Problem 102 - Chemistry The Science in Context

Question

The heavier hydrocarbons in white gas are hexanes \(\left(\mathrm{C}_{6} \mathrm{H}_{14}\right)\) a. Calculate the fuel value of \(C_{6} H_{14},\) given that \(\Delta H_{\mathrm{comb}}^{\circ}=-4163 \mathrm{kJ} / \mathrm{mol}\) b. How much energy is released during the combustion of \(1.00 \mathrm{kg}\) of \(\mathrm{C}_{6} \mathrm{H}_{14} ?\) c. How many grams of \(\mathrm{C}_{6} \mathrm{H}_{14}\) are needed to heat \(1.00 \mathrm{kg}\) of water from \(25.0^{\circ} \mathrm{C}\) to \(85.0^{\circ} \mathrm{C} ?\) Assume that all the energy released during combustion is used to heat the water. d. Assume white gas is \(25 \%\) C \(_{5}\) hydrocarbons and \(75 \% \mathrm{C}_{6}\) hydrocarbons; how many grams of white gas are needed to heat \(1.00 \mathrm{kg}\) of water from \(25.0^{\circ} \mathrm{C}\) to \(85.0^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

Answer: The fuel value of hexane is 4163 kJ/mol. To heat 1.00 kg of water from 25.0°C to 85.0°C, 6.94 g of white gas is needed.

Step 1: Identify the given value, the combustion enthalpy

In this part of the exercise, we are given the combustion enthalpy of hexane as \(\Delta H_{\mathrm{comb}}^{\circ}=-4163 \mathrm{kJ/mol}\).

Step 2: Calculate the fuel value

The fuel value of a substance is the amount of energy that can be obtained from combusting a certain amount of that substance. In this case, we can express the fuel value of hexane as energy per mole. Since the combustion enthalpy is given in kJ/mol, the fuel value is the same as the absolute value of combustion enthalpy: Fuel value of hexane = \(\Delta H_{\mathrm{comb}}^{\circ}\) = |-4163 kJ/mol| #a. Conclusion#

Step 3: Report the fuel value

The fuel value of hexane is 4163 kJ/mol. #b. Energy released during the combustion of 1 kg of hexane#

Step 1: Convert mass of hexane to moles

Given the molecular formula of hexane, we can find its molar mass as: Molar mass of \(\mathrm{C}_{6}\mathrm{H}_{14}\) = \(6(12.01 \mathrm{g/mol}) + 14(1.01 \mathrm{g/mol})\) = \(72.06 + 14.14\) = \(86.20 \mathrm{g/mol}\) Now, convert 1.00 kg of hexane (1000 g) to moles using the molar mass: moles of hexane = \(\frac{1000 \mathrm{g}}{86.20 \mathrm{g/mol}} = 11.60 \mathrm{moles}\)

Step 2: Calculate the energy released

To find the energy released during the combustion of 1.00 kg of hexane, multiply the moles of hexane by the fuel value: Energy released = moles of hexane × fuel value = \(11.60 \, \text{moles} \times 4163 \, \text{kJ/mol} = 48,291.8 \, \text{kJ}\) #b. Conclusion#

Step 3: Report the energy released

During the combustion of 1.00 kg of hexane, 48,291.8 kJ of energy is released. #c. Grams of hexane needed to heat 1.00 kg of water from 25.0°C to 85.0°C#

Step 1: Calculate the energy required to heat the water

To calculate the energy required to heat 1.00 kg of water from 25.0°C to 85.0°C, we use the formula: Energy required = mass × specific heat capacity × temperature change Energy required = \(1.00\, \text{kg} \times 4.186 \, \text{kJ/(kg °C)} \times (85.0 - 25.0) °\mathrm{C} = 251.6 \, \mathrm{kJ}\)

