Problem 78 - Chemistry

Question

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 \(-117.3^{\circ} \mathrm{C}\) and a boiling point of \(78.5^{\circ} \mathrm{C}\), but on your new scale calibrated in units of degrees ethanol, \({ }^{\circ} \mathrm{E}\), you define ethanol to melt at \(0{ }^{\circ} \mathrm{E}\) and boil at \(200^{\circ} \mathrm{E}\). (a) How does your ethanol degree compare in size with a Celsius degree? (b) How does an ethanol degree compare in size with a Fahrenheit degree? (c) What are the melting and boiling points of water on the ethanol scale? (d) What is normal human body temperature \(\left(98.6^{\circ} \mathrm{F}\right)\) on the ethanol scale? (e) If the outside thermometer reads \(130{ }^{\circ} \mathrm{E}\), how would you dress to go out?

Step-by-Step Solution

Answer

(a) 1°E = 0.979°C, (b) 1°E = 1.762°F, (c) Water melts at 119.85°E and boils at 222.19°E, (d) Body temperature is 157.77°E, (e) Wear warm clothing – it's about 11°C.

Step 1: Determine the Size of an Ethanol Degree Compared to Celsius Degree

To compare the two scales, calculate the range between the melting and boiling points in Celsius and Ethanol scales for ethanol. Celsius: from -117.3°C to 78.5°C. The range is:\[\Delta C = 78.5 - (-117.3) = 195.8\,^{\circ}C.\]Ethanol scale: from 0°E to 200°E. The range is:\[\Delta E = 200 - 0 = 200\,^{\circ}E.\]Thus, the size of one ethanol degree compared to one Celsius degree is:\[\text{Size of 1°E in °C} = \frac{\Delta C}{\Delta E} = \frac{195.8}{200} = 0.979\,^{\circ}C.\]

Step 2: Determine the Size of an Ethanol Degree Compared to Fahrenheit Degree

Convert the ethanol melting and boiling points from Celsius to Fahrenheit to find the scale range in Fahrenheit.Convert -117.3 °C and 78.5 °C into Fahrenheit:\[-117.3°C = (-117.3 \times \frac{9}{5}) + 32 = -179.14°F,\]\[78.5°C = (78.5 \times \frac{9}{5}) + 32 = 173.3°F.\]The range in Fahrenheit is:\[\Delta F = 173.3 - (-179.14) = 352.44\,^{\circ}F.\]Thus, the size of one ethanol degree compared to one Fahrenheit degree is:\[\text{Size of 1°E in °F} = \frac{\Delta F}{\Delta E} = \frac{352.44}{200} = 1.762\,^{\circ}F.\]

Step 3: Find Melting and Boiling Points of Water on Ethanol Scale

The melting point of water is 0°C and the boiling point is 100°C. Use the conversion from Celsius to Ethanol:Use the formula for conversion:\[E = \frac{200}{195.8}(C + 117.3).\]For 0°C (melting point of water):\[E = \frac{200}{195.8}(0 + 117.3) = 119.85\,^{\circ}E.\]For 100°C (boiling point of water):\[E = \frac{200}{195.8}(100 + 117.3) = 222.19\,^{\circ}E.\]

Step 4: Convert Human Body Temperature from Fahrenheit to Ethanol Scale

First, convert 98.6°F to Celsius:\[C = \frac{5}{9}(98.6 - 32) = 37°C.\]Then, use the conversion formula from Celsius to Ethanol:\[E = \frac{200}{195.8}(37 + 117.3) = 157.77\,^{\circ}E.\]

Step 5: Interpret 130°E Temperature and Dress Accordingly

Convert 130°E to Celsius to better understand the temperature:Use the conversion formula:\[C = \frac{195.8}{200}(130 - 117.3) = 11.03°C.\]Thus, 130°E is cool, around 11°C, so you should wear warm clothing.

Key Concepts

Ethanol Temperature Scale

Creating a temperature scale using ethanol involves selecting two fixed points: the melting and boiling points of ethanol. On the Celsius scale, ethanol melts at -117.3°C and boils at 78.5°C. For the new scale, define ethanol to melt at 0°E and boil at 200°E. This custom range allows us to design a scale lining up nearly with natural ethanol phase shifts.

The chosen points dictate the degree size on the ethanol scale relative to other systems. Such a design aligns closer to those unique properties compared to water, offering an alternative perspective for temperature measurement. Clear understanding of this specialized scale development lays the foundation for further conversion work in comparing to Celsius and Fahrenheit scales.

Such scales are potentially useful when focused on ethanol-specific applications, like chemical processes or educational tools.

Celsius and Fahrenheit Comparison

Converting between temperature systems assists in comparing ethanol degrees to Celsius and Fahrenheit scales. To compare ethanol degrees with Celsius:

• Calculate the temperature range on Celsius from -117.3°C to 78.5°C, equating to a difference of 195.8°C.
• Meanwhile, in ethanol degrees, this difference is defined as 200°E.

Therefore, one ethanol degree translates to approximately 0.979°C.

Bridge to Fahrenheit requires converting -117.3°C and 78.5°C first:

• -117.3°C converts to -179.14°F.
• 78.5°C translates to 173.3°F.

Thus, the difference becomes 352.44°F on the Fahrenheit scale, making one ethanol degree roughly 1.762°F.

These conversions provide understanding across various contexts where different temperature scales might be needed, such as meteorology, cooking, or engineering.

Temperature Scale Design

Designing temperature scales involves a thoughtful methodology to define meaningful temperature points. A typical design choice like that in ethanol scale involves:

• Selecting appropriate reference points, like freezing and boiling points.
• Defining the intervals between these fixed points to establish degree sizes.

In creating the ethanol temperature scale, the ethanol melting point at 0°E and boiling point at 200°E sets the scale's scope. This scope is then split into 200 equal segments, linking the physical changes in ethanol to human-interpretable units.

The conversion formula \[E = \frac{200}{195.8}(C + 117.3)\] allows conversions, making the knowledge of the selected physical constants vital for creating accurate cross-scale calculations.

Melting and Boiling Points

Melting and boiling points are fundamental markers for understanding temperature changes in substances. They make great reference points for scaling.
For the ethanol scale, transferring water's known points involves finding where 0°C (water's melting point) and 100°C (boiling point) land on this new scale.

• Using formula \[E = \frac{200}{195.8}(C + 117.3)\], the melting point for water converts to about 119.85°E.
• Similarly, boiling water converts to 222.19°E.

These values represent water's phase changes in terms of ethanol-based degree units.

Understanding these conversions fosters comparisons across household cannons or lab instruments where such phase transitions of water are symbolic."