Problem 4 - 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

Fahrenheit and Celsius agree at -40 degrees; Fahrenheit and Kelvin agree at 574.59 degrees.

Step 1: Understanding the Fahrenheit-Celsius Relationship

The relationship between degrees Fahrenheit and degrees Celsius can be expressed with the formula \( F = \frac{9}{5}C + 32 \). Where \( F \) is the temperature in Fahrenheit, and \( C \) is the temperature in Celsius.

Step 2: Setting Fahrenheit Equal to Celsius

To find the temperature where Fahrenheit and Celsius agree, set \( F = C \). Substitute \( C \) for \( F \) in the formula, resulting in \( C = \frac{9}{5}C + 32 \).

Step 3: Solving the Equation for Celsius

Solve the equation \( C = \frac{9}{5}C + 32 \). Rearranging terms gives \( C - \frac{9}{5}C = 32 \). This simplifies to \(-\frac{4}{5}C = 32 \). Multiply both sides by \(-5/4\) to solve for \( C \):\[ C = 32 \times \left(-\frac{5}{4}\right) = -40 \]

Step 4: Verifying the Solution

Convert \( -40 \) Celsius using the formula: \( F = \frac{9}{5}(-40) + 32 = -40 \). Both thermometers show \(-40\), confirming the solution.

Step 5: Understanding the Fahrenheit-Kelvin Relationship

The conversion formula between Fahrenheit and Kelvin is \( F = \frac{9}{5}(K - 273.15) + 32 \). Where \( F \) is the degrees Fahrenheit and \( K \) is the temperature in Kelvin.

Step 6: Setting Fahrenheit Equal to Kelvin

To find the temperature where Fahrenheit and Kelvin agree, set \( F = K \). Substitute \( K \) for \( F \) in the formula: \( K = \frac{9}{5}(K - 273.15) + 32 \).

Step 7: Solving the Equation for Kelvin

Solve the equation \( K = \frac{9}{5}(K - 273.15) + 32 \). Distribute and rearrange:\[ K = \frac{9}{5}K - \frac{9}{5} \times 273.15 + 32 \]This simplifies to:\[ K - \frac{9}{5}K = - \frac{9 \times 273.15}{5} + 32 \]\[ -\frac{4}{5}K = -491.67 + 32 \]\[ -\frac{4}{5}K = -459.67 \]Multiply both sides by \(-5/4\) to solve for \( K \):\[ K = -459.67 \times \left(-\frac{5}{4}\right) = 574.59 \]

Step 8: Verifying the Solution

Convert \(574.59 \) Kelvin using the formula: \( F = \frac{9}{5}(574.59 - 273.15) + 32 \) which simplifies to \(574.59 \) agreeing with the Kelvin value.

Key Concepts

Fahrenheit and Celsius

The relationship between Fahrenheit and Celsius is fundamental to understanding temperature conversion. When you want to convert from Celsius to Fahrenheit, you use the formula:

• \( F = \frac{9}{5}C + 32 \)

Here, \( F \) represents the temperature in degrees Fahrenheit and \( C \) represents the temperature in degrees Celsius.
To find the temperature where the two scales agree, meaning both thermometers show the same reading, we set the two equal:

• \( F = C \)

This results in the equation:

• \( C = \frac{9}{5}C + 32 \)

Solving this, we rearrange and simplify to find that \( C = -40 \).
At this temperature, \(-40\)°C indeed converts to \(-40\)°F when checked using the formula. Understanding this special intersection helps solidify grasp of how Fahrenheit and Celsius differ and align.

Fahrenheit and Kelvin

The relationship between Fahrenheit and Kelvin is slightly different and less intuitive than Celsius. The conversion formula is:

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

Where \( F \) is in Fahrenheit and \( K \) is in Kelvin.
To determine where Fahrenheit matches Kelvin exactly, set \( F = K \):

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

Solving for \( K \), you simplify the equation to eventually find that \( K = 574.59 \).
When you backtrack by converting \(574.59\) Kelvin to Fahrenheit using the original formula, it equals \(574.59\)°F, verifying the equivalence. This intersection temperature reveals an interesting interplay between these systems, emphasizing the complexity of temperature conversion.

Temperature Equivalence

Temperature equivalence occurs when two different measures show the same number value for a specific point. Understanding this concept requires navigating through precise equations.
When solving problems of equivalence, first set the formulas equal to each other, given that they represent the same physical situation.

• Fahrenheit and Celsius example: \(-40\) is the intersection point.
• Fahrenheit and Kelvin take a different mathematical path, resulting in \(574.59\).

Finding these equivalent points encourages not only practicing algebra but also deepening one's grasp of how different systems are interconnected.
Having these skills is essential for students to solve real-world problems where temperature conversion is necessary across different scientific domains.