Question

Suppose you decide to define your own temperature scale with units of \(\mathrm{O}\), using the freezing point \(\left(13^{\circ} \mathrm{C}\right)\) and boiling point \(\left(360^{\circ} \mathrm{C}\right)\) of oleic acid, the main component of olive oil. If you set the freezing point of oleic acid as \(0^{\circ} \mathrm{O}\) and the boiling point as \(100^{\circ} \mathrm{O},\) what is the freezing point of water on this new scale?

Short answer

The freezing point of water in the new temperature scale O can be found using the conversion formula \(O = m \cdot x_C + b\) based on the given freezing and boiling points of oleic acid in Celsius and the new scale. After calculating the slope (m) and y-intercept (b), we can substitute the freezing point of water in Celsius, \(0^{\circ} \mathrm{C}\), as \(x_C\) in the formula to find its equivalent value in the O scale.

Step-by-step solution

Identify the known conversion points

We are given the information that: Freezing point of oleic acid: \(13^{\circ} \mathrm{C} = 0^{\circ} \mathrm{O}\) Boiling point of oleic acid: \(360^{\circ} \mathrm{C} = 100^{\circ} \mathrm{O}\)

Find the linear relationship between Celsius and the new scale

We have two points on the coordinate plane that correspond to the known equivalences between the Celsius and O scales. Let's denote the celsius values as \(x_C\) and the O values as \(O\). Using the slope-intercept equation \(O = m \cdot x_C + b\), we have: Point 1: \( (x_{C1}, O_1) = (13, 0) \); Point 2: \( (x_{C2}, O_2) = (360, 100) \). Now, we'll find the slope and y-intercept for this linear relationship. Slope \(m = \frac{O_2 - O_1}{x_{C2} - x_{C1}} = \frac{100-0}{360-13}\) Use Point 1 to find the y-intercept (b): \( O_1 = m \cdot x_{C1} + b\), solve for \(b\).

Establish the conversion formula

Once we have found the values of \(m\) and \(b\), the linear conversion formula between Celsius and the new temperature scale O will be: \(O = m \cdot x_C + b\)

Find the freezing point of water in the new scale

We know that the freezing point of water in Celsius is \(0^{\circ} \mathrm{C}\). We will use this value as \(x_C\) in our conversion formula to find the equivalent value, \(O\), in the new temperature scale.

Key concepts

Oleic Acid

Oleic Acid is a fascinating organic compound that plays a pivotal role in many household and industrial applications. It is a monounsaturated fatty acid, commonly found in olive oil, which lends it the reputation of being heart-healthy. But beyond its nutritional benefits, oleic acid is an important scientific component. Its structure consists of a long carbon chain with a single double bond, which influences its physical properties.

When observing oleic acid in the context of temperature, its behaviors at specific temperatures, such as freezing and boiling points, are key. These benchmarks are used in scientific studies to understand the stability and usability of the acid in various environments. It's fascinating how oleic acid can be a benchmark to create new temperature scales, highlighting its consistency and reliability.

Freezing Point

The concept of a freezing point is crucial in understanding how substances change state from liquid to solid. For oleic acid, this occurs at a relatively warm temperature of 13°C. The freezing point is the temperature at which a substance loses its thermal energy enough for its molecules to arrange into a fixed structure, leaving the fluid state behind.

In everyday life, freezing points help us decide how to store food and other perishables. In science, understanding a substance's freezing point can lead to innovations in how we process and utilize materials under extreme temperatures. In the case of oleic acid, using its freezing point to create a new temperature scale offers a method to reinterpret our understanding of cold based on specific applications.

Boiling Point

Boiling points mark another critical temperature threshold where a liquid turns into vapor. For oleic acid, this takes place at 360°C. At this high temperature, energy increases so that the liquid molecules break free into the gaseous state. This transition involves surmounting intermolecular forces that are significantly stronger at this elevated temperature compared to the liquid state.

Boiling points are essential not only for cooking and everyday thermochemistry but also play vital roles in industrial processes where precise temperature regulation is necessary. They can signify the purity and behavior of substances. By setting oleic acid's boiling point as part of a new temperature scale, we create a controlled reference zone to gauge other thermal interactions, especially at higher temperatures.