Question

Perform the indicated temperature conversions. a. \(275 \mathrm{K}\) to \(^{\circ} \mathrm{C}\) b. \(82^{\circ} \mathrm{F}\) to \(^{\circ} \mathrm{C}\) c. \(-21^{\circ} \mathrm{C}\) to \(^{\circ} \mathrm{F}\) d. \(-40^{\circ} \mathrm{F}\) to \(^{\circ} \mathrm{C}\) (Notice anything unusual about your answer?)

Short answer

a. \(1.85 °C\) b. \(27.78 °C\) c. \(-5.8 °F\) d. \(-40 °C\) (Notice: -40°F is equal to -40°C)

Step-by-step solution

a. Convert 275 K to °C

To convert from Kelvin to Celsius, we will use the formula \(T(°C) = T(K) - 273.15\). \(T(°C) = 275 K - 273.15 = 1.85 °C\)

b. Convert 82°F to °C

To convert from Fahrenheit to Celsius, we will use the formula \(T(°C) = \frac{5}{9} \times (T(°F) - 32)\). \(T(°C) = \frac{5}{9} \times (82°F - 32) = \frac{5}{9} \times 50 = 27.78 °C\)

c. Convert -21°C to °F

To convert from Celsius to Fahrenheit, we will use the formula \(T(°F) = \frac{9}{5} \times T(°C) + 32\). \(T(°F) = \frac{9}{5} \times (-21°C) + 32 = -5.8 °F\)

d. Convert -40°F to °C

To convert from Fahrenheit to Celsius, we will again use the formula \(T(°C) = \frac{5}{9} \times (T(°F) - 32)\). \(T(°C) = \frac{5}{9} \times (-40°F - 32) = \frac{5}{9} \times -72 = -40 °C\) Notice that the final conversion (-40°F to -40°C) has the same result in both temperature scales. This is an unusual coincidence, as the Fahrenheit and Celsius scales are not typically equal at any point except for the one mentioned in this exercise.

Key concepts

Kelvin to Celsius conversion

Understanding the shift from Kelvin to Celsius is like learning a simple subtraction. Imagine you have a thermometer that only shows Kelvin, and you want to know the temperature in the more commonly used Celsius scale. To do this, you would take the Kelvin temperature and subtract 273.15 from it. Let's use a number as an example: if your thermometer reads 300 Kelvin, you would calculate it as follows:

\( T(^\text{o}C) = T(K) - 273.15 \)
\( T(^\text{o}C) = 300K - 273.15 = 26.85^\text{o}C \)
Therefore, a balmy 300 Kelvin day is about 27 degrees Celsius. Why subtract 273.15? Because that's the difference in degrees between the two scales' zero points. 0 Kelvin, known as absolute zero, is equivalent to -273.15 degrees Celsius. This is the theoretical lowest temperature possible, where particles have minimal thermal movement.

Fahrenheit to Celsius conversion

Switching from Fahrenheit to Celsius is akin to converting miles to kilometers; it involves a bit of multiplication and addition (or subtraction). American friends often talk in Fahrenheit and to understand what they mean in Celsius, you'd use the following formula:

\( T(^\text{o}C) = \left(\frac{5}{9}\right) \times (T(^\text{o}F) - 32) \)
If your American cousin says it's 90°F outside, you'd convert it like this:
\( T(^\text{o}C) = \left(\frac{5}{9}\right) \times (90^\text{o}F - 32) = \left(\frac{5}{9}\right) \times 58 = 32.22^\text{o}C \)
That 90°F suddenly turns into a hot 32°C day. The magic numbers here are 5/9 and 32, which act as the 'exchange rate' between these temperature 'currencies.' 32 is used because that's the temperature in Fahrenheit when water freezes, which is 0 degrees in Celsius.

Celsius to Fahrenheit conversion

Going from Celsius back to Fahrenheit is like baking a cake; you need the right ingredients and a little heat! To explain this to your friend who only talks in Celsius when you’re used to Fahrenheit, you’ll need to multiply their Celsius temperature by 9/5 and then add 32. So when they tell you it's 25°C, you would convert it as such:

\( T(^\text{o}F) = \left(\frac{9}{5}\right) \times T(^\text{o}C) + 32 \)
\( T(^\text{o}F) = \left(\frac{9}{5}\right) \times 25^\text{o}C + 32 = 45 + 32 = 77^\text{o}F \)
Now, a pleasant 25°C equals a warm 77°F. The addition of 32 in this 'recipe' aligns the scales so that the point at which water freezes (0°C) is the same as 32°F.

Temperature scales

Temperature scales are like rulers for measuring how hot or cold something is, but depending on where you live or what you're measuring, you'll use a different kind of 'ruler.' The three main scales are Celsius, Fahrenheit, and Kelvin. Celsius is like a metric ruler, used mostly around the world for daily weather and in scientific contexts alongside Kelvin. Fahrenheit is the old imperial ruler, which is still used in the United States for most day-to-day temperature readings. Kelvin, on the other hand, is the ruler for scientists, especially in physics, because it starts at zero where theoretically nothing could be colder (absolute zero).

• Celsius (°C) scale is based on the properties of water, with 0°C as the freezing point and 100°C as the boiling point at sea level.
• Fahrenheit (°F) scale also uses the properties of water but sets the freezing point at 32°F and boiling at 212°F.
• Kelvin (K) scale is an absolute scale where 0 K is absolute zero, and it increases incrementally in the same size as Celsius degrees.

Understanding these scales and how to convert between them is crucial for scientific studies, cooking recipes, or simply talking about the weather with friends across the globe.