Question

Both dimethylhydrazine, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NNH}_{2},\) and methylhydrazine, \(\mathrm{CH}_{3} \mathrm{NHN} \mathrm{H}_{2},\) have been used as rocket fuels. When dinitrogen tetroxide \(\left(\mathrm{N}_{2} \mathrm{O}_{4}\right)\) is used as the oxidizer, the products are \(\mathrm{H}_{2} \mathrm{O}, \mathrm{CO}_{2},\) and \(\mathrm{N}_{2} .\) If the thrust of the rocket depends on the volume of the products produced, which of the substituted hydrazines produces a greater thrust per gram total mass of oxidizer plus fuel? (Assume that both fuels generate the same temperature and that \(\mathrm{H}_{2} \mathrm{O}(g)\) is formed.)

Short answer

Dimethylhydrazine generates a greater thrust per gram total mass of oxidizer plus fuel compared to methylhydrazine. This is because it produces a higher number of moles of gas products per gram of fuel and oxidizer, resulting in a greater thrust. Specifically, dimethylhydrazine produces \(0.0576\, \mathrm{moles/(g\, \mathrm{fuel} + g\, \mathrm{oxidizer})}\) while methylhydrazine produces \(0.0456\, \mathrm{moles/(g\, \mathrm{fuel} + g\, \mathrm{oxidizer})}\).

Step-by-step solution

Write the balanced chemical equations for both fuels

First, we will write the balanced chemical equations for the reactions of dimethylhydrazine and methylhydrazine with dinitrogen tetroxide. For dimethylhydrazine: \[\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NNH}_{2} + 3\mathrm{N}_{2}\mathrm{O}_{4} \rightarrow 2\mathrm{CO}_{2} + 4\mathrm{H}_{2}\mathrm{O} + 6\mathrm{N}_{2} \] For methylhydrazine: \[\mathrm{CH}_{3} \mathrm{NHN} \mathrm{H}_{2} + 2\mathrm{N}_{2}\mathrm{O}_{4} \rightarrow \mathrm{CO}_{2} + 3\mathrm{H}_{2}\mathrm{O} + 4\mathrm{N}_{2} \]

Calculate the molar mass of each fuel and the oxidizer

In order to compare the thrust generated by each fuel, we need to find the molar mass of each fuel and the oxidizer. Dimethylhydrazine: \[\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NNH}_{2}\]Molar mass = \(2\times(12.01 + 3\times1.01) + 14.01 + 2\times1.01 = 46.09\, g/mol\) Methylhydrazine: \[\mathrm{CH}_{3} \mathrm{NHN} \mathrm{H}_{2}\]Molar mass = \(12.01 + 3\times1.01 + 2\times(14.01) + 3\times1.01 = 60.10\, g/mol\) Dinitrogen tetroxide: \[\mathrm{N}_{2} \mathrm{O}_{4}\]Molar mass = \(2\times14.01 + 4\times16.00 = 92.02\, g/mol\)

Calculate the grams of oxidizer required for each fuel

To find out the grams of oxidizer required for each fuel, we need to use the balanced chemical equations obtained in step 1. For dimethylhydrazine, 3 moles of \(\mathrm{N}_{2}\mathrm{O}_{4}\) react with 1 mole of fuel: \[3\times92.02\, g\, \mathrm{N}_{2}\mathrm{O}_{4} / 46.09\, g\, \mathrm{fuel}\] For methylhydrazine, 2 moles of \(\mathrm{N}_{2}\mathrm{O}_{4}\) react with 1 mole of fuel: \[2\times92.02\, g\, \mathrm{N}_{2}\mathrm{O}_{4} / 60.10\, g\, \mathrm{fuel}\]

Calculate the moles of gas products produced per gram of fuel and oxidizer

We will use the stoichiometry of the balanced equations to calculate the moles of gas products produced per gram of fuel and oxidizer. For dimethylhydrazine: \[\frac{2\,\mathrm{moles}\, \mathrm{CO}_{2} + 4\,\mathrm{moles}\, \mathrm{H}_{2}\mathrm{O} + 6\, \mathrm{moles}\, \mathrm{N}_{2}}{46.09\, g\, \mathrm{fuel} + 3\times92.02\, g\, \mathrm{N}_{2}\mathrm{O}_{4}}\] For methylhydrazine: \[\frac{1\, \mathrm{moles}\, \mathrm{CO}_{2} + 3\,\mathrm{moles}\, \mathrm{H}_{2}\mathrm{O} + 4\, \mathrm{moles}\, \mathrm{N}_{2}}{60.10\, g\, \mathrm{fuel} + 2\times92.02\, g\, \mathrm{N}_{2}\mathrm{O}_{4}}\]

