Question

From the following data for three prospective fuels, calculate which could provide the most energy per unit volume:$$\begin{array}{lccc}& \begin{array}{c}\text{ Density }\\ \text{ at }{20}^{\circ }\mathrm{C}\\ \left(\mathrm{g}/{\mathrm{cm}}^3\right)\end{array}& \begin{array}{c}\text{ Molar Enthalpy }\\ \text{ of Combustion }\\ \text{ Fuel }\end{array}& \mathrm{kJ}/\mathrm{mol}\\ \text{ Nitroethane, }{\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NO}}_2(l)& 1.052& -1368\\ \text{ Ethanol, }{\mathrm{C}}_2{\mathrm{H}}_5\mathrm{OH}(l)& 0.789& -1367\\ \text{ Methylhydrazine, }{\mathrm{CH}}_6{\mathrm{N}}_2(l)& 0.874& -1305\end{array}$$

Short answer

Based on the calculations, Methylhydrazine provides the most energy per unit volume with an energy content of approximately -24.74 kJ/cm³, followed by Ethanol (-23.62 kJ/cm³) and Nitroethane (-19.32 kJ/cm³).

Step-by-step solution

Calculate the molar mass of each fuel.

To calculate the molar mass of each fuel, we can use the following formula: Molar mass = (number of carbon atoms * 12.01) + (number of hydrogen atoms * 1.008) + (number of nitrogen atoms * 14.01) + (number of oxygen atoms * 16.00) For Nitroethane (C2H5NO2): Molar mass = (2 * 12.01) + (5 * 1.008) + (1 * 14.01) + (2 * 16.00) ≈ 75.07 g/mol For Ethanol (C2H5OH): Molar mass = (2 * 12.01) + (6 * 1.008) + (1 * 16.00) ≈ 46.07 g/mol For Methylhydrazine (CH6N2): Molar mass = (1 * 12.01) + (6 * 1.008) + (2 * 14.01) ≈ 46.09 g/mol So we have the molar masses of the fuels: Nitroethane = 75.07 g/mol Ethanol = 46.07 g/mol Methylhydrazine = 46.09 g/mol

Calculate the energy per unit volume.

Now using the formula: Energy per unit volume = (Molar enthalpy of combustion) / (Molar mass) * (Density) For Nitroethane: Energy per unit volume = (-1368 kJ/mol) / (75.07 g/mol) * (1.052 g/cm³) ≈ -19.32 kJ/cm³ For Ethanol: Energy per unit volume = (-1367 kJ/mol) / (46.07 g/mol) * (0.789 g/cm³) ≈ -23.62 kJ/cm³ For Methylhydrazine: Energy per unit volume = (-1305 kJ/mol) / (46.09 g/mol) * (0.874 g/cm³) ≈ -24.74 kJ/cm³

Compare the energy per unit volume.

Now, we compare the energy per unit volume for the three fuels: Nitroethane = -19.32 kJ/cm³ Ethanol = -23.62 kJ/cm³ Methylhydrazine = -24.74 kJ/cm³ Since Methylhydrazine has the highest energy per unit volume among the three fuels, Methylhydrazine provides the most energy per unit volume.

Key concepts

Molar Mass Calculation

Understanding molar mass is crucial when evaluating different fuels. Molar mass is the mass of one mole of a substance and is calculated by adding the atomic masses of all atoms in a molecule. Here's how it's done:

• Atomic Weights: Use standard atomic weights (e.g., Carbon = 12.01, Hydrogen = 1.008, Nitrogen = 14.01, Oxygen = 16.00).
• Multiply and Add: Multiply the atomic weight by the number of atoms of that element in the molecule. Then, sum these values for all elements in the molecule.

For instance, the molar mass of Nitroethane (C₂H₅NO₂) is calculated as follows:

• Carbon: (2 atoms) × 12.01
• Hydrogen: (5 atoms) × 1.008
• Nitrogen: (1 atom) × 14.01
• Oxygen: (2 atoms) × 16.00

Add these all together to find that Nitroethane's molar mass is approximately 75.07 g/mol. This step is repeated similarly for each fuel using their respective molecular formulas.

Molar Enthalpy of Combustion

Molar enthalpy of combustion refers to the amount of energy released when one mole of a substance is burned completely. It's an exothermic process, meaning energy is released, and is usually expressed in kilojoules per mole (kJ/mol).

In this context, we evaluate how much energy each fuel releases:

• Negative Values: The negative sign indicates the energy is being released. So, in essence, the more negative the value, the more energy is expelled.
• Reference Fuels: For Nitroethane, Ethanol, and Methylhydrazine, their molar enthalpies are approximately -1368, -1367, and -1305 kJ/mol, respectively.

This measurement helps us predict the overall energy output when comparing fuels. It's an essential factor in determining the fuel's efficiency in energy generation.

Energy per Unit Volume Calculation

Calculating the energy per unit volume is key to determining which fuel provides the highest energy efficiency by volume. Here's how the calculation works:

• Formula Used: Energy per unit volume = (Molar enthalpy of combustion) / (Molar mass) × (Density).
• Densities Matter: Energy per volume takes density into account. A fuel with a high density and molar enthalpy will naturally have a higher energy output per volume.
• Division and Multiplication: First, divide the molar enthalpy by the molar mass to get energy per unit mass, and then multiply by the density to convert it to energy per unit volume.

For example, Methylhydrazine with a density of 0.874 g/cm³, molar mass of 46.09 g/mol, and molar enthalpy of -1305 kJ/mol, results in approximately -24.74 kJ/cm³, making it the most energy-efficient by volume among the three fuels. This comparison provides crucial insights for selecting fuels based on volume efficiency.