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. \(7 \times 10^{7}\) F, temperature required to initiate fusion reactions in the sun

Short answer

a. -459°F ≈ -273.33°C and 0K b. -40°F = -40°C and 233.15K c. 68°F = 20°C and 293.15K d. \( 7 \times 10^{7} \)°F ≈ \( 3.89 \times 10^{7} \)°C and \( 3.89 \times 10^{7} \)K

Step-by-step solution

Convert Fahrenheit to Celsius

\ Using the formula, we have: \( C = \frac{5}{9} (-459 - 32) = \frac{5}{9} (-491) = -273.33 \)

Convert Celsius to Kelvin

\ Now, using the second formula: \( K = -273.33 + 273.15 = 0.18 \approx 0 \) So, -459°F is approximately equal to -273.33°C and 0K. b. Convert -40°F to Celsius and Kelvin:

Convert Fahrenheit to Celsius

\ Using the formula, we have: \( C = \frac{5}{9} (-40 - 32) = \frac{5}{9} (-72) = -40 \)

Convert Celsius to Kelvin

\ Now, using the second formula: \( K = -40 + 273.15 = 233.15 \) So, -40°F is equal to -40°C and 233.15K. c. Convert 68°F to Celsius and Kelvin:

Convert Fahrenheit to Celsius

\ Using the formula, we have: \( C = \frac{5}{9} (68 - 32) = \frac{5}{9} (36) = 20 \)

Convert Celsius to Kelvin

\ Now, using the second formula: \( K = 20 + 273.15 = 293.15 \) So, 68°F is equal to 20°C and 293.15K. d. Convert \( 7 \times 10^{7} \)°F to Celsius and Kelvin:

Convert Fahrenheit to Celsius

\ Using the formula, we have: \( C = \frac{5}{9} (7 \times 10^{7} - 32) = \frac{5}{9} (7 \times 10^{7} - 32) \approx 3.89 \times 10^{7} \)

Convert Celsius to Kelvin

\ Now, using the second formula: \( K = 3.89 \times 10^{7} + 273.15 \approx 3.89 \times 10^{7} \) So, \( 7 \times 10^{7} \)°F is approximately equal to \( 3.89 \times 10^{7} \)°C and \( 3.89 \times 10^{7} \)K.

Key concepts

Fahrenheit to Celsius

Temperature conversion between Fahrenheit and Celsius requires using a specific formula. This formula helps us understand the relationship between these two temperature scales. To convert a temperature from Fahrenheit to Celsius, we use the formula:\[ C = \frac{5}{9} (F - 32) \]Here, "C" stands for Celsius and "F" stands for Fahrenheit. By following this calculation, you are essentially adjusting the Fahrenheit temperature by subtracting 32 and then scaling it by \( \frac{5}{9} \). This scaling reflects the difference in the size of degrees between the two scales.

• Subtract 32 from your Fahrenheit temperature. This step gives you the zero-reference adjustment.
• Multiply the result by \( \frac{5}{9} \). This step converts the scale from Fahrenheit to Celsius.

It's important for students to understand this conversion because both Fahrenheit and Celsius are widely used in everyday life.

Celsius to Kelvin

Once you have a temperature in Celsius, converting it to Kelvin is straightforward. The Kelvin scale is the scientific standard for measuring temperature. Unlike Celsius or Fahrenheit, Kelvin doesn't use degrees, but is instead an absolute scale starting from absolute zero.To convert Celsius to Kelvin, you simply use:\[ K = C + 273.15 \]This equation adds 273.15 to the Celsius temperature. This is because zero Kelvin is absolute zero, which is -273.15 on the Celsius scale.

• Add 273.15 to your Celsius temperature.
• This gives you the temperature in Kelvin, which better suits scientific calculations and formulas.

The Kelvin scale is particularly useful in scientific contexts where understanding extreme temperatures and conditions, like those found in outer space or in fusion reactions, is crucial.

Temperature Scales

Temperature measurements use various scales, with the most common being Celsius, Fahrenheit, and Kelvin. Each of these scales serves different purposes and uses different reference points.

• Celsius is widely used in most of the world and is especially common in weather reports and cooking. Its water freezing point is 0°C and boiling point is 100°C.
• Fahrenheit is primarily used in the United States. Water freezes at 32°F and boils at 212°F on this scale, making it more aligned with everyday temperature experiences.
• Kelvin is used extensively in the scientific community. This scale starts at absolute zero (0 K), the conceptual point at which molecular movement ceases. It's crucial for scientific equations and understanding matters at extreme temperatures.

Understanding these scales helps in converting temperatures and grasping the principles of heat and energy.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers more conveniently. It’s widely used in science and engineering to simplify these numbers.For example, the temperature of \( 7 \times 10^{7} \)°F is in scientific notation. It represents a very high temperature, such as those conditions needed to initiate nuclear fusion in the sun. Instead of writing 70,000,000, scientific notation shortens it to \( 7 \times 10^{7} \).

• The part before the multiplication sign (coefficients like 7) represents the significant figures.
• The exponent (like \( 10^{7} \)) tells you how many spaces to move the decimal point.

Understanding scientific notation allows students to work with extreme values in a more manageable format, which is vital across various scientific disciplines, from astronomy to physics.