Problem 2 - Chemistry and Global Awareness

Question

a. Convert \(0^{\circ} \mathrm{F}\) to degrees Celsius. b. Convert \(212^{\circ} \mathrm{C}\) to degrees Fahrenheit.

Step-by-Step Solution

Answer

a. -17.78°C; b. 413.6°F.

Step 1: Understand the Conversion Formula for Fahrenheit to Celsius

The formula used to convert temperatures from Fahrenheit to Celsius is given by:\[C = \frac{5}{9}(F - 32)\]where \(C\) is the temperature in degrees Celsius and \(F\) is the temperature in degrees Fahrenheit.

Step 2: Substitute 0°F into the Conversion Formula

We need to convert \(0^{\circ} \mathrm{F}\) to Celsius. Substitute \(F = 0\) into the formula for converting Fahrenheit to Celsius:\[C = \frac{5}{9}(0 - 32)\]

Step 3: Calculate the Celsius Temperature

Evaluate the expression:\[C = \frac{5}{9}(-32) = \frac{-160}{9} \approx -17.78^{\circ} \mathrm{C}\]Thus, \(0^{\circ} \mathrm{F}\) is approximately \(-17.78^{\circ} \mathrm{C}\).

Step 4: Understand the Conversion Formula for Celsius to Fahrenheit

The formula used to convert temperatures from Celsius to Fahrenheit is given by:\[F = \frac{9}{5}C + 32\]where \(F\) is the temperature in degrees Fahrenheit and \(C\) is the temperature in degrees Celsius.

Step 5: Substitute 212°C into the Conversion Formula

We are now tasked with converting \(212^{\circ} \mathrm{C}\) to Fahrenheit. Substitute \(C = 212\) into the formula above:\[F = \frac{9}{5} \times 212 + 32\]

Step 6: Calculate the Fahrenheit Temperature

Evaluate the expression:\[F = \frac{9}{5} \times 212 + 32 = 381.6 + 32 = 413.6^{\circ} \mathrm{F}\]Thus, \(212^{\circ} \mathrm{C}\) is equivalent to \(413.6^{\circ} \mathrm{F}\).

Key Concepts

Fahrenheit to Celsius Conversion

Converting temperatures from Fahrenheit to Celsius is a straightforward process using a specific formula. To make this conversion, use the formula:

• \( C = \frac{5}{9}(F - 32) \)

Here, \(C\) represents the temperature in degrees Celsius, and \(F\) is the temperature in degrees Fahrenheit. This formula involves two main steps:

• Subtract 32 from the Fahrenheit temperature.
• Multiply the result by \(\frac{5}{9}\).

This process helps in translating the Fahrenheit scale, used mainly in the United States, to the Celsius scale, which is widely used worldwide and in scientific settings.

Celsius to Fahrenheit Conversion

To convert a temperature from Celsius to Fahrenheit, a different formula is used. This gives us insight into how temperature scales vary around the world. The formula is:

• \( F = \frac{9}{5}C + 32 \)

This equation showcases two key operations:

• Multiply the Celsius temperature by \(\frac{9}{5}\).
• Add 32 to the result.

By performing these steps, we translate Celsius temperatures, commonly used in countries other than the United States, into the Fahrenheit scale.

Temperature Calculation

Calculating temperature using these formulas involves substitution and arithmetic operations. Let's look at a step-by-step scenario:

• Select the correct formula based on the conversion needed.
• Substitute the known temperature value into the formula.
• Perform the arithmetic to find the converted temperature.

Take, for example, converting \(0^{\circ} \mathrm{F}\) to Celsius. By plugging \(0\) into the Fahrenheit to Celsius formula, we perform the operations to find \(-17.78^{\circ} \mathrm{C}\). This process involves subtraction and fraction multiplication for precise calculation.

Conversion Formulas

The conversion formulas play an essential role in understanding and calculating temperature changes between Celsius and Fahrenheit. Here's a quick comparison of these formulas:

• Fahrenheit to Celsius: \( C = \frac{5}{9}(F - 32) \)
• Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \)

Both formulas are designed to adjust for the differences in the degree intervals and starting points of each temperature scale. Fahrenheit to Celsius focuses on subtracting and scaling down, while Celsius to Fahrenheit emphasizes scaling up and offsetting. Mastery of these formulas allows the prediction and understanding of temperature in various scientific, culinary, and everyday applications, ensuring clarity and consistency in temperature-related calculations.