Problem 41 - General Chemistry: Principles and Modern Applications

Question

We want to mark off a thermometer in both Celsius and Fahrenheit temperatures. On the Celsius scale, the lowest temperature mark is at \(-10^{\circ} \mathrm{C},\) and the highest temperature mark is at \(50^{\circ} \mathrm{C} .\) What are the equivalent Fahrenheit temperatures?

Step-by-Step Solution

Answer

-10 degrees Celsius is equal to 14 degrees Fahrenheit and 50 degrees Celsius is equal to 122 degrees Fahrenheit.

Step 1: Convert -10 degrees Celsius to Fahrenheit

Apply the formula \( F = (C * 9/5) + 32 \) by replacing C with -10 to get the Fahrenheit equivalent. The calculation is as follows: \( F = (-10 * 9/5) + 32 \).

Step 2: Simplify the conversion of -10 degrees Celsius to Fahrenheit

Perform the calculations to get the Fahrenheit equivalent. This comes to \( F = -18 + 32 \) which simplifies further to \( F = 14 \). Therefore, -10 degrees Celsius is equal to 14 degrees Fahrenheit.

Step 3: Convert 50 degrees Celsius to Fahrenheit

Again use the formula \( F = (C * 9/5) + 32 \), this time replacing C with 50. You will get this equation: \( F = (50 * 9/5) + 32 \). Remember to perform the multiplication before the addition.

Step 4: Simplify the conversion of 50 degrees Celsius to Fahrenheit

Perform the calculations to find the Fahrenheit equivalent. This comes to \( F = 90 + 32 \) which further simplifies to \( F = 122 \). Therefore, 50 degrees Celsius equals 122 degrees Fahrenheit.

Key Concepts

Celsius to Fahrenheit

Understanding how to convert temperatures between Celsius and Fahrenheit is essential for many scientific and daily applications. The key to this conversion is the relationship between these two units.

Celsius, often used worldwide, sets 0°C as the freezing point and 100°C as the boiling point of water. In contrast, Fahrenheit, mainly used in the United States, sets 32°F as the freezing point and 212°F as the boiling point.

To convert a Celsius temperature to Fahrenheit, use the formula:

• \( F = (C \times \frac{9}{5}) + 32 \)

This formula works by first multiplying the Celsius temperature by 9/5, which accounts for the difference in the number of degrees between boiling and freezing points on each scale. Then, add 32 to realign the starting points of the scales.

Mathematical Formula

The conversion formula, \( F = (C \times \frac{9}{5}) + 32 \), is a simple yet powerful tool. It's derived from the differences between the scales and their starting points. Let's break it down further:

The term \( C \times \frac{9}{5} \) converts Celsius into a Fahrenheit "rate," because there are 1.8 (or 9/5) Fahrenheit degrees for each Celsius degree.

By adding 32, you're adjusting for the point at which the two scales align. Remember that Celsius and Fahrenheit start counting from different base points (0°C vs. 32°F), hence the addition of 32.

Here's how you might use it:

• Start with the Celsius temperature.
• Multiply it by 9/5.
• Add 32 to the result.

This sequence of operations ensures you accurately convert temperatures between scales.

Thermometer Calibration

Thermometer calibration is the process of marking a thermometer so it correctly reflects Celsius and Fahrenheit readings. Calibration is crucial to ensure accurate and reliable temperature readings.

When calibrating, you must consider both scales and their conversion formula. Start by determining known reference points, such as the freezing and boiling points of water:

• At 0°C (or 32°F), the thermometer should read freezing.
• At 100°C (or 212°F), it should read boiling.

These fixed points help in aligning the scales properly across the thermometer.

Regular calibration checks, especially in scientific environments, ensure the thermometer's accuracy. This involves comparing its readings against these standards and adjusting as needed to maintain precision.