Problem 103 - Chemistry: The Central Science in SI Units, Global Edition

Question

Both dimethylhydrazine, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NNH}_{2}\), and methylhydrazine, \(\mathrm{CH}_{3} \mathrm{NHNH}_{2}\), 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.)

Step-by-Step Solution

Answer

Methylhydrazine produces a greater thrust per gram total mass of oxidizer plus fuel, with a thrust per gram value of \(0.0446 \frac{moles}{g}\), compared to dimethylhydrazine, which has a thrust per gram value of \(0.0328 \frac{moles}{g}\).

Step 1: Write and balance the chemical equations for the two hydrazines

We need to have a balanced chemical equation for both the dimethylhydrazine and methylhydrazine reactions with dinitrogen tetroxide. For Dimethylhydrazine: \((CH_{3})_{2}NNH_{2} + N_{2}O_{4} \rightarrow H_{2}O + CO_{2} + N_{2}\) Balanced equation: \[2(CH_{3})_{2}NNH_{2} + 3N_{2}O_{4} \rightarrow 6H_{2}O + 2CO_{2} + 5N_{2}\] For Methylhydrazine: \(CH_{3}NHNH_{2} + N_{2}O_{4} \rightarrow H_{2}O + CO_{2} + N_{2}\) Balanced equation: \[4CH_{3}NHNH_{2} + 5N_{2}O_{4} \rightarrow 12H_{2}O + 4CO_{2} + 9N_{2}\]

Step 2: Calculate the molar mass of each reactant and product

Now, we will determine the molar mass of each reactant and product in the chemical equations: Dimethylhydrazine: \(2\times12.01 + 2\times1.01 + 14.01 + 2\times1.01 = 60.14 \, g/mol\) Methylhydrazine: \(12.01 + 3\times1.01 + 14.01 + 2\times1.01 = 46.08 \, g/mol\) Dinitrogen tetroxide: \(2\times14.01 + 4\times16.00 = 92.02 \, g/mol\) Water: \(2\times1.01 + 16.00 = 18.02 \, g/mol\) Carbon dioxide: \(12.01 + 2\times16.00 = 44.01 \, g/mol\) Dinitrogen: \(2\times14.01 = 28.02 \, g/mol\)

Step 3: Calculate the thrust per gram for each substituted hydrazine

According to the balanced equations, we have the following relationships: Per mole of dimethylhydrazine: - Fuel: \(2(CH_{3})_{2}NNH_{2}\) - Oxidizer: \(3N_{2}O_{4}\) - Products: \(6H_{2}O + 2CO_{2} + 5N_{2}\) Total mass of reactants for dimethylhydrazine (fuel + oxidizer): \(2 \times 60.14 + 3 \times 92.02 = 396.34 \, g\) Total moles of products produced: \(6 + 2 + 5 = 13\, moles\) Thrust per gram for dimethylhydrazine: \(\frac{13 \, moles}{396.34 \, g} = 0.0328 \frac{moles}{g}\) Per mole of methylhydrazine: - Fuel: \(4CH_{3}NHNH_{2}\) - Oxidizer: \(5N_{2}O_{4}\) - Products: \(12H_{2}O + 4CO_{2} + 9N_{2}\) Total mass of reactants for methylhydrazine (fuel + oxidizer): \(4 \times 46.08 + 5 \times 92.02 = 560.34 \, g\) Total moles of products produced: \(12 + 4 + 9 = 25\, moles\) Thrust per gram for methylhydrazine: \(\frac{25 \, moles}{560.34 \, g} = 0.0446 \frac{moles}{g}\)

Step 4: Compare the thrust per gram for each substituted hydrazine

Now that we have the thrust per gram for both dimethylhydrazine (0.0328 moles/g) and methylhydrazine (0.0446 moles/g), we can see that methylhydrazine produces a greater thrust per gram total mass of oxidizer plus fuel.

Key Concepts

Chemical Equations

In rocket propulsion, chemical reactions between a fuel and an oxidizer generate thrust. Understanding and using balanced chemical equations is essential to predict the amount of products that will form. When both dimethylhydrazine and methylhydrazine act as fuels with dinitrogen tetroxide being the oxidizer, they produce water (\(\text{H}_2\text{O}\)), carbon dioxide (\(\text{CO}_2\)), and nitrogen (\(\text{N}_2\)) as combustion products.

The balanced chemical equation for dimethylhydrazine is:

• \(2(CH_3)_2NNH_2 + 3N_2O_4 \rightarrow 6H_2O + 2CO_2 + 5N_2\)

For methylhydrazine, it is:

• \(4CH_3NHNH_2 + 5N_2O_4 \rightarrow 12H_2O + 4CO_2 + 9N_2\)

Balancing these equations ensures the conservation of mass and allows us to account for the number of moles of each component involved, providing a necessary step before calculating thrust.

Molar Mass Calculations

Calculating molar masses of reactants and products helps in assessing the efficiency of the chemical reaction.

For dimethylhydrazine \([\text{C}_2\text{H}_8\text{N}_2]\), the molar mass is calculated as follows:

• Carbon: \(2 \times 12.01 = 24.02\)
• Hydrogen: \(8 \times 1.01 = 8.08\)
• Nitrogen: \(2 \times 14.01 = 28.02\)
• Total: 60.14 g/mol

For methylhydrazine \([\text{CH}_6\text{N}_2]\), the molar mass is:

• Carbon: \(12.01\)
• Hydrogen: \(6 \times 1.01 = 6.06\)
• Nitrogen: \(2 \times 14.01 = 28.02\)
• Total: 46.08 g/mol

The oxidizer dinitrogen tetroxide \([\text{N}_2\text{O}_4]\) has a molar mass of \(92.02\) g/mol. Understanding these masses aids in determining the mass of reactants needed and evaluating their contribution to the rocket's thrust.

Thrust Calculation

Thrust is the force that propels a rocket and is influenced by the mass and velocity of the combustion gases. The more extensive the volume of gases, the greater the thrust.This exercise examines which hydrazine fuel produces more thrust per gram of reactants used.

For dimethylhydrazine, the equation and reactants give:

• Total reactant mass: \((2 \times 60.14) + (3 \times 92.02) = 396.34\) g
• Products formed: \(13\) moles
• Thrust efficiency: 0.0328 \(\frac{moles}{g}\)

For methylhydrazine, the values are:

• Total reactant mass: \((4 \times 46.08) + (5 \times 92.02) = 560.34\) g
• Products formed: \(25\) moles
• Thrust efficiency: 0.0446 \(\frac{moles}{g}\)

Thus, methylhydrazine is more efficient, generating a higher thrust per gram of fuel and oxidizer, making it preferable for maximizing rocket propulsion.