Question

Perform the following conversions: (a) \(68^{\circ} \mathrm{F}\) (a pleasant spring day) to \({ }^{\circ} \mathrm{C}\) and \(\mathrm{K}\) (b) \(-164^{\circ} \mathrm{C}\) (the boiling point of methane, the main component of natural gas) to \(\mathrm{K}\) and \({ }^{\circ} \mathrm{F}\) (c) \(0 \mathrm{~K}\) (absolute zero, theoretically the coldest possible temperature) to \({ }^{\circ} \mathrm{C}\) and \({ }^{\circ} \mathrm{F}\)

Short answer

(a) 68°F = 20°C and 293.15K, (b) -164°C = 109.15K and -231.2°F, (c) 0K = -273.15°C and -459.67°F

Step-by-step solution

Convert 68°F to °C

Use the formula \(\frac{5}{9}(\text{F} - 32)\). Substituting the value:\[ \frac{5}{9} (68 - 32) = \frac{5}{9} \times 36 = 20^{\circ} \text{C} \]

Convert 68°F to K

First convert 68°F to °C as previously calculated (20°C). Then use the conversion formula \(\text{C} + 273.15\). Substituting the value: \[20 + 273.15 = 293.15\ K \]

Convert -164°C to K

Use the formula \(\text{C} + 273.15\). Substituting the value: \[-164 + 273.15 = 109.15\ K \]

Convert -164°C to °F

Use the formula \(\frac{9}{5} \text{C} + 32\). Substituting the value: \[\frac{9}{5} \times -164 + 32 = -263.2 + 32 = -231.2^{\circ}\text{F} \]

Convert 0 K to °C

Use the formula \(\text{K} - 273.15\). Substituting the value: \[0 - 273.15 = -273.15^{\circ} \text{C} \]

Convert 0 K to °F

First convert 0 K to °C as previously calculated (-273.15°C). Then use the formula \(\frac{9}{5} \text{C} + 32\). Substituting the value: \[\frac{9}{5} \times (-273.15) + 32 = -459.67^{\circ} \text{F} \]

Key concepts

Fahrenheit to Celsius conversion

Converting from Fahrenheit to Celsius involves a straightforward formula. The formula to use is: \( \text{C} = \frac{5}{9} ( \text{F} - 32 ) \). Here, \( \text{F} \) represents the temperature in Fahrenheit, and \( \text{C} \) represents the temperature in Celsius.
To better understand, let's work through an example. If you have 68°F and want to find the equivalent Celsius temperature, substitute 68 for \( \text{F} \) in the formula:
\[ \text{C} = \frac{5}{9} ( 68 - 32 ) \]
This simplifies to:
\[ \frac{5}{9} \times 36 = 20^{\circ} \text{C} \]
So, 68°F converts to 20°C. Remember this formula as it is commonly used in temperature conversions. Here's a quick summary:

• Subtract 32 from the Fahrenheit temperature
• Multiply the result by \( \frac{5}{9} \)
• The final result is the temperature in Celsius

Celsius to Kelvin conversion

Converting from Celsius to Kelvin is perhaps the simplest of all temperature conversions. The formula is: \( \text{K} = \text{C} + 273.15 \). Here, \( \text{C} \) stands for Celsius, and \( \text{K} \) stands for Kelvin.
Temperature in Kelvin is always 273.15 units higher than in Celsius. Let's go through an example using 20°C:
\[ \text{K} = 20 + 273.15 = 293.15 \]
So, 20°C equals 293.15 K. This formula helps you understand two critical points:

• The Celsius and Kelvin scales have the same unit size
• Changing temperatures on one scale affects the other equally by a shift of 273.15

Remember, Kelvin is often used in scientific calculations due to its absolute nature.

Kelvin to Celsius conversion

To convert Kelvin to Celsius, you just reverse the formula: \( \text{C} = \text{K} - 273.15 \). This is very straightforward, as you are essentially removing the added value of 273.15 from the Kelvin temperature.
Consider you have a temperature of 0 K (absolute zero). To convert it to Celsius:
\[ \text{C} = 0 - 273.15 = -273.15^{\circ} \text{C} \]
So, 0 K equals -273.15°C, which is the lowest possible temperature.

• Subtract 273.15 from the Kelvin temperature to get Celsius
• It is used often in scientific contexts where precision is key

Understanding this conversion ensures you can flip between Kelvin and Celsius with ease.

Celsius to Fahrenheit conversion

For converting Celsius to Fahrenheit, the formula is: \( \text{F} = \frac{9}{5} \text{C} + 32 \).
\( \text{C} \) is the temperature in Celsius, and \( \text{F} \) is the temperature in Fahrenheit.
Suppose you want to convert -164°C to Fahrenheit. Plugging -164 into the formula:
\[ \text{F} = \frac{9}{5} (-164) + 32 = -295.2 + 32 = -231.2^{\circ}\text{F} \]
Therefore, -164°C equals -231.2°F. Remember these steps for conversion:

• Multiply the Celsius temperature by \( \frac{9}{5} \)
• Add 32 to the result
• The final result is the temperature in Fahrenheit

By remembering this formula, you'll be able to quickly switch between Celsius and Fahrenheit.