Problem 106 - Chemistry

Question

At a certain point, the Celsius and Fahrenheit scales "cross" giving the same numerical value on both. At what temperature does this crossover occur?

Step-by-Step Solution

Answer

Celsius and Fahrenheit are equal at -40 degrees.

Step 1: Understand the relationship between Celsius and Fahrenheit

The formula to convert Celsius (C) to Fahrenheit (F) is given by \( F = \frac{9}{5}C + 32 \). We need to find the temperature where the Celsius and Fahrenheit readings are the same, thus \( F = C \).

Step 2: Set up the equation where Celsius equals Fahrenheit

In this situation, since we want to find when the values are the same, we replace \( F \) with \( C \) in the formula: \[ C = \frac{9}{5}C + 32 \].

Step 3: Solve for the temperature

To solve the equation \( C = \frac{9}{5}C + 32 \), first subtract \( \frac{9}{5}C \) from both sides: \[ C - \frac{9}{5}C = 32 \]. This simplifies to \( \frac{5}{5}C - \frac{9}{5}C = 32 \), or \( -\frac{4}{5}C = 32 \).

Step 4: Simplify and solve for C

Multiply both sides by \(-\frac{5}{4}\) to isolate \( C \): \[ C = 32 \times -\frac{5}{4} \]. Simplifying \( 32 \times -\frac{5}{4} \) gives \( C = -40 \).

Step 5: Verify the solution makes sense

Substituting \( C = -40 \) back into the original formula \( F = \frac{9}{5}C + 32 \), you get \( F = \frac{9}{5}(-40) + 32 = -72 + 32 = -40\), confirming that the temperature is the same for both Celsius and Fahrenheit at -40.

Key Concepts

Celsius to Fahrenheit conversion

Converting Celsius to Fahrenheit involves understanding the relationship between these two temperature scales. The formula to convert a temperature from Celsius (C) to Fahrenheit (F) is: \[ F = \frac{9}{5}C + 32 \] This equation means that to find the temperature in Fahrenheit, you multiply the Celsius temperature by \( \frac{9}{5} \) and then add 32. This adjustment accounts for the different starting points and divisions in each scale compared to each other. Understanding this formula helps us see how temperature readings translate on both scales and aids in performing accurate conversions. Moreover, knowing this conversion formula is essential for those who work in fields where both scales are common, such as science and engineering. When using this formula, always remember:

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

This straightforward process will lead you to correct Fahrenheit values.

Temperature scales

Temperature scales are systems used to measure temperature. The two most common scales are Celsius and Fahrenheit. These scales are widely used across different regions and fields.

• Celsius Scale: Used mainly in most parts of the world, this scale defines 0°C as the freezing point and 100°C as the boiling point of water, under standard atmospheric conditions.
• Fahrenheit Scale: Primarily used in the United States, it sets the freezing point of water at 32°F and the boiling point at 212°F.

The difference in scale intervals is a result of different base points for each system. Celsius is based on the properties of water, while Fahrenheit uses a different system initially based on salt mixed with water's freezing and body temperature approximations. It's important to note that each scale is suited to differing climatic conditions and cultural preferences, which can affect international communication regarding temperature. Understanding both allows for accurate scientific communication.

Solving equations

Equations are mathematical statements that assert the equality of two expressions. Solving equations involves finding the values of the unknowns that make the equation true. In this particular problem, we need to find the temperature at which Celsius and Fahrenheit readings are equal. The key is setting the equation correctly. Use the conversion formula and equate Celsius (C) to Fahrenheit (F) because we seek the point where both scales read the same numerically: \[ C = \frac{9}{5}C + 32 \] Here's how we solve the equation step by step:

• Subtract \( \frac{9}{5}C \) from both sides: This simplifies to \( \frac{5}{5}C - \frac{9}{5}C = 32 \)
• Simplify further: This leads to \( -\frac{4}{5}C = 32 \)
• Isolate C by multiplying by \( -\frac{5}{4} \): You'll find \( C = 32 \times -\frac{5}{4} \) which calculates to \(-40\)

Through solving equations, algebraic manipulation is used to isolate the variable and reach the solution. The steps involve simplifying expressions and systematically performing operations on both sides of the equation to maintain balance. Each step must be done carefully to ensure accuracy in the final solution.