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 \((400 \mathrm{K}) ;\) (b) the temperature at the tops of the clouds in the atmosphere of Saturn \((95 \mathrm{K}) ;(\mathrm{c})\) the temperature at the center of the sun \(\left(1.55 \times 10^{7} \mathrm{K}\right)\).

Step-by-Step Solution

Answer

(a) 126.85°C, 260.33°F; (b) -178.15°C, -288.67°F; (c) 1.549972685×10⁷°C, 2.78995083×10⁷°F.

Step 1: Convert Kelvin to Celsius

To convert Kelvin to Celsius, use the formula \( C = K - 273.15 \). We have three temperatures to convert: (a) For the moon, \( K = 400 \). Thus, \[ C = 400 - 273.15 = 126.85 \](b) For Saturn's clouds, \( K = 95 \). Thus, \[ C = 95 - 273.15 = -178.15 \](c) For the Sun's center, \( K = 1.55 \times 10^7 \). Thus, \[ C = 1.55 \times 10^7 - 273.15 = 1.549972685 \times 10^7 \].

Step 2: Convert Celsius to Fahrenheit

To convert Celsius to Fahrenheit, use the formula \( F = \frac{9}{5}C + 32 \). Calculate for each Celsius temperature from Step 1:(a) For the moon, \( C = 126.85 \): \[ F = \frac{9}{5} \times 126.85 + 32 = 260.33 \](b) For Saturn's clouds, \( C = -178.15 \): \[ F = \frac{9}{5} \times (-178.15) + 32 = -288.67 \](c) For the Sun's center, \( C = 1.549972685 \times 10^7 \): \[ F = \frac{9}{5} \times 1.549972685 \times 10^7 + 32 \approx 2.78995083 \times 10^7 \].

Key Concepts

Kelvin to Celsius

Understanding how to convert temperatures from Kelvin to Celsius is essential for various scientific disciplines. The Kelvin scale is an absolute temperature scale, frequently used in scientific contexts, where 0 K is the absolute zero—the point at which molecular motion ceases. To convert Kelvin to Celsius, you utilize a simple subtraction formula:

• Formula: \(C = K - 273.15\)
• Example: For a temperature of 400 K, the conversion would be \(400 - 273.15 = 126.85 \degree C\).

The Kelvin to Celsius conversion is straightforward because both scales increment by the same amount for each degree. The only difference is the starting point, with Celsius beginning at the freezing point of water and Kelvin starting at absolute zero. Remember this simple conversion formula, and you'll easily navigate between these two scales in any scientific context.

Celsius to Fahrenheit

The Celsius to Fahrenheit conversion is crucial for day-to-day weather understanding, as Fahrenheit is primarily used in the United States. Converting Celsius to Fahrenheit involves a multiplication and then an addition:

• Formula: \(F = \frac{9}{5}C + 32\)
• Example: For a temperature of 126.85 °C, the calculation is \(F = \frac{9}{5} \times 126.85 + 32 = 260.33 \degree F\).

This method transforms Celsius into Fahrenheit by expanding the Celsius scale to match Fahrenheit's larger scale units, and then shifting by 32 degrees to align with the freezing point of water in Fahrenheit. This conversion is vital for accurate communication of temperatures across regions where different units are used.

Temperature Scales

Understanding temperature scales is fundamental to measuring and comparing temperatures in various contexts. Three primary scales are Kelvin, Celsius, and Fahrenheit, each defined by different zero points and increments:

• Kelvin Scale: Starts at absolute zero (0 K), where no thermal energy remains. It is used in scientific calculations for its direct relation to thermodynamic laws.
• Celsius Scale: Sets zero at the freezing point of water and 100 at the boiling point, making it ideal for everyday use and scientific research involving water-based processes.
• Fahrenheit Scale: Uses 32 °F as the freezing point of water and 212 °F as the boiling point, typically used in the United States for meteorological purposes.

Each scale has its applications, and knowing how and when to use these scales allows for effective communication and understanding in both scientific and everyday contexts.