{NaCN} = n_\text{NaCN} \cdot M_\text{NaCN}\). #Part c#

08

1. Calculate the mass of HCN in the fibers of the rug

Using the given mass of Acrilan fibers, we can calculate the mass of HCN present in the fibers: \(m_\text{HCN} = 850 \mathrm{~g} \cdot 0.509\), where \(0.509\) is the fraction of HCN by mass in Acrilan.

09

2. Determine the mass of HCN generated when the rug burns

Now we can find the mass of HCN generated when the rug burns, taking into account the yield and the amount of carpet consumed: \(m_\text{HCN,generated} = m_\text{HCN} \cdot 0.20 \cdot 0.50\), where \(0.20\) is the yield of HCN and \(0.50\) is the fraction of carpet consumed.

10

3. Compare the mass of HCN generated to the lethal dose

Finally, compare the mass of HCN generated to the lethal dose found in part (a). If \(m_\text{HCN,generated} \ge \mathrm{Lethal\:dose\:of\:HCN}\), then a lethal dose of HCN will be generated in the room when the rug burns.

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