Suppose you were dissatisfied with both Celsius and Fahrenheit units and wanted to design your own temperature scale based on ethyl alcohol (ethanol). On the Celsius scale, ethanol has a melting point of -117.3 °C and a boiling point of 78.5 °C, but on your new scale calibrated in units of degrees ethanol, °E, you define ethanol to melt at 0 °E and boil at 200 °E. (c) What are the melting and boiling points of water on the ethanol scale?

Verified step by step guidance

1. insert step 1: Identify the known points on both the Celsius and ethanol scales. Ethanol melts at -117.3 °C and boils at 78.5 °C, which correspond to 0 °E and 200 °E, respectively.
2. insert step 2: Determine the relationship between the Celsius scale and the ethanol scale. Calculate the difference in temperature between the melting and boiling points on both scales.
3. insert step 3: Calculate the conversion factor between the Celsius and ethanol scales by dividing the temperature range in °E by the temperature range in °C.
4. insert step 4: Use the conversion factor to convert the melting point of water (0 °C) to the ethanol scale.
5. insert step 5: Use the conversion factor to convert the boiling point of water (100 °C) to the ethanol scale.

Key Concepts

Temperature Scales

Temperature scales, such as Celsius and Fahrenheit, provide a way to quantify thermal energy. Each scale has defined reference points, like the freezing and boiling points of water. Understanding how to convert between these scales is essential for solving problems involving temperature, especially when creating a new scale based on different substances.

Phase Changes

Phase changes refer to the transitions between solid, liquid, and gas states of matter, which occur at specific temperatures. For example, water freezes at 0 °C and boils at 100 °C under standard atmospheric pressure. Knowing the melting and boiling points of substances like ethanol and water is crucial for determining their behavior on a new temperature scale.

Linear Transformation of Temperature

When creating a new temperature scale, a linear transformation can be used to relate the new scale to existing ones. This involves establishing a formula that converts temperatures from one scale to another based on known reference points. In this case, the melting and boiling points of ethanol serve as anchors for defining the new scale in degrees ethanol, °E.