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

Step-by-Step Solution

Answer

(a) 126.85°C, 260.33°F; (b) -178.15°C, -288.67°F; (c) 1.54999728×10⁷°C, 2.789995104×10⁷°F.

Step 1: Convert from Kelvin to Celsius

To convert a temperature from Kelvin (K) to Celsius (°C), use the formula: \[ °C = K - 273.15 \]. Let's apply this formula to the given temperatures: (a) 400 K, (b) 95 K, (c) 1.55 \times 10^7 K.(a) \[400 \, \mathrm{K} - 273.15 = 126.85 \, ^{\circ} \mathrm{C}\](b) \[95 \, \mathrm{K} - 273.15 = -178.15 \, ^{\circ} \mathrm{C}\](c) \[1.55 \times 10^7 \, \mathrm{K} - 273.15 = 1.54999728 \times 10^7 \, ^{\circ} \mathrm{C}\]

Step 2: Convert from Celsius to Fahrenheit

To convert from Celsius (°C) to Fahrenheit (°F), use the formula: \[ °F = °C \times \frac{9}{5} + 32 \]. Apply this to the Celsius values found in Step 1:(a) \[126.85 \, ^{\circ} \mathrm{C} \times \frac{9}{5} + 32 = 260.33 \, ^{\circ} \mathrm{F} \](b) \[-178.15 \, ^{\circ} \mathrm{C} \times \frac{9}{5} + 32 = -288.67 \, ^{\circ} \mathrm{F} \](c) \[1.54999728 \times 10^7 \, ^{\circ} \mathrm{C} \times \frac{9}{5} + 32 = 2.789995104 \times 10^7 \, ^{\circ} \mathrm{F} \]

Key Concepts

Kelvin to Celsius

The Kelvin and Celsius temperature scales are both widely used in scientific contexts. The Kelvin scale is the absolute temperature scale and has its zero point at absolute zero. To convert a temperature from Kelvin to Celsius, you need to know a simple formula: \[ °C = K - 273.15 \]

• Subtract 273.15 from the Kelvin temperature to get the Celsius equivalent.
• This conversion is straightforward as both scales use the same size for their degrees.

For example, in our exercise, we converted 400 K to Celsius:

• 400 K - 273.15 gives us 126.85 °C, equivalent to the moon's midday temperature.
• 95 K, representing the temperature at Saturn's cloud tops, becomes -178.15 °C.
• Finally, the extremely high temperature at the sun's center, 1.55 \times 10^7 K, converts to approximately 1.55 \times 10^7 °C.

Remember, this formula is essential for many scientific calculations that require temperature measures in Celsius.

Celsius to Fahrenheit

Once you have temperatures in Celsius, converting them to Fahrenheit involves another simple formula. Fahrenheit is often used in everyday settings, especially in the United States. The conversion formula between Celsius and Fahrenheit is: \[ °F = °C \times \frac{9}{5} + 32 \]

• To convert, multiply the Celsius value by \( \frac{9}{5} \) and then add 32 to the result.
• This formula reflects the different sizes of degrees and zero points in the two scales.

Applying this formula to our converted Celsius values gives us temperatures in Fahrenheit:

• 126.85 °C becomes approximately 260.33 °F for the lunar surface temperature.
• -178.15 °C converts to about -288.67 °F at Saturn's cloud tops.
• 1.55 \times 10^7 °C results in roughly 2.79 \times 10^7 °F for the sun's core.

Understanding these calculations is helpful for translating temperature data across different scales used in various fields.

Temperature Scales

To appreciate the conversions, it's crucial to understand the concept of temperature scales. Different scales are used depending on the context and geographic location:

• Kelvin: The Kelvin scale is used primarily in science, particularly for thermodynamic calculations. It starts at absolute zero, where all molecular movement stops.
• Celsius: Widely used in most parts of the world for everyday temperature measurement. It sets the freezing and boiling points of water at 0 °C and 100 °C, respectively.
• Fahrenheit: Used primarily in the United States for weather forecasts and other applications. It places the freezing point of water at 32 °F and the boiling point at 212 °F.

Each scale has its advantages and common usage areas. Kelvin's advantage is the directness it offers in scientific research, which is often necessary when dealing with very low or very high temperatures. Meanwhile, Celsius and Fahrenheit provide practical scalability for daily weather and environmental observations. Understanding these differences is key to properly converting and using temperatures across these scales.