Problem 103 - Chemistry Matter and Change

Question

Analyze When nitroglycerin \(\left(C_{3} H_{5} N_{3} O_{9}\right)\) explodes, it decomposes into the following gases: \(CO_{2}, N_{2}, NO\) , and \(H_{2} O\) . If 239 \(\mathrm{g}\) of nitroglycerin explodes, what volume will the mixture of gaseous products occupy at 1.00 \(\mathrm{atm}\) pressure and \(2678^{\circ} C ?\)

Step-by-Step Solution

Answer

The volume of the gaseous mixture produced after the explosion of 239 g of nitroglycerin, under the given conditions of 1.00 atm pressure and 2678°C temperature, is approximately 1757.9 L.

Step 1: Write the balanced chemical equation for the decomposition of nitroglycerin

The balanced chemical equation for the decomposition of nitroglycerin is: \(4 C_{3}H_{5}N_{3}O_{9} ⟶ 6 N_{2} + 10 H_{2}O + 12 CO_{2} + O_{2}\) Now we have the stoichiometric coefficients to determine the number of moles of gases produced.

Step 2: Calculate the number of moles of nitroglycerin

Given that we have 239 g of nitroglycerin, we first need to calculate the molar mass of nitroglycerin, which is: Molar mass = 3(12.01 g/mol C) + 5(1.01 g/mol H) + 3(14.01 g/mol N) + 9(16.00 g/mol O) = 227.09 g/mol Now we can find the number of moles of nitroglycerin: Number of moles = mass / molar mass = \(239g / 227.09 g/mol = 1.052\) moles of nitroglycerin

Step 3: Calculate the number of moles of each gas produced

Using the stoichiometric coefficients from the balanced chemical reaction, we can find the number of moles of each gas produced: For \(N_2\): Number of moles = 1.052 moles nitroglycerin * (6 moles \(N_2\) / 4 moles nitroglycerin) = 1.578 moles of \(N_2\) For \(H_2O\): Number of moles = 1.052 moles nitroglycerin * (10 moles \(H_2O\) / 4 moles nitroglycerin) = 2.630 moles of \(H_2O\) For \(CO_2\): Number of moles = 1.052 moles nitroglycerin * (12 moles \(CO_2\) / 4 moles nitroglycerin) = 3.156 moles of \(CO_2\) For \(O_2\): Number of moles = 1.052 moles nitroglycerin * (1 moles \(O_2\) / 4 moles nitroglycerin) = 0.263 moles of \(O_2\)

Step 4: Calculate the total number of moles of gases produced

Now that we've found the number of moles for each gas, we can calculate the total number of moles of gases produced: Total moles of gases produced = 1.578 moles of \(N_2\) + 2.630 moles of \(H_2O\) + 3.156 moles of \(CO_2\) + 0.263 moles of \(O_2\) = 7.627 moles

Step 5: Use the ideal gas law to find the volume of the gaseous mixture

We're given that the pressure is 1.00 atm, and the temperature is 2678 °C. However, the temperature needs to be converted to Kelvin. Temperature in K = 2678 °C + 273.15 = 2951.15 K Now we can use the ideal gas law (PV = nRT) to find the volume: Volume = nRT / P = (7.627 moles)(0.0821 L atm/(mol K))(2951.15 K) / (1.00 atm) = 1757.9 L The volume of the gaseous mixture produced after the explosion of 239 g of nitroglycerin, under the given conditions, is approximately 1757.9 L.

Key Concepts

Ideal Gas Law

The ideal gas law is critical in predicting the behavior of gases under different conditions. It is expressed as PV = nRT, where P stands for pressure, V represents volume, n indicates the number of moles, R is the universal gas constant, and T is the temperature in Kelvin. This key principle allows us to understand the relationships between these variables.

For instance, in the given problem, once we've calculated the total number of moles of gases produced from the decomposition of nitroglycerin, the ideal gas law enables us to compute the volume of the gas mixture at a given pressure and temperature. This calculation requires converting the temperature to Kelvin (K) from Celsius (°C); one can do so by adding 273.15 to the Celsius value. By knowing the ideal gas constant (R = 0.0821 L atm/(mol K) for pressure in atmospheres and volume in liters), we can easily calculate the final volume of the gases.

Molar Mass Calculation

Molar mass calculation is foundational in stoichiometry because it allows converting the mass of a substance to the amount in moles. The molar mass of a compound is calculated by summing the atomic masses of each element in the compound, multiplied by the number of atoms of that element in the formula.

In our exercise, we determine the molar mass of nitroglycerin \( C_{3}H_{5}N_{3}O_{9} \) by adding up the masses of 3 carbon atoms, 5 hydrogen atoms, 3 nitrogen atoms, and 9 oxygen atoms, based on their respective molar masses (given in grams per mole). Once the molar mass is known, dividing the total mass of nitroglycerin by its molar mass gives us the number of moles, a critical step for further calculations in stoichiometry.

Chemical Equation Balancing

Chemical equations must be balanced to obey the Law of Conservation of Mass, which states that matter cannot be created or destroyed. Balancing an equation involves matching the number of atoms of each element on both sides of the reaction. Each molecule and compound is represented by a chemical formula, and the number of molecules is indicated by a stoichiometric coefficient placed in front.

In the provided exercise, the balanced chemical equation for the decomposition of nitroglycerin was achieved by ensuring that the number of atoms for each element on the reactant side is equal to the number of atoms on the product side. Correctly balancing the equation is crucial for accurately determining the stoichiometry of the reaction, which directly affects the calculation of the number of moles of product formed.

Stoichiometric Coefficients

Stoichiometric coefficients are the numbers written in front of compounds within a balanced chemical equation. They tell us how many moles of each reactant and product participate in the reaction. It's the key to converting between moles of different substances in a reaction.

In our scenario, once the chemical equation is balanced, we can use the stoichiometric coefficients to calculate the exact number of moles of each gas produced from the decomposition of nitroglycerin. These coefficients form the necessary ratios for converting moles of the reactant into moles of each product. Understanding and applying these coefficients are essential for accurate calculations and predictions involving chemical reactions.