Problem 86 - Chemistry Principles and Practice

Question

At what temperature does a Celsius thermometer give the same numerical reading as a Fahrenheit thermometer?

Step-by-Step Solution

Answer

The same reading occurs at -40 degrees.

Step 1: Understanding the Problem

We need to find the temperature at which the Celsius and Fahrenheit scales show the same numerical value. For this, we need to set the two temperature expressions equal to each other.

Step 2: Temperature Conversion Formula

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

Step 3: Setting Up the Equation

Since we want the Celsius and Fahrenheit readings to be equal, we set \( F = C \). So, the equation becomes: \( C = \frac{9}{5}C + 32 \).

Step 4: Isolating the Variable

To find \( C \), we need to isolate it on one side of the equation. Start by subtracting \( \frac{9}{5}C \) from both sides: \( C - \frac{9}{5}C = 32 \).

Step 5: Simplifying the Equation

Combine the terms on the left: \( \frac{5}{5}C - \frac{9}{5}C = 32 \), which simplifies to \( \frac{-4}{5}C = 32 \).

Step 6: Solving for \( C \)

Multiply both sides by \(-\frac{5}{4}\) to solve for \( C \): \( C = 32 \times -\frac{5}{4} \).

Step 7: Final Calculation

Perform the multiplication: \( C = -40 \). Both the Celsius and Fahrenheit scales give a reading of -40.

Key Concepts

Celsius Scale

The Celsius scale is a metric system temperature scale, used widely across the globe for scientific and daily temperature measurements. Named after the Swedish astronomer Anders Celsius, this scale is based on two fixed points: the freezing point of water at 0 degrees Celsius and the boiling point of water at 100 degrees Celsius.

It is characterized by its simplicity and consistency with the metric system. Each degree Celsius represents the same temperature change as a degree on the Kelvin scale, which is specifically used in scientific contexts.

• Water freezes at 0°C
• Water boils at 100°C

Moreover, the Celsius scale is linear, which means the temperature increment between each degree is uniform. This makes it user-friendly and easy to understand, especially when compared to other scales that are not decimal based.

Fahrenheit Scale

The Fahrenheit scale, unlike the Celsius scale, is primarily used in the United States for weather forecasts and some industries. It was developed by Daniel Gabriel Fahrenheit, a physicist from Poland. This scale sets the freezing point of water at 32 degrees Fahrenheit and the boiling point at 212 degrees Fahrenheit.

This setup creates a 180-degree span between these two crucial points, offering a different sense of temperature changes compared to Celsius. One key aspect to remember is:

• Water freezes at 32°F
• Water boils at 212°F

Understanding the Fahrenheit scale can initially be a bit perplexing because it doesn't work on a straightforward base-10 system. Despite this, it remains useful in contexts that require a detailed reading due to its finer granulation between degree increments.

Temperature Equivalence

Finding the point at which two temperatures on different scales are equivalent can be quite fascinating! In the case of Celsius and Fahrenheit, they coincide at -40 degrees. This is perhaps the most famous example of temperature equivalence between the two scales.

How does one arrive at this point? By setting up the relationship between Celsius (C) and Fahrenheit (F) with the formula \( F = \frac{9}{5}C + 32 \), where F equals C at this specific junction. By mathematically isolating the variable \( C \), we solve:

• Equation: \( C = \frac{9}{5}C + 32 \)
• Solve \( C = -40 \)
• Verification through calculation shows both scales agree.

This special intersection at -40 is intriguing because it allows for the direct translation of temperatures where both scales express the numerical reading without conversion, an occurrence not often found given the scales' varying intervals and units.