Problem 4 - University Physics with Modern Physics

Question

Two beakers of water, \(A\) and \(B\) , initially are at the same temperature. The temperature of the water in beaker \(A\) is increased \(10 F^{\circ},\) and the temperature of the water in beaker \(B\) is increased 10 \(\mathrm{K}\) . After these temperature changes, which beaker of water has the higher temperature? Explain.

Step-by-Step Solution

Answer

Beaker B will have the higher temperature.

Step 1: Understanding Temperature Scales

The problem involves two different temperature scales: Fahrenheit and Kelvin. It's essential to compare the increments in the same scale to determine which beaker has the higher final temperature.

Step 2: Convert Kelvin Increase to Fahrenheit

The conversion from Kelvin to Fahrenheit requires understanding the relationship between these temperature scales. The formula to convert a temperature change from Kelvin to Fahrenheit is \( \Delta T_{F} = \Delta T_{K} \times \frac{9}{5} \). Applying this gives \( 10\,\text{K} \times \frac{9}{5} = 18\,^{\circ}F \).

Step 3: Comparison of Temperature Increases

Beaker \(A\) has an increase of \(10^{\circ}F\), while beaker \(B\) has an equivalent increase of \(18^{\circ}F\). Therefore, beaker \(B\) experiences a larger temperature increase.

Step 4: Determine the Higher Temperature

Since both beakers started at the same initial temperature, the beaker with the higher temperature increase will end up at a higher final temperature. Thus, beaker \(B\) will have a higher temperature compared to beaker \(A\) after the changes.

Key Concepts

Fahrenheit to Kelvin

Fahrenheit and Kelvin are two different units used to measure temperature. While Fahrenheit is predominantly used in the United States, the Kelvin scale is widely used in scientific contexts. Converting between these two scales involves a precise mathematical formula because they have different zero points and increments per degree.

To understand the conversion from an increase in Fahrenheit to Kelvin, consider the conversion formula between these temperature scales. The general formula for converting a temperature change from Fahrenheit to Kelvin is:

• \( \Delta T_{K} = \Delta T_{F} \times \frac{5}{9} \)

This formula arises from the fact that a degree Fahrenheit is smaller than a degree Kelvin by a factor of \( \frac{5}{9} \).

By understanding this conversion, you can easily determine how temperature changes in Fahrenheit correspond to changes in Kelvin, which is crucial for solving problems that involve multiple temperature scales.

Temperature Scales

Temperature scales are systems for measuring temperature, each with their unique baselines and increments. The three most commonly used temperature scales are Celsius, Kelvin, and Fahrenheit. Each scale has its own use and application, helping to provide precise temperature readings for varying purposes.

**Fahrenheit Scale**: This scale uses 32°F as the freezing point of water and 212°F as the boiling point. It is mostly used in the United States for everyday temperature measurements.

**Kelvin Scale**: Used predominantly in science, the Kelvin scale starts at absolute zero (0 K), the theoretically coldest possible temperature, and uses increments that are equivalent to the Celsius scale. Water freezes at 273.15 K and boils at 373.15 K.

Understanding these differences helps when converting measurements or deciding which scale to use for various applications. The Kelvin scale's absolute nature makes it essential for scientific calculations, while Fahrenheit, with its finer gradation, is practical for daily weather reporting.

Temperature Increase Comparison

In comparing temperature changes across scales, converting all changes to the same unit is crucial. This allows for an accurate assessment of which change is greater. In the scenario, beaker A experiences a 10°F increase, while beaker B gets a 10 K increase.

Because Kelvin increases are larger than Fahrenheit ones (as illustrated by the conversion factor), the corresponding Fahrenheit increase for beaker B is calculated as:

• \(10 \text{ K} \times \frac{9}{5} = 18^{\circ}\text{F} \)

This conversion shows that beaker B's temperature rise is equivalent to 18°F, which is larger than beaker A's 10°F increase.

By understanding these increments and converting appropriately, we ensure we are accurately comparing increases and determining which substance or object has experienced a greater change in temperature. In our case, beaker B ends up with the higher final temperature because its temperature increase was greater.