Question

Which air temperature feels coldest? a) \(-40^{\circ} \mathrm{C}\) b) \(-40^{\circ} \mathrm{F}\) c) \(233 \mathrm{~K}\) d) All three are equal.

Short answer

Answer: d) All three are equal.

Step-by-step solution

Convert temperatures

First, we need to convert all temperatures to the same scale. Let's convert them all to Celsius. a) \(-40^{\circ} \mathrm{C}\) is already in Celsius, so no conversion needed. b) To convert \(-40^{\circ} \mathrm{F}\) to Celsius, we will use the formula: \(C = \frac{5}{9} (F - 32)\) $$C = \frac{5}{9}(-40 - 32) = \frac{5}{9}(-72) = -40^{\circ} \mathrm{C}$$ c) To convert \(233 \mathrm{~K}\) to Celsius, we will use the formula: \(C = K - 273.15\) $$C = 233 - 273.15 = -40^{\circ} \mathrm{C}$$

Compare temperatures

Now we have all temperatures in Celsius: a) \(-40^{\circ} \mathrm{C}\) b) \(-40^{\circ} \mathrm{C}\) c) \(-40^{\circ} \mathrm{C}\) All three temperatures are equal, each at -40 degrees Celsius.

Choose the answer

Since all three temperatures are equal (-40 degrees Celsius), the correct answer is: d) All three are equal.

Key concepts

Celsius to Fahrenheit Conversion

Understanding how to convert Celsius to Fahrenheit is essential when comparing temperatures across different regions or scientific contexts. The formula for converting from Celsius to Fahrenheit is as follows:
\[F = \left( \frac{9}{5} \times C \right) + 32\]
For example, if you have a temperature of 0 degrees Celsius and want to convert it to Fahrenheit, you would perform the following calculation:
\[F = \left( \frac{9}{5} \times 0 \right) + 32 = 32^{\circ} \mathrm{F}\]
This formula helps us grasp that water freezes at 32 degrees Fahrenheit and boils at 212 degrees Fahrenheit, illustrating a common point of reference for everyday life and scientific observations. It's important to remember the numbers 9, 5, and 32 in the formula as these play a key role in the conversion process.

Kelvin to Celsius Conversion

The Kelvin scale is another temperature scale used primarily in the scientific community because it starts at absolute zero, the lowest theoretically possible temperature. Converting Kelvin to Celsius allows us to align scientific measurements with more commonly used temperature scales.
To convert Kelvin to Celsius, you subtract 273.15 from the Kelvin value:
\[C = K - 273.15\]
Let’s say you're given a temperature of 298 K which you want to express in Celsius. By applying the conversion formula, you get:
\[C = 298 - 273.15 = 24.85^{\circ} \mathrm{C}\]
This simple subtraction operation helps bridge the gap between scientific understanding and everyday temperature references. It’s essential for any study involving heat transfer, thermodynamics, or even weather reporting.

Comparing Temperatures

When comparing temperatures, it’s important to convert them to a common scale. Remarkably, at \(-40\) degrees, Celsius and Fahrenheit scales intersect, meaning this temperature is the same in both units.
However, in most cases to effectively compare you must convert all temperatures into either Celsius, Fahrenheit, or Kelvin using the above mentioned conversion formulas. After conversion, you can directly compare the numerical values. For instance, to compare room temperature with body temperature, you would need to express both in the same unit—such as 20°C and 37°C respectively—for an accurate comparison.
Additionally, knowing that the Kelvin scale does not use degrees and starts at absolute zero, which is \(-273.15^{\circ} \mathrm{C}\) or \(0 \mathrm{K}\), is critical. It's intriguing how temperature scales have been designed to serve different purposes but can be translated into one another to simplify our understanding of heat and energy.