Determine which fuel generates a greater thrust

Now we will compare the moles of gas products produced per gram of fuel and oxidizer for the two fuels. The fuel with a greater number of moles of gas products per gram will produce a greater thrust. Dimethylhydrazine: \[\frac{2 + 4 + 6}{46.09 + 3\times92.02} = 0.0576\, \mathrm{moles/(g\, \mathrm{fuel} + g\, \mathrm{oxidizer})}\] Methylhydrazine: \[\frac{1 + 3 + 4}{60.10 + 2\times92.02} = 0.0456\, \mathrm{moles/(g\, \mathrm{fuel} + g\, \mathrm{oxidizer})}\] The results show that dimethylhydrazine produces a higher number of moles of gas products per gram of fuel and oxidizer and thus a greater thrust, compared to methylhydrazine.

Key concepts

Dimethylhydrazine

Dimethylhydrazine, \((\mathrm{CH}_{3})_{2} \mathrm{NNH}_{2}\), is a chemical compound often used as a rocket fuel. It is composed of two methyl groups (\(\mathrm{CH}_3\)) attached to a hydrazine molecule (\(\mathrm{NNH}_{2}\)). This combination gives it unique chemical properties that are highly effective for propulsion systems.
One of the reasons dimethylhydrazine is popular in rocket engines is its high energy density. It releases a significant amount of energy upon combustion, which is key to generating thrust. This characteristic makes it an efficient choice as a fuel.
Moreover, in the given exercise, dimethylhydrazine reacts with dinitrogen tetroxide to produce water, carbon dioxide, and nitrogen gas. This reaction releases a substantial amount of gaseous products, which contribute directly to the rocket's thrust.*

Methylhydrazine

Methylhydrazine, represented by the chemical formula \(\mathrm{CH}_{3} \mathrm{NHN} \mathrm{H}_{2}\), is another hydrazine derivative commonly used in rocket propulsion. It includes one methyl group (\(\mathrm{CH}_3\)) and enhances the fuel's volatility and energy content.
The single methyl group in methylhydrazine contributes to its distinct chemical behavior compared to dimethylhydrazine. It is less dense and has a higher molar mass, meaning it requires different considerations in rocket engine designs.
In a typical reaction with dinitrogen tetroxide oxidizer, methylhydrazine efficiently produces combustion products that are vital for thrust generation. Although it generally produces fewer moles of gas per gram compared to dimethylhydrazine, it still serves as a crucial player in various propulsion systems.*

Oxidizer

An oxidizer is a chemical substance that provides oxygen for the fuel to combust. In rocket engines, the role of an oxidizer is crucial for maintaining and controlling the combustion process. Dinitrogen tetroxide (\(\mathrm{N}_{2} \mathrm{O}_{4}\)) is a common oxidizer used with both dimethylhydrazine and methylhydrazine.
The oxidizer reacts with the fuel in a chemical reaction that produces heat and gaseous products necessary for propulsion. Since space does not have sufficient oxygen for combustion, like in the Earth's atmosphere, carrying an oxidizer is essential for rockets to operate.
Dinitrogen tetroxide splits into nitric oxide (\(\mathrm{NO}_2\)), which reacts rapidly with hydrazine fuels, ensuring a controlled and consistent reaction. This assists in generating the maximum possible thrust from the fuel-oxidizer combination.*

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a crucial concept when working with chemical equations, as it helps determine the proportions of reactants and products in chemical reactions.
Calculating the molar mass involves adding up the atomic masses of all atoms present in a molecule. For instance, calculating the molar mass of dimethylhydrazine involves adding the masses of carbon, hydrogen, and nitrogen atoms present in its formula. Likewise, the same process applies to methylhydrazine and the oxidizer, dinitrogen tetroxide.
Understanding molar mass is vital for comparing the efficiency of different fuels and oxidizers since it directly impacts how much of each is needed to produce a specific amount of thrust in a rocket engine.*

Rocket Fuels

Rocket fuels are materials that store energy in chemical form and release it during combustion, creating thrust. They are categorized into two main types: liquid fuels and solid fuels. Dimethylhydrazine and methylhydrazine are examples of liquid fuels, often paired with liquid oxidizers like dinitrogen tetroxide.
The selection of a rocket fuel involves balancing various factors, such as energy content, stability, handling safety, and the compatibility with the chosen oxidizer. Liquid fuels like hydrazines are favored for their high energy density and ease of control during combustion.
Rocket fuels are engineered to maximize thrust while minimizing mass, a critical consideration for any space mission. Combustion reactions involving rocket fuels need to produce large volumes of hot gases that are expelled backward, resulting in the forward movement of the rocket according to Newton's third law of motion.*