Question

Nitroglycerin decomposes into four different gases when detonated: $$4 \mathrm{C}_{3} \mathrm{H}_{5}\left(\mathrm{NO}_{3}\right)_{3}(\ell) \rightarrow 6 \mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})+ 12 \mathrm{CO}_{2}(\mathrm{g})+10 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) $$ The detonation of a small quantity of nitroglycerin produces a total pressure of 4.2 atm at a temperature of \(450^{\circ} \mathrm{C}\) (a) What is the partial pressure of \(\mathrm{N}_{2} ?\) (b) If the gases occupy a volume of \(1.5 \mathrm{L}\), what mass of nitroglycerin was detonated?

Short answer

(a) Partial pressure of N2 is approximately 0.869 atm. (b) Mass of nitroglycerin detonated is approximately 3.34 g.

Step-by-step solution

Analyze the stoichiometry

First, we break down the balanced chemical equation. For 4 moles of nitroglycerin, the products are 6 moles of \(\mathrm{N}_2\), 1 mole of \(\mathrm{O}_2\), 12 moles of \(\mathrm{CO}_2\), and 10 moles of \(\mathrm{H}_2\mathrm{O}\). These sum up to a total of 29 moles of gas.

Calculate the mole fraction of N2

The mole fraction of \(\mathrm{N}_2\) in the gas mixture is calculated using: \[\text{Mole fraction of } \mathrm{N}_2 = \frac{\text{moles of } \mathrm{N}_2}{\text{total moles of gas}} = \frac{6}{29}.\]

Calculate the partial pressure of N2

The partial pressure of a gas is given by: \[P_{\mathrm{N}_2} = \text{Mole fraction of } \mathrm{N}_2 \times \text{Total Pressure}.\]Substitute the values to find:\[P_{\mathrm{N}_2} = \frac{6}{29} \times 4.2 \, \text{atm} \approx 0.869 \, \text{atm}.\]

Calculate moles of gas using PV=nRT

Use the ideal gas law to find the total moles of gas produced. Remember to convert \(450^\circ \mathrm{C}\) to Kelvin:\[ T = 450 + 273.15 = 723.15 \, \text{K}.\]The ideal gas law is \(n = \frac{PV}{RT}\). With \(P = 4.2 \, \text{atm}\), \(V = 1.5 \, \text{L}\), \(R = 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1}\), and \(T = 723.15 \, \text{K}\), we find:\[n = \frac{4.2 \times 1.5}{0.0821 \times 723.15} \approx 0.1067 \, \text{moles of gas}.\]

Calculate moles of nitroglycerin from moles of gas

Since 29 moles of gas are produced from 4 moles of nitroglycerin, we can set up a proportion:\[\frac{\text{moles of gas}}{29} = \frac{\text{moles of nitroglycerin}}{4}.\]Substitute the value of moles of gas:\[\frac{0.1067}{29} = \frac{x}{4},\]solving for \(x\), the moles of nitroglycerin:\[x \approx \frac{0.1067 \times 4}{29} \approx 0.0147 \, \text{moles of nitroglycerin}.\]

Calculate mass of nitroglycerin

Finally, calculate the mass of nitroglycerin using its molar mass. The molar mass of nitroglycerin \(\mathrm{C}_3\mathrm{H}_5\left(\mathrm{NO}_3\right)_3\) is 227.09 g/mol. Therefore, \[\text{mass} = 0.0147 \, \text{moles} \times 227.09 \, \text{g/mol} \approx 3.34 \, \text{g}.\]

Key concepts

Partial Pressure

When a mixture of gases is present, each gas contributes to the total pressure. This contribution is known as the partial pressure. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture equals the sum of the partial pressures of its components. To find the partial pressure of a specific gas like \(_2\), you can use its mole fraction and the total pressure of the gas mixture.
The mole fraction represents the ratio of moles of a specific gas to the total moles of all gases combined. In the given exercise, the partial pressure of \(_2\) is calculated by multiplying its mole fraction with the total pressure, providing crucial insight into the behavior of gas mixtures in chemical reactions.

Ideal Gas Law

The Ideal Gas Law is a powerful tool in chemistry, expressed by the equation \(PV = nRT\). It relates pressure (\(P\)), volume (\(V\)), moles of gas (\(n\)), the ideal gas constant (\(R\)), and temperature (\(T\)) in Kelvin.
This law is useful for calculating unknown values in a gas system. For instance, knowing that the gases occupy 1.5 liters at a certain pressure and temperature allows for the calculation of the total moles produced after nitroglycerin decomposition. Remember to convert temperature from Celsius to Kelvin by adding 273.15. The simplicity and universality of the Ideal Gas Law make it a cornerstone for analyzing gaseous reactions.

Mole Fraction

Mole fraction is an essential concept in stoichiometry, especially when dealing with gas mixtures. It is the ratio of the number of moles of a component to the total number of moles in the mixture. For example, in the decomposition reaction of nitroglycerin, there are 29 moles of gaseous products. To find the mole fraction of \(_2\), simply divide the moles of \(_2\) by the total moles of gas.
Mole fraction can help predict how each component gas affects the total pressure. This fractional representation simplifies the study of mixtures and allows chemical equations to be accessible and meaningful, especially when calculating partial pressures.

Chemical Decomposition

Chemical decomposition is a reaction where a compound is broken down into simpler substances. It requires energy input, such as heat, light, or electricity, to break the bonds within the compound. In the case of nitroglycerin, it decomposes explosively into four different gases: \(N_2\), \(O_2\), \(CO_2\), and \(H_2O\).
This reaction is highly exothermic, releasing significant energy and contributing to its use in explosions. Understanding decomposition reactions helps in risk assessment, energy calculations, and chemical engineering processes. It provides insight into the transition of complex compounds to simpler elements, making it a pivotal concept in chemistry.