Question

The complete combustion of octane, \(\mathrm{C}_{8} \mathrm{H}_{18}\), produces \(5470 \mathrm{~kJ}\) of heat. Calculate how many grams of octane is required to produce \(20,000 \mathrm{~kJ}\) of heat.

Short answer

Approximately 3.66 grams of octane is required to produce 20,000 kJ of heat.

Step-by-step solution

Write down the known information

We are given the following pieces of information: 1. Heat produced by complete combustion of octane: 5470 kJ 2. We need to calculate the grams of octane required to produce 20,000 kJ of heat.

Setting up the proportion

Since we know the heat produced by the complete combustion of octane, we can set up a proportion that relates the heat produced (in kJ) to the grams of octane required. Let x be the grams of octane required to produce 20,000 kJ of heat. Then, we can set up the following proportion: \(\frac{Heat \, produced \, by \, octane}{Grams \, of \, octane} = \frac{Heat \, needed}{Grams \, of \, octane \, needed}\)

Plugging in the known values

Now we will plug in the known values into the proportion: \(\frac{5470 \, kJ}{1 \, g \, of \, C_{8}H_{18}} = \frac{20,000 \, kJ}{x \, g \, of \, C_{8}H_{18}}\)

Solve for x

We can now solve for x, which represents the grams of octane required to produce 20,000 kJ of heat. 1. Cross-multiply: \(5470 \, kJ \times x = 20,000 \, kJ \times 1 \, g\) 2. Simplify: \(5470x = 20,000\) 3. Divide by 5470: \(x = \frac{20,000}{5470}\) 4. Calculate the result: \(x ≈ 3.66\)

Conclusion

The grams of octane required to produce 20,000 kJ of heat is approximately 3.66 grams.

Key concepts

Heat Production

When octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) undergoes complete combustion, it reacts with oxygen to produce carbon dioxide and water. This process releases energy, often in the form of heat. The amount of heat provided by such a reaction is specific and is an important characteristic of the substance. For octane, complete combustion produces 5470 kJ of heat per gram. This predictable energy release makes octane a common choice as a fuel in engines.
Understanding how much energy a substance releases during combustion is key to using it effectively as a fuel. By knowing the energy content of a compound, you can match fuel requirements to energy needs for various applications such as heating or powering engines. Knowledge of heat production allows us to harness this energy efficiently for practical use.

Energy Calculation

Energy calculations in chemistry often involve converting known quantities of a substance into energy produced or required. To do this, we use the property we call 'enthalpy,' which is essentially the heat measured at constant pressure. For combustion problems, it's vital to know the heat produced per unit of substance. To find out how many grams of octane are needed to produce 20,000 kJ of heat, knowing that every gram of octane provides 5470 kJ upon combustion, we set up an equation based on this information. Calculations like these are straightforward once you understand the relationship between the mass of the substance and the energy released—and this is done through proportions. This kind of energy calculation is critical for industries that require precise energy outputs.

Proportional Relationships

In chemistry, proportional relationships help us determine unknown quantities based on known comparable information. When it comes to our octane combustion problem, we use a proportion to find out how many grams of octane are needed to meet a specific energy demand.The general form of a proportion is \(\frac{A}{B} = \frac{C}{D}\), where these terms represent two related pairs of quantities. In our context:

• \(A = 5470\) (kJ of heat produced per gram of octane)
• \(B = 1\) (gram of octane)
• \(C = 20,000\) (target heat in kJ)
• \(D = x\) (grams of octane needed)

By solving this proportion, we find the value of \(x\), which will tell us how many grams of octane are required. Using proportional relationships allows for a clear method to translate given data into required quantities, making them a valuable tool in problem-solving.