Problem 22 - Chemistry and Chemical Reactivity

Question

The enthalpy of combustion of isooctane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\) one of the many hydrocarbons in gasoline, is \(5.45 \times 10^{3} \mathrm{kJ} / \mathrm{mol} .\) Calculate the enthalpy change per gram of isooctane and per liter of isooctane \((d=0.688 \mathrm{g} / \mathrm{mL})\)

Step-by-Step Solution

Answer

\(47.72 \, \text{kJ/g}\) and \(32817.36 \, \text{kJ/L}\)

Step 1: Find the Molar Mass of Isooctane

Calculate the molar mass of isooctane \(\text{C}_8\text{H}_{18}\). Carbon (\(\text{C}\)) has a molar mass of \(12.01 \, \text{g/mol}\) and hydrogen (\(\text{H}\)) has a molar mass of \(1.01 \, \text{g/mol}\). Thus, the molar mass of isooctane is:\[\text{Molar Mass} = (8 \times 12.01) + (18 \times 1.01) = 114.22 \, \text{g/mol}\]

Step 2: Calculate Enthalpy Change per Gram

Use the enthalpy of combustion provided and the molar mass to calculate the enthalpy change per gram of isooctane:\[\text{Enthalpy change per gram} = \frac{5.45 \times 10^3 \, \text{kJ/mol}}{114.22 \, \text{g/mol}} = 47.72 \, \text{kJ/g}\]

Step 3: Convert Density to Volume

Convert the density from \(\text{g/mL}\) to \(\text{g/L}\) to work with liters. Since \(\text{d} = 0.688 \, \text{g/mL}\), we have:\[\text{Density in g/L} = 0.688 \, \text{g/mL} \times 1000 \, \text{mL/L} = 688 \, \text{g/L}\]

Step 4: Calculate Enthalpy Change per Liter

Use the density of isooctane to find the enthalpy change per liter:\[\text{Enthalpy change per liter} = 47.72 \, \text{kJ/g} \times 688 \, \text{g/L} = 32817.36 \, \text{kJ/L}\]

Key Concepts

Molar Mass Calculation

To understand the energy changes when burning a substance like isooctane, it's essential to begin with calculating its molar mass. This value helps in connecting the amount of substance with its energy-related properties. For isooctane, which is a hydrocarbon with the molecular formula \( ext{C}_8 ext{H}_{18}\), we calculate its molar mass as follows:

• Carbon (\( ext{C}\)) contributes with a molar mass of \(12.01 \, \text{g/mol}\) and there are 8 carbon atoms in isooctane, so \(8 \times 12.01\).
• Hydrogen (\( ext{H}\)) has a molar mass of \(1.01 \, \text{g/mol}\), and there are 18 hydrogen atoms, so \(18 \times 1.01\).

Adding these amounts gives the molar mass of isooctane:\[\text{Molar Mass} = (8 \times 12.01) + (18 \times 1.01) = 114.22 \, \text{g/mol}\]This molar mass is used in further calculations to determine energy changes per gram or per liter of the substance.

Enthalpy Change Per Gram

Once the molar mass is known, it's easier to calculate how much energy is released when burning each gram of the substance. This value, known as the enthalpy change per gram, is derived using the provided enthalpy of combustion and the molar mass. The formula involves dividing the enthalpy of combustion by the molar mass:\[\text{Enthalpy change per gram} = \frac{5.45 \times 10^3 \, \text{kJ/mol}}{114.22 \, \text{g/mol}} = 47.72 \, \text{kJ/g}\]This calculation tells us that every gram of isooctane releases approximately 47.72 kJ of energy when combusted. Understanding this value is crucial in practical applications, such as calculating energy outputs for engines and similar systems.

Density Conversion

Density conversions are indispensable when moving from calculations per gram to those per liter. In this problem, we begin with the density of isooctane given in \(\text{g/mL}\) and need to convert it to \(\text{g/L}\). This conversion is straightforward:

• Since there are 1000 milliliters in a liter, you multiply the given density by 1000.

For isooctane, given the density (\(d\)) as 0.688 \(\text{g/mL}\), the conversion is:\[\text{Density in g/L} = 0.688 \, \text{g/mL} \times 1000 \, \text{mL/L} = 688 \, \text{g/L}\]Now, this density value can be used to determine how much energy is associated with one liter of isooctane.

Enthalpy Change Per Liter

With the density known in \(\text{g/L}\), calculating the enthalpy change per liter gives a more practical measure for bulk quantities. We use the enthalpy change per gram calculated earlier and multiply by the density:\[\text{Enthalpy change per liter} = 47.72 \, \text{kJ/g} \times 688 \, \text{g/L} = 32817.36 \, \text{kJ/L}\]This calculation shows that when burned, each liter of isooctane releases approximately 32,817.36 kJ of energy. This value is particularly useful in contexts where fuel efficiency and total energy output are of interest, such as in the gasoline industry. Understanding these kinds of conversions also supports understanding environmental impacts and energy budgeting for various applications.