Problem 3 - Physics

Question

On the moon the surface temperature ranges from \(375 \mathrm{~K}\) during the day to \(1.00 \times 10^{2} \mathrm{~K}\) at night. What are these temperatures on the (a) Celsius and (b) Fahrenheit scales?

Step-by-Step Solution

Answer

(a) Day: 101.85 °C, 215.33 °F; (b) Night: -173.15 °C, -279.67 °F.

Step 1: Understanding the Temperature Conversion

The problem requires converting temperatures from Kelvin (K) to both Celsius (°C) and Fahrenheit (°F). The conversion between these scales involves specific formulas. To convert from Kelvin to Celsius, use the formula: \[ °C = K - 273.15 \]

Step 2: Convert Day Temperature from Kelvin to Celsius

Convert the day temperature of 375 K to Celsius using the formula from Step 1.Substitute 375 for K: \[ °C = 375 - 273.15 = 101.85 °C \]

Step 3: Convert Night Temperature from Kelvin to Celsius

Convert the night temperature of \(1.00 \times 10^2\) K to Celsius. This is 100 K.Substitute 100 for K:\[ °C = 100 - 273.15 = -173.15 °C \]

Step 4: Convert Day Temperature from Kelvin to Fahrenheit

Convert 375 K to Fahrenheit using the formula: \[ °F = (K - 273.15) \times \frac{9}{5} + 32 \]Substitute 375 for K:\[ °F = (375 - 273.15) \times \frac{9}{5} + 32 \]\[ °F = 101.85 \times \frac{9}{5} + 32 \]\[ °F = 183.33 + 32 = 215.33 °F \]

Step 5: Convert Night Temperature from Kelvin to Fahrenheit

Convert 100 K to Fahrenheit using the formula:\[ °F = (100 - 273.15) \times \frac{9}{5} + 32 \]\[ °F = -173.15 \times \frac{9}{5} + 32 \]\[ °F = -311.67 + 32 = -279.67 °F \]

Step 6: Conclusion

The temperatures on the moon are: (a) Day: 101.85 °C (or 215.33 °F) (b) Night: -173.15 °C (or -279.67 °F)

Key Concepts

Kelvin to Celsius Conversion

To understand the conversion from Kelvin to Celsius, it's important to know what each scale represents. The Kelvin scale is often used in scientific settings because it is an absolute scale. It starts at absolute zero, or 0 K, where all molecular motion stops. The Celsius scale, which is more common in everyday use, is based on the freezing and boiling points of water at sea level, set at 0 °C and 100 °C respectively.

• The conversion from Kelvin to Celsius is straightforward: you simply subtract 273.15 from the Kelvin temperature.
• The formula for conversion is: \[ °C = K - 273.15 \]
  For example, a Kelvin temperature of 375, when converted to Celsius is: \[ 375 - 273.15 = 101.85 °C \]
• Similarly, if the temperature is at 100 K: \[ 100 - 273.15 = -173.15 °C \]

Understanding this conversion helps in interpreting temperatures in scientific and engineering contexts, where Kelvin is predominantly used.

Kelvin to Fahrenheit Conversion

The Fahrenheit scale, commonly used in the United States, is known for its slightly more complex relationship with other temperature scales. The conversion from Kelvin to Fahrenheit needs two steps. You first convert Kelvin to Celsius, and then Celsius to Fahrenheit.
The formula for converting from Kelvin directly to Fahrenheit spells out these steps as:\[ °F = (K - 273.15) \times \frac{9}{5} + 32 \] Here's how you can apply this formula:

• For a day temperature on the moon of 375 K: \[ °F = (375 - 273.15) \times \frac{9}{5} + 32 = 215.33 °F \]
• For a colder night temperature of 100 K: \[ °F = (100 - 273.15) \times \frac{9}{5} + 32 = -279.67 °F \]

This conversion is useful when working with data that integrates both scientific measurements (Kelvin) with more localized usage (Fahrenheit).

Temperature Scales

Before diving into conversions, it's beneficial to understand the temperature scales commonly encountered in science and daily life. Each scale has its unique reference points and usage.

• Kelvin: Often used in scientific contexts, Kelvin is the SI unit for temperature. Absolute zero, the point at which there is theoretically no molecular motion, is at 0 K, making Kelvin very useful in physical sciences.
• Celsius: Widely applied across most of the world, Celsius is based on the properties of water. The scale sets the freezing point at 0 °C and boiling point at 100 °C under standard atmospheric conditions.
• Fahrenheit: Used primarily in the United States, this scale is based on older, less precise references. It sets water's freezing and boiling points at 32 °F and 212 °F respectively.

Understanding these scales' applications and points of reference is crucial for effectively dealing with temperature-related problems in various fields. Recognizing these differences helps in interpreting temperatures correctly in both everyday and scientific settings.