Question

Suppose you decide to define your own temperature scale 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 on the Oleic scale is approximately \(-3.75^{\circ}\mathrm{O}\).

Step-by-step solution

Define the given information in the problem

We are given the freezing and boiling points of oleic acid in both Celsius and Oleic scales: Freezing point of oleic acid: \(13^{\circ}\mathrm{C} = 0^{\circ}\mathrm{O}\) Boiling point of oleic acid: \(360^{\circ}\mathrm{C} = 100^{\circ}\mathrm{O}\) Additionally, we know the freezing point of water in Celsius: \(0^{\circ}\mathrm{C}\) Our task is to find the freezing point of water in Oleic scale, which we'll represent as \(x^{\circ}\mathrm{O}\).

Set up a proportion relating the Celsius and Oleic scales

To find the relationship between the Celsius and Oleic scales, we can set up a proportion: \[\frac{C - C_{fo}}{C_{bo} - C_{fo}} = \frac{O}{100}\] Here, \(C\) = temperature in Celsius scale \(C_{fo}\) = freezing point of oleic acid in Celsius scale (\(13^{\circ}\mathrm{C}\)) \(C_{bo}\) = boiling point of oleic acid in Celsius scale (\(360^{\circ}\mathrm{C}\)) \(O\) = temperature in Oleic scale

Plug in the known values and solve for the freezing point of water in Oleic scale

We want to find the freezing point of water in Oleic scale, so we'll plug in the freezing point of water in Celsius (\(0^{\circ}\mathrm{C}\)) for \(C\) and solve for \(O\): \[\frac{0 - 13}{360 - 13} = \frac{O}{100}\] Now, we can solve for \(O\) by cross-multiplying: \[-13 \times 100 = O \times (360 - 13)\] Divide both sides by (360 - 13), which is 347: \[- \frac{13 \times 100}{347} = O\] Now, it is a matter of simplification: \[- \frac{1300}{347} = O \approx -3.75\] Therefore, the freezing point of water on the Oleic scale is approximately \(-3.75^{\circ}\mathrm{O}\).

Key concepts

Proportional Relationships

Understanding proportional relationships is critical in various fields, especially in sciences like Physics and Chemistry, as well as in daily life. In essence, two quantities are proportional if they increase or decrease at the same rate. You can think of it as two friends going up an escalator side by side — as one steps up, so does the other, maintaining their relative position.

In the context of temperature scales, the concept of proportional relationships comes into play when we try to translate a temperature reading from one scale to another, like Celsius to Fahrenheit or, in the textbook exercise, Celsius to a custom Oleic scale. To achieve this, we establish a proportion—a mathematical equation that states that two ratios are equivalent.

For the Oleic scale problem, we know the freezing and boiling points on both the Celsius and Oleic scales. Using the formula for proportions, which essentially states \( \frac{a}{b} = \frac{c}{d} \), we can find any temperature value on one scale when given its counterpart on the other scale.

Temperature Unit Conversion

The process involved in temperature unit conversion is about translating a temperature value from one unit of measurement to another. This is crucial, as different regions and scientific disciplines use various temperature scales, and the ability to convert between these scales ensures proper communication and accuracy.

To convert temperatures, we often use formulas designed for specific scale conversions like Celsius to Fahrenheit or Kelvin. However, these conversions are based on known fixed points, such as the freezing and boiling points of water. In our textbook exercise, we need to convert between the Celsius and a custom Oleic scale. Here, too, we base our conversion on the freezing and boiling points, but this time, it's of oleic acid. By establishing a proportional relationship, we create a conversion formula fitting for our unique scale.

Once the proportional relationship has been determined, it can be applied to any temperature in the Celsius scale to determine its Oleic scale counterpart, as we have with the freezing point of water.

Freezing and Boiling Points of Substances

Freezing and boiling points are physical properties of substances that indicate the temperatures at which they transition between solid, liquid, and gas phases. These points are unique to each substance and remain consistent under a given set of conditions.

For instance, the freezing point of water is universally recognized as \(0^\circ\mathrm{C}\) and its boiling point as \(100^\circ\mathrm{C}\) at sea level. The textbook exercise introduces oleic acid, which has different freezing (\(13^\circ\mathrm{C}\)) and boiling (\(360^\circ\mathrm{C}\)) points. When creating a new temperature scale, these points are often used as reference or fixed points. They become benchmarks to which all other temperatures are related.

In converting temperature units or in calibrating thermometers, these points are indispensable. They provide the necessary landmarks to ensure the scale is accurate and universally applicable, as showed through the Oleic scale problem where the freezing and boiling points of oleic acid served as the foundational parameters for our custom temperature scale.