Problem 8 - University Physics with Modern Physics

Question

(a) Calculate the one temperature at which Fahrenheit and Celsius thermometers agree with each other. (b) Calculate the one temperature at which Fahrenheit and Kelvin thermometers agree with each other.

Step-by-Step Solution

Answer

(a) -40 degrees, (b) 576.09 Kelvin

Step 1: Understanding the Relationship Between Fahrenheit and Celsius

The relationship between Fahrenheit (F) and Celsius (C) is given by the conversion formula: \[ F = \frac{9}{5}C + 32 \]. To find the temperature at which Fahrenheit and Celsius are equal, we set \( F = C \).

Step 2: Set Up the Equation for Fahrenheit and Celsius Agreement

Replace \( F \) with \( C \) in the equation: \[ C = \frac{9}{5}C + 32 \].

Step 3: Solve for Celsius in the Fahrenheit-Celsius Equation

Subtract \( \frac{9}{5}C \) from both sides to get \[ C - \frac{9}{5}C = 32 \]. This simplifies to \[ -\frac{4}{5}C = 32 \]. Multiply both sides by \(-\frac{5}{4}\) to solve for \( C \): \[ C = -40 \].

Step 4: Conclude the Agreement of Fahrenheit and Celsius

The temperature at which both Fahrenheit and Celsius thermometers read the same is \(-40\) degrees.

Step 5: Understanding the Relationship Between Fahrenheit and Kelvin

The relationship between Fahrenheit (F) and Kelvin (K) can be established using Celsius as an intermediary: \( C = K - 273.15 \) and \( F = \frac{9}{5}C + 32 \). So we need \( F = K \).

Step 6: Set Up the Equation for Fahrenheit and Kelvin Agreement

Substitute \( C = K - 273.15 \) in the Fahrenheit equation: \[ F = \frac{9}{5}(K - 273.15) + 32 \]. Simplify it to \( F = K \): \[ K = \frac{9}{5}(K - 273.15) + 32 \].

Step 7: Solve for Kelvin in the Fahrenheit-Kelvin Equation

Distribute and simplify the equation: \[ K = \frac{9}{5}K - \frac{9}{5} \times 273.15 + 32 \]. This simplifies to: \[ K = \frac{9}{5}K - 491.67 + 32 \]. Rearrange to get: \[ 5K = 9K - 2304.35 \]. Simplifying further: \[ -4K = -2304.35 \]. Solve for \( K \): \[ K = 576.0875 \].

Step 8: Conclude the Agreement of Fahrenheit and Kelvin

The temperature at which both Fahrenheit and Kelvin thermometers read the same is approximately \( 576.09 \) Kelvin.

Key Concepts

Fahrenheit and Celsius relationship

Many people wonder about the point where Fahrenheit and Celsius temperature scales meet. It's fascinating because these two scales are widely used around the world. Each has its unique way of measuring temperature.
The relationship between these scales is governed by the formula:

• \[ F = \frac{9}{5}C + 32 \]

This equation represents the conversion method from Celsius (\( C \)) to Fahrenheit (\( F \)). To find the temperature where both Fahrenheit and Celsius equal each other, you set \( F = C \).
By substituting in the equation, it simplifies to:

• \[ -\frac{4}{5}C = 32 \]

From here, solving gives us \( C = -40 \).
Thus, at \-40 degrees, both Fahrenheit and Celsius thermometers agree.

Fahrenheit and Kelvin relationship

Have you ever wondered at which point Fahrenheit and Kelvin scales show the same number? Although Fahrenheit and Kelvin might seem unrelated, they can be connected through Celsius.
Here's how:

• Start with the Celsius-Kelvin relationship: \( C = K - 273.15 \)
• Utilize the Fahrenheit conversion formula \( F = \frac{9}{5}C + 32 \)

Now, for Fahrenheit and Kelvin to match, we set \( F = K \).
Substituting leads to:

• \[ K = \frac{9}{5}(K - 273.15) + 32 \]

By solving this equation for \( K \), it simplifies down to \( K = 576.09 \).
Therefore, both Fahrenheit and Kelvin thermometers show the same temperature at approximately 576.09 Kelvin.

Solving temperature equations

Temperature equations can sometimes be a head-scratcher for students.
Understanding how to solve these equations is essential when moving between temperature scales like Fahrenheit, Celsius, and Kelvin.

• Begin by learning the key conversion formulas among these scales.
• Set the scale you want to equal another, like \( F = C \) or \( F = K \).

Apply algebraic manipulation, such as substituting and combining like terms, to isolate the variable you are solving for.
For example, solving \( K = \frac{9}{5}K - 491.67 + 32 \) involves rearranging and simplifying terms.
Being methodical with each step ensures accuracy and a deeper understanding of how these temperature conversion relationships work.