Question

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 \(27^{\circ} \mathrm{C}\) 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?

Short answer

The partial pressure of nitrogen gas produced is \(7.52 \: \mathrm{atm}\), and the total pressure in the tank is approximately \(22.57 \: \mathrm{atm}\).

Step-by-step solution

Calculate the moles of dimethylhydrazine

First, we need to convert the given mass of dimethylhydrazine to moles. The molar mass of dimethylhydrazine \((\left(\mathrm{CH}_{3}\right)_{2} \mathrm{~N}_{2} \mathrm{H}_{2})\) is approximately \(2 * 12.01 (C) + 6 * 1.01 (H) + 2 * 14.01 (N) = 60.12 \: \mathrm{g/mol}. \) The moles of dimethylhydrazine can be calculated as follows: Moles of dimethylhydrazine = \(\dfrac{150 \: \mathrm{g}}{60.12 \: \mathrm{g/mol}} = 2.5 \: \mathrm{mol}\)

Determine the moles of nitrogen gas formed

Using the balanced chemical equation, we can determine the stoichiometric ratio between moles of dimethylhydrazine and nitrogen gas: 1 mol dimethylhydrazine: 3 mol nitrogen gas From Step 1, we know that there are 2.5 moles of dimethylhydrazine. Therefore, the moles of nitrogen gas formed are: Moles of nitrogen gas = \(2.5 \: \mathrm{mol\: dimethylhydrazine} * \dfrac{3 \: \mathrm{mol\: nitrogen\: gas}}{1 \: \mathrm{mol\: dimethylhydrazine}} = 7.5 \: \mathrm{mol\: nitrogen\: gas}\)

Calculate the partial pressure of nitrogen gas

Using the ideal gas law (PV=nRT), we can find the partial pressure of nitrogen gas. We are given the volume (V) as 250 L, the number of moles (n) as 7.5 mol, and the temperature (T) as \(27^{\circ}\mathrm{C}\) which needs to be converted to Kelvin (K): T = \(27+273.15 = 300.15 \: \mathrm{K}\) The gas constant (R) is given as 0.0821 L atm/mol K. The ideal gas law then becomes: Partial pressure of nitrogen gas (P) = \(\dfrac{n * R * T}{V} = \dfrac{7.5 \: \mathrm{mol} * 0.0821 \: \mathrm{L\: atm/mol \: K} * 300.15 \: \mathrm{K}}{250 \: \mathrm{L}} = 7.52 \: \mathrm{atm}\)

Determine the moles of other gases produced

Using stoichiometry from the balanced chemical equation, we can determine the moles of water vapor and carbon dioxide produced: 2.5 mol dimethylhydrazine: 4 mol water vapor: 2 mol carbon dioxide Moles of water vapor = \(2.5 \: \mathrm{mol\: dimethylhydrazine} * \dfrac{4 \: \mathrm{mol\: water\: vapor}}{1 \: \mathrm{mol\: dimethylhydrazine}} = 10 \: \mathrm{mol\: water\: vapor}\) Moles of carbon dioxide = \(2.5 \: \mathrm{mol\: dimethylhydrazine} * \dfrac{2 \: \mathrm{mol\: carbon\: dioxide}}{1 \: \mathrm{mol\: dimethylhydrazine}} = 5 \: \mathrm{mol\: carbon\: dioxide}\)

Calculate the total moles of gas formed

The total moles of gas formed is the sum of moles of nitrogen gas, water vapor, and carbon dioxide: Total moles of gas = 7.5 mol nitrogen gas + 10 mol water vapor + 5 mol carbon dioxide = 22.5 mol

Calculate the total pressure in the tank

Using the ideal gas law (PV=nRT), we can find the total pressure in the tank. We have the total moles (n) as 22.5 mol, the volume (V) as 250 L, and the temperature (T) as 300.15 K: Total pressure (P) = \(\dfrac{n * R * T}{V} = \dfrac{22.5 \: \mathrm{mol} * 0.0821 \: \mathrm{L\: atm/mol \: K} * 300.15 \: \mathrm{K}}{250 \: \mathrm{L}} = 22.57 \: \mathrm{atm}\) Thus, the partial pressure of nitrogen gas produced is 7.52 atm, and the total pressure in the tank is approximately 22.57 atm.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry used to relate the pressure, volume, temperature, and number of moles of a gas. It is expressed as \( PV = nRT \). Here:

• \( P \) represents the pressure of the gas in atmospheres (atm).
• \( V \) is the volume in liters (L).
• \( n \) is the number of moles of the gas.
• \( R \) is the ideal gas constant, which is 0.0821 L atm/mol K.
• \( T \) is the temperature in Kelvin (K).

To use the Ideal Gas Law, we first ensure that all values are in the correct units. For example, temperature must be in Kelvin, and if given in Celsius, it can be converted using the formula \( T(K) = T(°C) + 273.15 \).
In a chemical reaction like the one presented, the Ideal Gas Law helps calculate the partial pressures of gases produced by converting known moles and conditions into measurable pressures.

Molar Mass

Molar mass is the mass of one mole of a substance and is usually expressed in grams per mole (g/mol). To calculate the molar mass, add up the atomic masses of all atoms present in a molecule.
For dimethylhydrazine, \((\mathrm{CH}_3)_2 \mathrm{N}_2 \mathrm{H}_2\), its molar mass is calculated as follows:

• Carbon (C): 2 atoms × 12.01 g/mol = 24.02 g/mol
• Hydrogen (H): 6 atoms × 1.01 g/mol = 6.06 g/mol
• Nitrogen (N): 2 atoms × 14.01 g/mol = 28.02 g/mol

The total is 58.10 g/mol.
Understanding molar mass is crucial because it allows chemists to convert between the mass of a substance and the number of moles, an essential step in stoichiometric calculations, and it is vital for precisely measuring and reacting chemicals.

Partial Pressure

Partial pressure refers to the pressure exerted by a single component of a mixture of gases. According to Dalton's Law of Partial Pressures, the total pressure of a mixture is the sum of the partial pressures of the individual gases.
In this exercise, we calculate the partial pressure of nitrogen gas produced using the Ideal Gas Law, accounting only for nitrogen while ignoring other gases that might be present.

• Find the number of moles of the gas of interest, which is nitrogen in this case.
• Use \( n \), \( V \), \( T \), and the constant \( R \) in the Ideal Gas Law equation to find the gas's contribution to the pressure.

Partial pressures are essential for understanding gas behavior in chemical reactions and processes, as they indicate how much of the total pressure each gas contributes.

Rocket Fuels

Rocket fuels are substances that propel rockets by undergoing a chemical reaction that releases a large amount of energy. The qualities of an effective rocket fuel include high energy release and low molar mass, enabling them to produce vigorous thrust.
The combination of dimethylhydrazine and dinitrogen tetroxide is an example of a hypergolic propellant. These fuels ignite spontaneously when they come into contact with each other, making them highly reliable for rapid ignition scenarios.

• Reaction: When the lightweight liquids like dimethylhydrazine react with an oxidizer such as dinitrogen tetroxide, gases like nitrogen, water vapor, and carbon dioxide are produced.
• Benefits: High thrust and low molecular weight improve propulsion efficiency.
• Application: This type of fuel was used effectively in missions like the Lunar Lander, where consistent and controlled thrust was crucial.

Understanding the properties of rocket fuels helps scientists and engineers design more efficient propulsion systems while ensuring safety and reliability.