Problem 6 - University Physics with Modern Physics

Question

Convert the following Kelvin temperatures to the Celsius and Fahrenheit scales: (a) the midday temperature at the surface of the moon (400 K); (b) the temperature at the tops of the clouds in the atmosphere of Saturn (95 K); (c) the temperature at the center of the sun \((1.55 \times 10{^7} K)\).

Step-by-Step Solution

Answer

400 K: 126.85 °C, 260.33 °F; 95 K: -178.15 °C, -288.67 °F; Sun's core: 1.5499973 x 10⁷ °C, 2.789995 x 10⁷ °F.

Step 1: Understanding Conversion Formulas

To convert Kelvin (K) to Celsius (°C), use the formula: \( °C = K - 273.15 \). To convert Kelvin (K) to Fahrenheit (°F), use the formula: \( °F = (K - 273.15) \times \frac{9}{5} + 32 \).

Step 2: Convert 400 K to Celsius

Using the formula \( °C = K - 273.15 \), calculate: \( 400 - 273.15 = 126.85 \ °C \).

Step 3: Convert 400 K to Fahrenheit

Using the formula \( °F = (400 - 273.15) \times \frac{9}{5} + 32 \), calculate: \( 126.85 \times \frac{9}{5} + 32 = 260.33 \ °F \).

Step 4: Convert 95 K to Celsius

Using the formula \( °C = K - 273.15 \), calculate: \( 95 - 273.15 = -178.15 \ °C \).

Step 5: Convert 95 K to Fahrenheit

Using the formula \( °F = (95 - 273.15) \times \frac{9}{5} + 32 \), calculate: \( -178.15 \times \frac{9}{5} + 32 = -288.67 \ °F \).

Step 6: Convert \(1.55 \times 10^7\) K to Celsius

Using the formula \( °C = K - 273.15 \), calculate: \( 1.55 \times 10^7 - 273.15 = 1.5499973 \times 10^7 \ °C \).

Step 7: Convert \(1.55 \times 10^7\) K to Fahrenheit

Using the formula \( °F = (1.55 \times 10^7 - 273.15) \times \frac{9}{5} + 32 \), calculate: \( 1.5499973 \times 10^7 \times \frac{9}{5} + 32 = 2.789995 \times 10^7 \ °F \).

Key Concepts

Kelvin to Celsius

Converting temperatures from Kelvin to Celsius is quite straightforward and follows a simple formula. The Kelvin and Celsius scales are directly related with a linear offset.
To convert a temperature from Kelvin to Celsius, just subtract 273.15 from the Kelvin temperature:

• Formula: \( °C = K - 273.15 \)
• This formula arises because 0 °C is equivalent to 273.15 K. The 273.15 is the freezing point of water in Kelvin, serving as the reference point between these scales.

This simplicity is due to Kelvin being simply an absolute scale of Celsius. Therefore, an increase or decrease in one degree Kelvin is exactly the same change in degrees as on the Celsius scale. For everyday applications, Kelvin is typically used in scientific contexts where absolute temperatures are needed, while Celsius is more common in everyday life.
For example, if you're converting the moon's midday temperature at 400 K, the calculation is:

• \( 400 - 273.15 = 126.85 \ \degree C \)

Always remember, when dealing with temperature conversions, the precise subtraction of 273.15 is crucial.

Kelvin to Fahrenheit

While converting from Kelvin to Fahrenheit involves a bit more math than converting to Celsius, it's still quite manageable with a methodical approach. Fahrenheit, unlike Celsius, does not directly align with Kelvin through a straightforward addition or subtraction due to its differing scale intervals.
Here's the step-by-step formula:

• First, convert Kelvin to Celsius: \( K - 273.15 \).
• Next, convert the Celsius temperature to Fahrenheit using: \( °F = (°C \times \frac{9}{5}) + 32 \).
• Combine these into a single formula: \( °F = (K - 273.15) \times \frac{9}{5} + 32 \).

Fahrenheit's different starting and end points (freezing point of water at 32 °F and boiling at 212 °F) make these conversions slightly more involved. For example, converting the midday temperature on the moon (400 K):

• \( °F = (400 - 273.15) \times \frac{9}{5} + 32 = 260.33 \ \degree F \)

Handling each step carefully ensures accurate conversion outcomes.

Temperature scales

Understanding temperature scales is essential for interpreting different temperature readings across scientific fields and everyday contexts. Primarily, three temperature scales are widely used: Kelvin, Celsius, and Fahrenheit.

• Kelvin: Absolute thermodynamic scale. It starts from absolute zero, the point where molecular motion stops, set at 0 K. Kelvin numbers share the same step increment as Celsius, making direct conversions straightforward.
• Celsius: Based on the freezing and boiling points of water, 0 °C and 100 °C respectively. Predominantly used globally for weather and most routine temperatures.
• Fahrenheit: Commonplace in the United States. It defines 32 °F as the freezing point of water and 212 °F as its boiling point. As such, smaller temperature intervals make changes appear more significant compared to Celsius or Kelvin.

Understanding these scales involves recognizing their origins and practical applications in differing contexts. Scientists often use Kelvin for precise measurements, essential in fields like astronomy and physics. Celsius is well-suited for daily weather reporting and various regional applications, whereas Fahrenheit, mainly used in the U.S., reflects a cultural preference rather than scientific necessity. Through mastering these scales, you can accurately interpret and convert temperatures across disciplines.