Question

At the Amundsen-Scott South Pole base station in Antarctica, when the temperature is \(-100.0^{\circ} \mathrm{F}\), researchers who live there can join the " 300 Club" by stepping into a sauna heated to \(200.0^{\circ} \mathrm{F}\) then quickly running outside and around the pole that marks the South Pole. What are these temperatures in \({ }^{\circ} \mathrm{C}\) ? What are these temperatures in \(\mathrm{K}\) ? If you measured the temperatures only in \({ }^{\circ} \mathrm{C}\) and \(\mathrm{K}\), can you become a member of the " 300 Club" (that is, is there a 300--degree difference between the temperature extremes when measured in \({ }^{\circ} \mathrm{C}\) and \(\mathrm{K}\) )?

Short answer

The temperatures in Celsius are -73.333°C and 93.333°C; in Kelvin, the temperatures are 199.817 K and 366.483 K. The difference between the two temperatures in both Celsius and Kelvin is 166.666 degrees, not 300 degrees. Therefore, one cannot become a member of the "300 Club" using temperatures measured only in Celsius or Kelvin.

Step-by-step solution

Convert Fahrenheit to Celsius

To convert the given Fahrenheit temperatures to Celsius, we will use the following formula: \(T_{\text{C}} = \frac{5}{9} (T_{\text{F}} - 32)\) Apply the formula for both temperatures: \(\text{-100.0}^{\circ} \text{F}\): \(T_{\text{C}} = \frac{5}{9} (-100.0 - 32)\) \(\text{200.0}^{\circ} \text{F}\): \(T_{\text{C}} = \frac{5}{9} (200.0 - 32)\)

Calculate Celsius temperatures

Calculate the Celsius temperatures for both values: \(\text{-100.0}^{\circ} \text{F}\): \(T_{\text{C}} = \frac{5}{9} (-132) = -73.333^{\circ} \text{C}\) \(\text{200.0}^{\circ} \text{F}\): \(T_{\text{C}} = \frac{5}{9} (168) = 93.333^{\circ} \text{C}\)

Convert Celsius to Kelvin

To convert the calculated Celsius temperatures to Kelvin, we will use the following formula: \(T_{\text{K}} = T_{\text{C}} + 273.15\) Apply the formula for both temperatures: \(-73.333^{\circ} \text{C}\): \(T_{\text{K}} = -73.333 + 273.15\) \(93.333^{\circ} \text{C}\): \(T_{\text{K}} = 93.333 + 273.15\)

Calculate Kelvin temperatures

Calculate the Kelvin temperatures for both values: \(-73.333^{\circ} \text{C}\): \(T_{\text{K}} = 199.817 \text{K}\) \(93.333^{\circ} \text{C}\): \(T_{\text{K}} = 366.483 \text{K}\)

Check for 300-degree difference

Finally, we will check if the difference in the two temperatures is 300 degrees when measured in Celsius and Kelvin. In Celsius: \(93.333 - (-73.333) = 166.666 \text{ degrees}\) In Kelvin: \(366.483 - 199.817 = 166.666 \text{ degrees}\) Since the difference in both Celsius and Kelvin is not 300 degrees, one cannot become a member of the "300 Club" using temperatures measured only in Celsius or Kelvin.

Key concepts

Fahrenheit to Celsius Conversion

Understanding the conversion process from Fahrenheit to Celsius is essential in various scientific disciplines where precise temperature control and communication are critical. To transition from Fahrenheit to Celsius, one must understand the mathematical relationship between these two scales. The formula used is:

\( T_{\text{C}} = \frac{5}{9} (T_{\text{F}} - 32) \)

In this formula, \(T_{\text{C}}\) represents the temperature in degrees Celsius, and \(T_{\text{F}}\) is the temperature in degrees Fahrenheit. The subtraction of 32 accounts for the difference in the zero points of the two scales, while the multiplication by \(\frac{5}{9}\) converts the scale of the interval. The South Pole exercise is a fascinating example of this, converting a chilly -100.0°F to a biting -73.333°C via this straightforward formula.

Celsius to Kelvin Conversion

The transformation from Celsius to Kelvin is a critical concept for scientific measurements, particularly in the fields of physics and chemistry. The conversion formula is simple, with no multiplication or division required; only a single addition operation is necessary.

\( T_{\text{K}} = T_{\text{C}} + 273.15 \)

Where \(T_{\text{K}}\) denotes the temperature in Kelvins and \(T_{\text{C}}\) is in degrees Celsius. This equation reflects the fact that 0 K (absolute zero) is equivalent to -273.15°C. When using this formula, a temperature of 93.333°C from our Antarctic sauna converts seamlessly to a steamy 366.483 K, demonstrating the ease with which scientists can switch between these scales.

Temperature Measurement in Science

Temperature measurement is a fundamental aspect of scientific research as it influences many experimental outcomes. Having a solid grasp of various temperature scales and conversions is indispensable for scientists. There are primarily three temperature scales used in science: Fahrenheit, Celsius, and Kelvin.

The Fahrenheit scale is commonly used in the United States but less so in scientific circles. The Celsius scale, favored in most of the world and in scientific communities, is based on the properties of water, with 0°C representing the freezing point and 100°C the boiling point at sea level. Lastly, the Kelvin scale is used extensively in the scientific community, especially in fields like physics.

Kelvin is the base unit for temperature in the International System of Units (SI) and is critical when it comes to absolute temperature measurements. Unlike Celsius or Fahrenheit, Kelvin starts at absolute zero—the coldest possible temperature where particles have minimal thermal motion. This makes the Kelvin scale invaluable for theoretical work and calculations requiring high accuracy.