Question

Both dimethylhydrazine, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NNH}_{2}\), and methylhydrazine, \(\mathrm{CH}_{3} \mathrm{NH} \mathrm{NH}_{2}\), 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 \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) is formed.)

Short answer

Comparing the volume per gram ratio for both dimethylhydrazine and methylhydrazine, we determine that the fuel which produces a greater thrust per gram total mass of oxidizer plus fuel depends on the volume ratio between these two. If the ratio is greater than 1, dimethylhydrazine is more efficient; if the ratio is less than 1, methylhydrazine is more efficient; and if equal, both fuels produce an equal thrust per gram.

Step-by-step solution

Write balanced reaction equations

First, we need to write balanced chemical equations for the reactions of both fuels with dinitrogen tetroxide: Reaction with dimethylhydrazine: \[\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NNH}_{2} + 4\mathrm{N}_{2} \mathrm{O}_{4} \longrightarrow 2\mathrm{CO}_{2} + 3\mathrm{N}_{2} + 4\mathrm{H}_{2}\mathrm{O}\] Reaction with methylhydrazine: \[\mathrm{CH}_{3} \mathrm{NH} \mathrm{NH}_{2} + 3\mathrm{N}_{2}\mathrm{O}_{4} \longrightarrow \mathrm{CO}_{2} + 4\mathrm{N}_{2} + 2\mathrm{H}_{2}\mathrm{O}\]

Calculate the molar volume of products in both reactions

Now we will calculate the molar volumes of products for each reaction: For dimethylhydrazine: In this reaction, we have 2 moles of CO2, 3 moles of N2, and 4 moles of H2O, for a total of 2+3+4=9 moles of product. For methylhydrazine: In this reaction, we have 1 mole of CO2, 4 moles of N2, and 2 moles of H2O, for a total of 1+4+2=7 moles of product.

Determine the amount of reactants in 1 g of total mass

Let's calculate the total molar mass of each reaction: Total molar mass of dimethylhydrazine reaction: \(M_{1} = M_{(CH_3)_2NNH_2} + 4M_{N_2O_4}\) Total molar mass of methylhydrazine reaction: \(M_{2} = M_{CH_3NHNH_2} + 3M_{N_2O_4}\) Now, calculate the amount (in moles) of reactants in 1 g of the total mass for each reaction: n[dimethylhydrazine] = 1 g / M_1 n[methylhydrazine] = 1 g / M_2

Calculate the volume of products per gram total mass

Now, we will calculate the volume of products per gram total mass of oxidizer plus fuel for both reactions using the ideal gas law: PV=nRT. Both rocket fuels generate the same temperature so we can assume that the temperature in both reactions is the same. Thus, the pressure and temperature can cancel out in our comparison: Volume per gram[dimethylhydrazine] = 9 * n[dimethylhydrazine] Volume per gram[methylhydrazine] = 7 * n[methylhydrazine]

Compare the thrust generated by each fuel

Now we will compare the volume per gram ratio for both fuels: If the volume ratio (Volume per gram[dimethylhydrazine] / Volume per gram[methylhydrazine]) is greater than 1, dimethylhydrazine produces a greater thrust per gram. If the volume ratio is less than 1, methylhydrazine produces a greater thrust per gram. If the volume ratio is equal to 1, both fuels produce an equal thrust per gram.

Key concepts

Chemical Reaction Stoichiometry

Understanding chemical reaction stoichiometry is crucial for solving problems related to chemical reactions, including those involving rocket fuels. Stoichiometry allows us to calculate the quantities of reactants and products in a chemical reaction. It involves balancing chemical equations to ensure that the law of conservation of mass is obeyed.

In the context of rocket fuels, calculating stoichiometry involves determining the amount of fuel and oxidizer required to produce the desired amount of product gases that generate thrust. When balanced, the stoichiometric coefficients in the chemical equation tell us the precise moles of reactants that react and the moles of products formed. For instance, the balanced equations for our rocket fuels show different ratios of reactants to products, which directly affect thrust.

Students often have difficulty with stoichiometry because it requires a systematic approach. Here's some advice to improve the learning experience:

• Begin with understanding the basic concepts of reactants, products, and the law of conservation of mass.
• Practice balancing simple chemical equations before attempting complex reactions.
• Use dimensional analysis to convert between units and ensure calculations are consistent.
• Relate stoichiometry to practical examples, like rocket propulsion, to visualize the applications.

It is by mastering stoichiometry that one can calculate which fuel provides more thrust in the rocket example.

Application of the Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry and physics that relates the pressure (P), volume (V), temperature (T), and amount (n) of moles of a gas. Mathematically, it is represented as PV = nRT, where R is the universal gas constant. This equation is especially useful for calculating the volume of gases at a certain temperature and pressure, which is pertinent to problems involving gaseous products, such as those generated by the combustion of rocket fuels.

In the rocket fuel problem, the ideal gas law allows us to predict the volume of gas produced from a certain mass of fuel and thereby estimate the thrust that can be generated. We assume conditions of constant temperature and pressure, which simplifies our calculations as these variables cancel out, focusing solely on the volume and the number of moles of gas. This plays into assessing rocket performance, as the thrust is directly related to the volume of gas expelled.

To improve grasp for students, explain that the ideal gas law is based on the behavior of ideal gases, which are hypothetical gases that perfectly follow the law. Real gases often behave similarly enough to ideal gases under many conditions, making this law a valuable tool for estimations. As students apply the law, remind them:

• Check if conditions approximate those of an ideal gas — high temperature and low pressure.
• Use the same units for temperature and pressure throughout.
• Always use the correct value for the gas constant R depending on the units in use.

By understanding the ideal gas law, students can better analyze problems involving gases, such as the production of thrust in rockets.

Molar Volume of Gases

The molar volume of a gas is the volume occupied by one mole of that gas at a given temperature and pressure. For gases under standard temperature and pressure (0°C and 1 atm), the molar volume is approximately 22.4 liters per mole for an ideal gas. However, real gases might deviate slightly from this value due to molecular interactions and the space occupied by gas particles.

In our rocket fuels scenario, the molar volume helps determine the thrust because it relates to the volume of gas products produced during the combustion of fuel. The total volume of gases can be inferred from the balanced stoichiometric equations once one knows the moles of each gas produced. More volume translates to potentially more thrust, assuming equal ejection speeds.

For educational purposes, illustrating this concept using graphical representations and animations of gas molecules in a container can help students visualize how volume, temperature, and pressure are interrelated. Also, real-life examples such as inflating a balloon in different temperatures can make the concept more tangible. Student comprehension can be aided by emphasizing:

• Standard molar volume is a useful benchmark but should be adjusted for non-standard conditions.
• The inverse relationship between pressure and volume, and direct relationship between temperature and volume.
• Practical applications of molar volume in calculating gas quantities in various chemical processes.

Molar volume is a foundational concept for understanding how gases interact and react in real-world applications, such as in the engineering of combustion engines and rockets.