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, and Fahrenheit and Kelvin agree at 574.59.

Step 1: Understand the Problem

We need to find two temperatures: one where the reading is the same in both the Fahrenheit and Celsius scales, and another where the reading is the same in both the Fahrenheit and Kelvin scales.

Step 2: Formulate the First Equation

For part (a), we start with the formula that relates Celsius (\( C \) ) and Fahrenheit (\( F \) ) temperatures: \[ F = \frac{9}{5}C + 32 \] Since we need the temperatures to be equal, we set \( F = C \) and substitute in the formula: \[ C = \frac{9}{5}C + 32 \].

Step 3: Solve the First Equation

Substituting into the equation \( C = \frac{9}{5}C + 32 \), we rearrange terms: \[ C - \frac{9}{5}C = 32 \] Simplify to get: \[ -\frac{4}{5}C = 32 \]. Multiply both sides by \(-\frac{5}{4}\) to solve for \( C \): \[ C = -40 \] Thus, \( F = C = -40 \).

Step 4: Formulate the Second Equation

For part (b), Fahrenheit (\( F \)) and Kelvin (\( K \)) thermometers need to agree, which happens when \( F = K \). We use the formulas: 1. \( F = \frac{9}{5}(K-273.15) + 32 \)2. Since we want \( F = K \), we equate: \( \frac{9}{5}(K - 273.15) + 32 = K \).

Step 5: Solve the Second Equation

Rearrange and solve \( \frac{9}{5}(K - 273.15) + 32 = K \): \[ \frac{9}{5}(K - 273.15) = K - 32 \] Multiply out: \[ 9K - 9 \times 273.15 = 5K - 160 \] Add like terms: \[ 4K = 2458.35 - 160 \] Solve for \( K \): \[ 4K = 2298.35 \] \[ K = 574.5875 \] Therefore, \( F = K = 574.59 \) (rounded to two decimal places).

Key Concepts

Fahrenheit and Celsius

Fahrenheit and Celsius are two different scales used to measure temperature. You might wonder why we need different scales. Well, historically, different countries and scientists developed their methods of measuring temperature which led to different scales.

The Celsius scale is based on the freezing and boiling points of water. Water freezes at 0°C and boils at 100°C at sea level. This makes the Celsius scale easy to use and understand for everyday temperatures.

On the other hand, the Fahrenheit scale is a bit different. On this scale, water freezes at 32°F and boils at 212°F. It was developed by Daniel Fahrenheit in the early 18th century.

When converting between these two scales, the formula is:

• Fahrenheit to Celsius: \( C = \frac{5}{9}(F - 32) \)
• Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \)

There is one temperature where Fahrenheit and Celsius agree with each other, which is \( -40 \, ^\circ \text{C} = -40 \, ^\circ \text{F}. \)

Fahrenheit and Kelvin

Let's talk about Fahrenheit and Kelvin. While Fahrenheit is common in the United States, Kelvin is used mostly in scientific contexts because it is an "absolute" scale. Here's why:

The Kelvin scale starts at absolute zero, the point where all molecular motion stops. This makes it handy for scientific purposes. In this scale, there are no negative numbers because Kelvin simply adds 273.15 to Celsius (\(K = C + 273.15\)). This means absolute zero is 0 K, equivalent to −273.15°C.

To convert from Fahrenheit to Kelvin, the formula is a bit more complex because it needs to account for the differences between the zero points and the scaling of each system:

• Convert to Celsius:\( C = \frac{5}{9}(F - 32) \)
• Then convert to Kelvin:\( K = C + 273.15 \)

Solving for when Fahrenheit equals Kelvin reveals one matching temperature, \( 574.59 \, K = 574.59 \, ^\circ \text{F} \)(rounded to two decimal places).

Temperature Conversion Formulas

Understanding temperature conversion formulas is crucial when dealing with multiple temperature scales. Let's break down the formulas you'll often use:

• Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \)
• Fahrenheit to Celsius: \( C = \frac{5}{9}(F - 32) \)
• Celsius to Kelvin: \( K = C + 273.15 \)
• Kelvin to Celsius: \( C = K - 273.15 \)
• Fahrenheit to Kelvin: First convert Fahrenheit to Celsius using \( C = \frac{5}{9}(F - 32) \), then use \( K = C + 273.15 \)
• Kelvin to Fahrenheit: First convert to Celsius \( C = K - 273.15 \), then to Fahrenheit \( F = \frac{9}{5}C + 32 \)

By mastering these formulas, you can easily convert any temperature between different scales. Remembering these will improve your understanding of temperature relations and conversions.