Problem 1 - Chemistry and Global Awareness

Question

a. What is \(98.6^{\circ} \mathrm{F}\) in degrees Celsius? b. What is \(25.0^{\circ} \mathrm{C}\) in degrees Fahrenheit?

Step-by-Step Solution

Answer

a. 37°C; b. 77°F.

Step 1: Understand the Fahrenheit to Celsius Formula

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

Step 2: Plug Fahrenheit Value into Formula

Using the formula from Step 1, substitute \(F = 98.6\) to convert to Celsius:\[C = \frac{5}{9} (98.6 - 32)\]

Step 3: Calculate the Celsius Temperature

First, perform the subtraction inside the parentheses: \(98.6 - 32 = 66.6\).Next, multiply by \(\frac{5}{9}\):\[C = \frac{5}{9} \times 66.6 \approx 37 \]Therefore, \(98.6^{\circ} \mathrm{F} = 37^{\circ} \mathrm{C}\).

Step 4: Understand the Celsius to Fahrenheit Formula

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

Step 5: Plug Celsius Value into Formula

Using the formula from Step 4, substitute \(C = 25.0\) to convert to Fahrenheit:\[F = \frac{9}{5} \times 25.0 + 32\]

Step 6: Calculate the Fahrenheit Temperature

First, multiply \(25.0\) by \(\frac{9}{5}\), which is \(45.0\).Next, add \(32\) to the result:\[F = 45.0 + 32 = 77.0\]Therefore, \(25.0^{\circ} \mathrm{C} = 77.0^{\circ} \mathrm{F}\).

Key Concepts

Fahrenheit to Celsius

The conversion from Fahrenheit to Celsius is a common task you may encounter in science or everyday life, especially if you are dealing with weather forecasts or recipes from different parts of the world. The formula used to convert Fahrenheit to Celsius is \[C = \frac{5}{9} (F - 32)\].
This equation shows that you first subtract 32 from the Fahrenheit value. This step is crucial because 32 is the freezing point of water in Fahrenheit.

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

By applying this formula, you can easily switch from Fahrenheit to Celsius, which is useful when trying to understand temperatures better in Celsius-oriented countries.

Celsius to Fahrenheit

Sometimes you might need to convert temperatures from Celsius to Fahrenheit. This is common in travel or science contexts when readings or instructions are provided in Celsius, but you are more familiar with Fahrenheit. The formula to use is \[F = \frac{9}{5} C + 32\].
Just like Celsius to Fahrenheit, you will need to handle the multiplication and addition:

• Multiply the Celsius temperature by \(\frac{9}{5}\).
• Add 32 to adjust for the starting point difference between the two scales.

This formula helps retain the accuracy while converting, ensuring you understand the warmth or coldness in a context familiar to you.

Temperature Formulas

Understanding these temperature conversion formulas is crucial for numerous scientific calculations and daily life activities. Having these formulas at your fingertips aids in accurately changing temperature readings.
For Fahrenheit to Celsius: - Use \(C = \frac{5}{9} (F - 32)\).
For Celsius to Fahrenheit: - Use \(F = \frac{9}{5} C + 32\).
These formulas not only help in conversion but also in understanding the mathematical relationship between the two temperature scales. They depict the linear relationship where each degree increment in Celsius or Fahrenheit corresponds to a fixed decimal increment in the other scale.

Mathematical Calculations

Mathematical calculations play a vital role in temperature conversions. The conversion process involves basic arithmetic operations such as subtraction, multiplication, and addition. Understanding these operations can help you navigate the conversion process smoothly.
When performing such calculations:

• Be attentive to the order of operations: subtraction comes before multiplication when using the Fahrenheit to Celsius formula.
• Always double-check your calculations to avoid errors.

These basic steps ensure precision, especially when temperatures need to be exact, such as in scientific experiments or culinary tasks. Mastering these calculations can make temperature conversion simple and straightforward.