Question

Convert the following Fahrenheit temperatures to the Celsius and Kelvin scales. a. \(-459^{\circ} \mathrm{F}\), an extremely low temperature b. \(-40 .{ }^{\circ} \mathrm{F}\), the answer to a trivia question c. \(68^{\circ} \mathrm{F}\), room temperature d. \(7 \times 10^{7}{ }^{\circ} \mathrm{F}\), temperature required to initiate fusion reactions in the sun

Short answer

a. \(-459^{\circ} \mathrm{F}\) is equal to \(-273.15^{\circ} \mathrm{C}\) and \(0\) Kelvin. b. \(-40^{\circ} \mathrm{F}\) is equal to \(-40^{\circ} \mathrm{C}\) and \(233.15\) Kelvin. c. \(68^{\circ} \mathrm{F}\) is equal to \(20^{\circ} \mathrm{C}\) and \(293.15\) Kelvin. d. \(7 \times 10^{7}{ }^{\circ} \mathrm{F}\) is equal to approximately \(3.89 \times 10^{7}^{\circ} \mathrm{C}\) and \(3.89 \times 10^{7}\) Kelvin.

Step-by-step solution

a. Convert \(-459^{\circ} \mathrm{F}\) to Celsius and Kelvin.

First, convert the Fahrenheit temperature to Celsius using the formula: \(C = \frac{5}{9}(F - 32)\) \(C = \frac{5}{9}(-459 - 32)\) \(C = -273.15^{\circ}\mathrm{C}\) Next, convert the Celsius temperature to Kelvin using the formula: \(K = C + 273.15\) \(K = -273.15 + 273.15\) \(K = 0\) So, \(-459^{\circ} \mathrm{F}\) is equal to \(-273.15^{\circ} \mathrm{C}\) and \(0\) Kelvin.

b. Convert \(-40^{\circ} \mathrm{F}\) to Celsius and Kelvin.

First, convert the Fahrenheit temperature to Celsius using the formula: \(C = \frac{5}{9}(F - 32)\) \(C = \frac{5}{9}(-40 - 32)\) \(C = -40^{\circ}\mathrm{C}\) Next, convert the Celsius temperature to Kelvin using the formula: \(K = C + 273.15\) \(K = -40 + 273.15\) \(K = 233.15\) So, \(-40^{\circ} \mathrm{F}\) is equal to \(-40^{\circ} \mathrm{C}\) and \(233.15\) Kelvin.

c. Convert \(68^{\circ} \mathrm{F}\) to Celsius and Kelvin.

First, convert the Fahrenheit temperature to Celsius using the formula: \(C = \frac{5}{9}(F - 32)\) \(C = \frac{5}{9}(68 - 32)\) \(C = 20^{\circ}\mathrm{C}\) Next, convert the Celsius temperature to Kelvin using the formula: \(K = C + 273.15\) \(K = 20 + 273.15\) \(K = 293.15\) So, \(68^{\circ} \mathrm{F}\) is equal to \(20^{\circ} \mathrm{C}\) and \(293.15\) Kelvin.

d. Convert \(7 \times 10^{7}{ }^{\circ} \mathrm{F}\) to Celsius and Kelvin.

First, convert the Fahrenheit temperature to Celsius using the formula: \(C = \frac{5}{9}(F - 32)\) \(C = \frac{5}{9}(7 \times 10^{7} - 32)\) \(C \approx 3.89 \times 10^{7}^{\circ}\mathrm{C}\) Next, convert the Celsius temperature to Kelvin using the formula: \(K = C + 273.15\) \(K = 3.89 \times 10^{7} + 273.15\) \(K \approx 3.89 \times 10^{7}\) So, \(7 \times 10^{7}{ }^{\circ} \mathrm{F}\) is equal to approximately \(3.89 \times 10^{7}^{\circ} \mathrm{C}\) and \(3.89 \times 10^{7}\) Kelvin.

Key concepts

Fahrenheit to Celsius

Converting temperatures from Fahrenheit to Celsius is a common task, especially in countries that use the metric system. The formula for this conversion is relatively straightforward:

• Use the formula: \( C = \frac{5}{9}(F - 32) \)
• This means to first subtract 32 from the Fahrenheit temperature
• Then multiply the result by \( \frac{5}{9} \) to get the Celsius temperature

This conversion is based on the relationship between the boiling and freezing points of water. For instance, water freezes at 32°F and boils at 212°F, which corresponds to 0°C and 100°C, respectively. Here are some practical examples:
- Temperature such as \(-40°F\) converts directly to \(-40°C\) because this is the unique point where the Celsius and Fahrenheit scales intersect.- If you have a temperature of \(68°F\) (like room temperature), the conversion will give you \(20°C\).Understanding how to convert between these two scales is useful for comparing weather forecasts, cooking temperatures, and scientific experiments.

Celsius to Kelvin

The Kelvin scale is particularly used in scientific contexts, and converting Celsius to Kelvin is a crucial step in understanding thermodynamics. Here's how you do it:

• The conversion formula is simple: \( K = C + 273.15 \)
• This means you add 273.15 to the Celsius temperature

Why do we add 273.15? The Kelvin scale starts at absolute zero, the theoretical point where all thermal motion ceases, which is equivalent to \(-273.15°C\). Since the size of one unit (or "degree") on the Kelvin scale is the same as on the Celsius scale, this straightforward addition converts the temperature. Here are some examples:
- If \(-273.15°C\) is converted, it becomes 0 K—absolute zero, a critical concept in physics.- A comfortable room temperature, which is around \(20°C\), converts to \(293.15K\).This conversion is important to ensure consistency in scientific calculations, especially those relating to gas laws and chemical reactions.

Kelvin scale

The Kelvin scale is an absolute temperature scale where zero Kelvin represents absolute zero, the lowest possible temperature. Unlike Celsius and Fahrenheit, the Kelvin scale doesn't have degrees; it is just "Kelvin." Here’s what you need to know:

• Invented by Lord Kelvin, this scale was designed to extend the concept of temperature down to absolute zero.
• On the Kelvin scale, water freezes at 273.15 K and boils at 373.15 K under standard atmospheric conditions.
• This scale is primarily used in scientific fields such as physics and chemistry.

A few points of interest:
- Because it is an absolute scale, zero Kelvin (0 K) is thought to be unattainably cold, a point at which molecular motion stops entirely. - The Kelvin scale is essential in the field of cryogenics, where researchers study materials at incredibly low temperatures. Understanding the Kelvin scale will enhance comprehension of energy and particle behavior at different temperatures, which is crucial in theoretical and applied sciences.