Problem 35 - Chemistry in Focus

Question

The coldest temperature ever measured in the United States is \(-80^{\circ} \mathrm{F}\) on January 23,1971 , in Prospect Creek, Alaska. Convert that temperature to Celsius and Kelvin.

Step-by-Step Solution

Answer

The temperature of \( -80^\circ \mathrm{F} \) converts to approximately \( -62.22^\circ \mathrm{C} \) or \( 210.93 \text{K} \)

Step 1: Convert Fahrenheit to Celsius

Use the formula to convert Fahrenheit (\text{F}) to Celsius (\text{C}): \( C = \frac{5}{9}(F - 32) \) Substitute \( F = -80 \) into the equation to find the Celsius temperature.

Step 2: Perform the Calculation

Plugging \( -80 \) into the formula gives \( C = \frac{5}{9}(-80 - 32) = \frac{5}{9}(-112) \) which simplifies to \( C = -62.2222... \) or \( C \approx -62.22^\circ \mathrm{C} \)

Step 3: Convert Celsius to Kelvin

Use the formula to convert Celsius to Kelvin (\text{K}): \( K = C + 273.15 \). Substitute the previously found Celsius value into this equation to find the Kelvin temperature.

Step 4: Perform the Conversion to Kelvin

Using the Celsius temperature, the Kelvin temperature is \( K = -62.22 + 273.15 = 210.93 \text{K} \).

Key Concepts

Fahrenheit to Celsius Conversion

Understanding how to convert temperature from Fahrenheit to Celsius is crucial in scientific studies, as Celsius is commonly used in most countries outside of the United States. To convert Fahrenheit to Celsius, we use the formula:
\( C = \frac{5}{9}(F - 32) \).
Why subtract 32? This is because the freezing point of water in Fahrenheit (32 degrees) corresponds to 0 degrees in Celsius. The fraction \(\frac{5}{9}\) is the ratio of the temperature scales between Celsius and Fahrenheit.
So, for the coldest temperature recorded in the U.S., which is \(-80^\circ \mathrm{F}\), the conversion goes like this:
\( C = \frac{5}{9}(-80 - 32) = \frac{5}{9}(-112) \), which simplifies to approximately \( C \approx -62.22^\circ \mathrm{C} \).
This method allows us to bridge the gap between different temperature scales and better understand climatic extremes across regions with different measurement standards.

Celsius to Kelvin Conversion

When converting from Celsius to Kelvin, we're moving from one metric temperature scale to another, which is simpler than our previous conversion. This is because both Celsius and Kelvin scales are based on the same interval size, but Kelvin starts at absolute zero. The conversion formula is:
\( K = C + 273.15 \).
Here, 273.15 is the difference in the starting points of the two scales. By adding this value to the Celsius reading, we effectively shift the starting point from the freezing point of water to absolute zero, the point where theoretical molecular motion stops.
Using the previously found Celsius temperature, \(-62.22^\circ \mathrm{C}\), converting to Kelvin would look like:
\( K = -62.22 + 273.15 = 210.93 \text{K} \).
This result represents the Kelvin temperature of the coldest day recorded in the U.S. and demonstrates how using Kelvin can help in fields such as physics where temperatures are often compared in relation to absolute zero.

Temperature Scales

Temperature scales provide a consistent way to measure thermal energy. The three most commonly used scales are Fahrenheit, Celsius, and Kelvin.

• Fahrenheit is often used in the United States for everyday temperatures and is represented by \( ^\circ\mathrm{F} \).
• Celsius is used in most of the world and is denoted by \( ^\circ\mathrm{C} \).
• Kelvin is the SI unit for temperature, used in scientific measurements, and has no degree symbol. It is represented simply as K.

Each scale has its own application area and conversion formula, as seen with the coldest day in the U.S. Being familiar with these scales is essential because they are used differently depending on the context. For example, scientists use Kelvin for theoretical physics and thermodynamics, while meteorologists might use Celsius for their weather reports. Understanding how to switch between these scales is a critical skill in both scientific and everyday contexts.