Step 2: Calculate the moles of hexane needed

Since we assume that all energy released during combustion is used to heat the water, the number of moles of hexane needed is found by dividing the energy required by the fuel value of hexane: moles of hexane = \(\frac{251.6 \, \mathrm{kJ}}{4163 \, \mathrm{kJ/mol}} = 0.0604 \, \mathrm{moles}\)

Step 3: Convert moles of hexane to grams

Now, we will convert the moles of hexane to grams using the molar mass: grams of hexane = \(0.0604 \, \mathrm{moles} \times 86.20 \, \mathrm{g/mol} = 5.204 \, \mathrm{g}\) #c. Conclusion#

Step 4: Report the grams of hexane needed

To heat 1.00 kg of water from 25.0°C to 85.0°C, 5.204 g of hexane is needed, assuming all the energy released during combustion is used to heat the water. #d. Grams of white gas needed to heat 1.00 kg of water from 25.0°C to 85.0°C#

Step 1: Calculate the grams of hexane in the white gas

Given that white gas is 75% hexane, the grams of hexane in the white gas is: grams of hexane in white gas = \(\frac{5.204 \, \mathrm{g}}{0.75} = 6.9387 \, \mathrm{g}\) #d. Conclusion#

Step 2: Report the grams of white gas needed

To heat 1.00 kg of water from 25.0°C to 85.0°C, 6.94 g of white gas is needed, assuming white gas is 25% \(\mathrm{C}_{5}\) hydrocarbons and 75% \(\mathrm{C}_{6}\) hydrocarbons, and all the energy released during combustion is used to heat the water.

Key Concepts

Combustion Enthalpy

Combustion enthalpy, often denoted as \(\Delta H_{\mathrm{comb}}^{\circ}\), is the energy change that occurs when one mole of a substance is burned in oxygen under standard conditions. In simpler terms, it tells us how much energy is either absorbed or released during the combustion process. This value is crucial for understanding how efficient a fuel is.
In the context of hexane, the combustion enthalpy is known to be \(-4163 \, \text{kJ/mol}\). This means that when one mole of hexane completes a combustion reaction, this exact amount of energy is released, making hexane a significant energy source. For students, understanding this concept helps in predicting how much energy a fuel can provide.

Fuel Value

Fuel value is a measure of energy output per mole of a specific fuel when it is burned. It is a practical projection of the combustion enthalpy and provides an idea of how much energy a particular fuel.
For hexane, the fuel value is the absolute magnitude of its combustion enthalpy, which is \(4163 \, \text{kJ/mol}\). This information is particularly useful for comparing different fuels and determining which is more energy-efficient theoretically. For those new to these concepts, consider it like comparing the miles-per-gallon of different cars; the higher the fuel value, the more energy it can provide.

Hexane Combustion

Burning hexane releases significant energy. Understanding hexane combustion involves knowing how to convert mass to moles, and use moles to calculate the total energy.
When you have \(1 \, \text{kg}\) of hexane, the first step is converting to moles using its molar mass \(86.20 \, \text{g/mol}\). This results in approximately \(11.60\) moles. Multiplying these moles by the fuel value gives the total energy output: \(48,291.8 \, \text{kJ}\).
This kind of calculation shows just how potent hexane is as a fuel source. Whether you're heating a house or running a generator, hexane's combustion potential is quite powerful.

Water Heating Calculation

To heat water using the energy from hexane's combustion, you need to understand the energy transfer process. The basic requirement to heat water is calculated through the formula: \text{Energy required} = ext{mass} \times ext{specific heat} \times ext{temperature change}\.
In the given scenario, heating \(1.00 \, \text{kg}\) of water from \(25.0^{\circ} \text{C}\) to \(85.0^{\circ} \text{C}\) requires \(251.6 \, \text{kJ}\). Knowing how much energy hexane releases, you can calculate the specific grams needed, here \(5.204 \, \text{g}\). This straightforward approach ensures that every joule of energy is accounted for when heating water or any other substance.