Some very effective rocket fuels are composed of lightweight liquids. The fuel composed of dimethylhydrazine \(\left[\left(\mathrm{CH}_{3}\right)_{2} \mathrm{~N}_{2} \mathrm{H}_{2}\right]\) mixed with dinitrogen tetroxide was used to power the Lunar Lander in its missions to the moon. The two components react according to the following equation: \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{~N}_{2} \mathrm{H}_{2}(l)+2 \mathrm{~N}_{2} \mathrm{O}_{4}(l) \longrightarrow 3 \mathrm{~N}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g)+2 \mathrm{CO}_{2}(g)\) If \(150 \mathrm{~g}\) dimethylhydrazine reacts with excess dinitrogen tetroxide and the product gases are collected at \(127^{\circ} \mathrm{C}\) in an evacuated 250-L tank, what is the partial pressure of nitrogen gas produced and what is the total pressure in the tank assuming the reaction has \(100 \%\) yield?

Balancing a chemical reaction is a fundamental step in the study of stoichiometry. It involves adjusting the coefficients (the numbers in front of the chemical formulas) to ensure that the number of atoms of each element is the same on both sides of the equation. This is based on the law of conservation of mass, which states that matter cannot be created or destroyed in a chemical reaction.

Let's consider the exercise of the rocket fuel composed of dimethylhydrazine reacting with dinitrogen tetroxide. The chemical equation provided is already balanced. It tells us that one molecule of dimethylhydrazine reacts with two molecules of dinitrogen tetroxide to produce three molecules of nitrogen gas, four molecules of water vapor, and two molecules of carbon dioxide. This information is crucial because it sets up the mole ratios needed for stoichiometric calculations.

Practical Tip:

When balancing equations, start by balancing atoms that appear only in one reactant and product first. Save hydrogen and oxygen for last, as they are often found in multiple compounds within the reaction.

The molar mass is the weight of one mole of a substance, typically expressed in grams per mole (g/mol). To calculate molar mass, one must sum the atomic masses of all atoms present in the molecular formula. The atomic mass of each element can be found on the periodic table.

In our exercise, the molar mass calculation for dimethylhydrazine is essential to convert the given mass to moles, which then allows for further stoichiometric computations. By multiplying the respective atomic masses by the number of atoms of each element in the molecular formula and adding them together, we obtain the molar mass.

Example:

For dimethylhydrazine \((CH_3)_2N_2H_2\), the calculation is as follows: the molar mass of carbon (C) is 12.01 g/mol, hydrogen (H) is 1.01 g/mol, and nitrogen (N) is 14.01 g/mol. Therefore, the molar mass calculation would be: \[M_{(CH_3)_2N_2H_2} = 2 \times (12.01) + 6 \times (1.01) + 2 \times (14.01) = 46.08 \, g/mol\]This calculated molar mass helps to establish a bridge between the mass of a substance and the number of moles, facilitating stoichiometric calculations.

The Ideal Gas Law is a critical equation in chemistry that relates the four variables of a gas: pressure (P), volume (V), number of moles (n), and temperature (T). It is represented by the formula: \[PV = nRT\] where R is the universal gas constant, \(0.0821 \, L\, atm / K\, mol\). This law allows us to calculate the amount, pressure, volume, and temperature of a gas under ideal conditions, where the gas particles are assumed to have no volume and do not interact with each other.

In the exercise, we use the Ideal Gas Law to determine the partial pressure of each gas produced in the chemical reaction. To do so, we need the number of moles of the gas, the volume of the container, and the temperature in Kelvin. For instance, the partial pressure of nitrogen can be found using the number of moles calculated from the balanced equation and the ideal gas constant.

Understanding Partial Pressures:

Each gas in a mixture exerts its pressure independently, known as its partial pressure. The total pressure of a gas mixture is the sum of the partial pressures of each component gas. In the given problem, once the partial pressures of nitrogen, water vapor, and carbon dioxide are calculated using their individual moles, the total pressure in the tank is found by adding these partial pressures together.