Problem 6 - College 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

400 K: 126.85 °C, 260.33 °F; 95 K: -178.15 °C, -288.67 °F; 1.55 × 10⁷ K: 1.54997 × 10⁷ °C, 2.78995 × 10⁷ °F.

Step 1: Understand the Kelvin to Celsius Conversion

The equation to convert a temperature from Kelvin to Celsius is given by: \[ T_{\text{Celsius}} = T_{\text{Kelvin}} - 273.15 \] This equation allows you to subtract the constant 273.15 from a temperature given in Kelvin to find the equivalent temperature in Celsius.

Step 2: Convert 400 K to Celsius

Using the equation from Step 1, convert 400 K to Celsius as follows:\[ T_{\text{Celsius}} = 400 - 273.15 = 126.85 \, \text{°C} \] Thus, 400 K is equivalent to 126.85 °C.

Step 3: Convert 400 K to Fahrenheit

First, convert the Kelvin temperature to Celsius using Step 1, then use the formula to convert Celsius to Fahrenheit:\[ T_{\text{Fahrenheit}} = \frac{9}{5}T_{\text{Celsius}} + 32 \] Apply this to the Celsius temperature from Step 2:\[ T_{\text{Fahrenheit}} = \frac{9}{5} \times 126.85 + 32 = 260.33 \, \text{°F} \]Thus, 400 K is equivalent to 260.33 °F.

Step 4: Convert 95 K to Celsius

Convert 95 K to Celsius using the Kelvin to Celsius formula:\[ T_{\text{Celsius}} = 95 - 273.15 = -178.15 \, \text{°C} \] So, 95 K is equivalent to -178.15 °C.

Step 5: Convert 95 K to Fahrenheit

First, convert the Kelvin temperature to Celsius using Step 4, then convert Celsius to Fahrenheit:\[ T_{\text{Fahrenheit}} = \frac{9}{5} \times (-178.15) + 32 = -288.67 \, \text{°F} \] Thus, 95 K is equivalent to -288.67 °F.

Step 6: Convert 1.55 × 10⁷ K to Celsius

Use the Kelvin to Celsius conversion formula for a large number:\[ T_{\text{Celsius}} = 1.55 \times 10^7 - 273.15 = 1.549972685 \times 10^7 \, \text{°C} \] So, 1.55 × 10⁷ K is equivalent to approximately 1.549972685 × 10⁷ °C.

Step 7: Convert 1.55 × 10⁷ K to Fahrenheit

Convert the Celsius temperature from Step 6 to Fahrenheit:\[ T_{\text{Fahrenheit}} = \frac{9}{5} \times 1.549972685 \times 10^7 + 32 = 2.789950833 \times 10^7 \, \text{°F} \] Thus, 1.55 × 10⁷ K is approximately 2.789950833 × 10⁷ °F.

Key Concepts

Kelvin to Celsius

The Kelvin to Celsius conversion is straightforward. We use the formula:

• \[ T_{\text{Celsius}} = T_{\text{Kelvin}} - 273.15 \]

This formula is derived from the definition of the Celsius scale, which is offset by 273.15 from the Kelvin scale.
For example, 400 K translates into 126.85 °C as shown in the solution. Simply subtract 273.15 from the given temperature in Kelvin.
This conversion is commonly used in scientific fields because Kelvin is the SI unit for temperature, but Celsius is more familiar in everyday contexts.

Kelvin to Fahrenheit

To convert Kelvin to Fahrenheit, first convert Kelvin to Celsius, then Celsius to Fahrenheit. The full conversion is:

• Use: \[ T_{\text{Celsius}} = T_{\text{Kelvin}} - 273.15 \]
• Then apply: \[ T_{\text{Fahrenheit}} = \frac{9}{5}T_{\text{Celsius}} + 32 \]

For instance, converting 400 K to Fahrenheit, first changes it to 126.85 °C, which is then converted to 260.33 °F.
This two-step process ensures accuracy by leveraging the familiar Celsius to Fahrenheit formula.

Physics Problem Solving

Physics problems often require converting temperatures across different scales. Understanding conversion formulas is vital.
Start by identifying the units provided and required. Apply the appropriate equations in sequence.
Simplicity is key:

• Use Kelvin for calculations related to physical laws due to its absolute nature.
• Convert to Celsius or Fahrenheit for readability or specific requirements.

Practice ensures that you become adept at these calculations, enabling smoother problem solving.

Celsius to Fahrenheit

Celsius and Fahrenheit are both common temperature scales. Their conversion helps in practical applications:

• The formula is: \[ T_{\text{Fahrenheit}} = \frac{9}{5}T_{\text{Celsius}} + 32 \]

This equation reflects the relationship between the two scales, where Fahrenheit has a finer granularity.
For example, 126.85 °C converts to 260.33 °F. Always perform calculations carefully to avoid mistakes in conversion, especially with decimals.
Remember, accurate conversions are essential for exact science applications.

Thermodynamic Temperature Scales

Thermodynamic temperature scales, like Kelvin and Rankine, are essential in scientific measurements. Kelvin is an absolute scale starting at absolute zero, the minimum theoretical temperature.
It’s pivotal in scientific fields like physics and chemistry due to its direct ties with energy calculations.
Key points:

• Kelvin avoids negative numbers by setting absolute zero as 0 K.
• Celsius, familiar in everyday life, is offset by 273.15 degrees from Kelvin.

Understanding these scales helps in converting and comparing temperatures accurately across various conditions.