Problem 2 - Physics

Question

Suppose you are hiking down the Grand Canyon. At the top, the temperature early in the morning is a cool \(3^{\circ} \mathrm{C}\). By late afternoon, the temperature at the bottom of the canyon has warmed to a sweltering \(34^{\circ} \mathrm{C}\). What is the difference between the higher and lower temperatures in (a) Fahrenheit degrees and (b) kelvins?

Step-by-Step Solution

Answer

Difference: 55.8°F and 31 K.

Step 1: Understanding the Problem

We need to find the difference between two temperatures when converted to Fahrenheit and Kelvin. The given temperatures are: Top temperature: \(3^{\circ} \mathrm{C}\), Bottom temperature: \(34^{\circ} \mathrm{C}\).

Step 2: Convert Temperatures to Fahrenheit

Use the formula to convert Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \).- Top temperature: \( F_1 = \frac{9}{5} \times 3 + 32 = 37.4^{\circ}F \).- Bottom temperature: \( F_2 = \frac{9}{5} \times 34 + 32 = 93.2^{\circ}F \).

Step 3: Calculate the Difference in Fahrenheit

Subtract the top temperature in Fahrenheit from the bottom temperature to find the difference: \( \Delta F = 93.2 - 37.4 = 55.8^{\circ}F \).

Step 4: Convert Temperatures to Kelvin

Use the formula to convert Celsius to Kelvin: \( K = C + 273.15 \).- Top temperature: \( K_1 = 3 + 273.15 = 276.15 \text{ K} \).- Bottom temperature: \( K_2 = 34 + 273.15 = 307.15 \text{ K} \).

Step 5: Calculate the Difference in Kelvin

Subtract the top temperature in Kelvin from the bottom temperature to find the difference: \( \Delta K = 307.15 - 276.15 = 31 \text{ K} \).

Key Concepts

Celsius to Fahrenheit conversion

Converting temperatures from Celsius to Fahrenheit is a common requirement in various contexts, such as weather forecasts and scientific measurements. It gives us an understanding of temperature in a scale that is widely used, especially in the United States. The formula to convert Celsius (C) into Fahrenheit (F) is:
\[F = \frac{9}{5}C + 32\]
This formula adjusts for the difference in starting points and the size of the Fahrenheit degree compared to a Celsius degree. To understand it better, let's break down the conversion.

• The ratio \(\frac{9}{5}\) accounts for the difference in size between the Fahrenheit degree and the Celsius degree, as there are 180 degrees between the freezing and boiling points of water in Fahrenheit, compared to 100 degrees in Celsius.
• Adding 32 adjusts for the zero points of the two scales: water freezes at 0°C and 32°F.

For example, given a temperature of 3°C, the conversion to Fahrenheit is \( \frac{9}{5} \times 3 + 32 = 37.4^{\circ}F \).
For a higher temperature, such as 34°C, the calculation is \( \frac{9}{5} \times 34 + 32 = 93.2^{\circ}F \). Knowing how to convert between these scales is helpful for comparing temperatures in different contexts.

Celsius to Kelvin conversion

Unlike the Fahrenheit scale, the Kelvin scale is an absolute temperature scale used primarily in scientific settings. It shifts the entire Celsius scale upwards by approximately 273.15 degrees, aligning the zero point of the Kelvin scale with absolute zero — the lowest temperature possible.
The formula for converting Celsius (C) to Kelvin (K) is straightforward:
\[K = C + 273.15\]
This simple addition reflects the close relationship between the Kelvin and Celsius scales, as they share the same degree size.

• An important distinction is that the Kelvin scale does not use the degree symbol, so temperatures are written as \(K\), not \(\text{°K}\).
• For scientific accuracy, note that this conversion formula directly reflects the hierarchy of temperature reference points, with absolute zero as the starting point.

For example, converting 3°C to Kelvin is \(3 + 273.15 = 276.15 \text{ K}\).
Similarly, converting 34°C gives us \(34 + 273.15 = 307.15 \text{ K}\). This ability to convert gives scientists a uniform temperature reference framework.

Temperature difference calculation

Calculating the difference in temperature is crucial in many areas, from daily weather analysis to complex scientific research. When managing temperature difference calculations, it's essential to understand that this difference is consistent across all measurement scales (i.e., Celsius, Fahrenheit, Kelvin).

• For Celsius to Fahrenheit conversions, once the individual temperatures are converted, the temperature difference remains unchanged in relation to the size of the Fahrenheit and Celsius degrees.
• In the original exercise, the difference between the top and bottom temperatures was 31°C. When converted, this remains a consistent difference of 55.8°F. This demonstrates the alignment of temperature scale operations as the difference in calculations doesn't include the addition of constants like 32, which is used in the conversion of individual temperature points.
• Similarly, with Kelvin, the conversion formula is a simple addition, not affecting the calculation of differences. Therefore, the temperature difference remains at 31 K.

Understanding and transitioning between these differences across scales help maintain consistency and reliability, ensuring clarity in what temperature difference signifies.