Question

What is the octane number of a mixture of \(35 \%\) heptane and \(65 \%\) isooctane?

Short answer

The octane number of a mixture of 35% heptane and 65% isooctane is 65, calculated using the weighted average equation.

Step-by-step solution

Set up the weighted average equation

We have a mixture of 35% heptane and 65% isooctane. The octane number of heptane is 0, while that of isooctane is 100. We'll use a weighted average equation to find the octane number of the mixture: Octane number of mixture = (percentage of heptane * octane number of heptane) + (percentage of isooctane * octane number of isooctane)

Plug in the given values

Now we plug in the values for the percentages and the octane numbers: Octane number of mixture = (0.35 * 0) + (0.65 * 100)

Calculate the octane number of the mixture

Now we simplify the equation and calculate the octane number of the mixture: Octane number of mixture = (0) + (65) = 65 The octane number of the given mixture of 35% heptane and 65% isooctane is 65.

Key concepts

Heptane and Isooctane Mixture

Understanding how different chemicals mix and the properties of resulting mixtures is at the core of chemical problem solving. In the context of fuels, such as gasoline, the octane rating is a crucial indicator of performance. The octane number measures a fuel's ability to resist engine knocking or pinging during combustion, caused by the air/fuel mixture detonating prematurely in the engine.

In the case of a heptane and isooctane mixture, these two hydrocarbons serve as benchmarks in determining the octane rating. Heptane tends to knock easily and is assigned an octane rating of 0. Conversely, isooctane resists knocking well, giving it a rating of 100. When mixed, the octane number of the resulting fuel can be found by considering the proportion of each component and their respective octane ratings. This is key for engines as it influences their efficiency and lifespan. The higher the octane number, the better the fuel's quality and performance.

Weighted Average Equation

The weighted average equation is a mathematical tool used to calculate the average of a set of numbers, where each number has a different weight or importance. In chemistry, this concept is often applied when mixing substances with different properties, such as in the octane number calculation.

The formula for a weighted average is: \[ Weighted \ Average = \frac{\text{Sum of (Weight × Value)}}{\text{Sum of Weights}} \] In simpler terms, this could be just the sum of the product of each weight with its corresponding value. The use of this equation ensures that the final average accurately reflects the proportion of each component in the mixture. It is not simply an arithmetic mean but accounts for the varying significance of each component's contribution to the overall property of the mixture.

Chemical Problem Solving

Solving chemical problems often involves a series of logical steps, each built upon fundamental principles of chemistry. In our octane number calculation example, the problem-solving process includes understanding the chemical properties (like the octane rating), setting up the correct weighted average equation, and then executing simple arithmetic.

This systematic approach is vital not only for academic exercises but also for real-world applications in industries such as fuel production. By developing the skills to break down complex problems into manageable steps and using appropriate equations, students can apply this methodology to a wide range of scientific and engineering challenges. The process encourages critical thinking and a deeper comprehension of how various components interact within a system, leading to accurate and practical solutions.