Problem 5 - University Physics with Modern Physics

Question

You put a bottle of soft drink in a refrigerator and leave it until its temperature has dropped 10.0 \(\mathrm{K}\) . What is its temperature change in (a) \(\mathrm{F}^{\circ}\) and (b) \(\mathrm{C}^{\circ} ?\)

Step-by-Step Solution

Answer

(a) 18.0 °F, (b) 10.0 °C

Step 1: Understanding Temperature Change in Celsius

Since temperature changes on the Kelvin and Celsius scales are equivalent, a change of 10.0 K is the same as a change of 10.0 °C.

Step 2: Calculating Temperature Change in Fahrenheit

To convert the temperature change from Celsius to Fahrenheit, use the formula: \( \Delta T_{°F} = \Delta T_{°C} \times \frac{9}{5} \). Substitute \( \Delta T_{°C} = 10.0 \) to get \( \Delta T_{°F} = 10.0 \times \frac{9}{5} = 18.0 \).

Key Concepts

Kelvin to Celsius

Temperature is one of the most fundamental physical quantities, and it's measured in different scales like Kelvin, Celsius, and Fahrenheit. When it comes to converting between these scales, it's important to understand the relationships between them. The Kelvin and Celsius scales are directly connected. In fact, a change of 1 Kelvin is exactly the same as a change of 1 degree Celsius. This is because both scales increment identically, differing only in their starting points.

To convert a temperature difference or change from Kelvin to Celsius, you simply use the fact that they are equivalent. For instance, when a substance cools down by 10 Kelvin, it also cools by 10 degrees Celsius. There's no conversion formula needed since the interval scales are the same.

If you're working on a problem involving changes in temperature, understanding that Kelvin and Celsius can interchange directly is especially useful. Remember that 0 Kelvin is known as absolute zero, which is the point where molecular energy is at a minimum, and this corresponds to approximately -273.15 °C.

Celsius to Fahrenheit

The Celsius and Fahrenheit scales are commonly used, especially in regions using the metric and imperial systems, respectively. However, they are not directly equivalent like Kelvin and Celsius. Because of this, you need to use a conversion formula to change from one to the other.

Understanding the Conversion Formula
For changes in temperature, the formula to convert Celsius to Fahrenheit is given by:
\[\Delta T_{°F} = \Delta T_{°C} \times \frac{9}{5}\] This formula emphasizes the fact that the Fahrenheit scale increments more finely than the Celsius scale.

For instance, if the temperature decreases by 10 degrees Celsius, you can calculate the equivalent drop in Fahrenheit as follows:

• Use the formula: \( 10.0 \times \frac{9}{5} \)
• Calculate to find a decrease of 18.0 degrees Fahrenheit.

Knowing this conversion is practical, especially when considering various real-world applications where different measurement systems are in use.

Temperature Scales

Temperature scales, such as Kelvin, Celsius, and Fahrenheit, are a way to measure how hot or cold something is. These scales have been developed for various practical and historical reasons and are used in different contexts around the world.

Key Characteristics:

• Kelvin: Used primarily in science, Kelvin is the absolute thermodynamic temperature scale starting at absolute zero (0 K), where no thermal energy remains. It doesn't use degrees and is related to Celsius by an offset of 273.15.


• Celsius: Widely used in daily life and scientific work, Celsius is based on the freezing (0 °C) and boiling points (100 °C) of water at sea level. It's convenient for tasks involving water or temperature changes close to ambient levels.


• Fahrenheit: Primarily used in the United States, Fahrenheit is based on a different scale, where the freezing point of water is 32 °F and the boiling point is 212 °F. It's an older scale but still prevalent in various industries and residential use.

Different scales have distinct uses. Kelvin is critical in scientific calculations, Celsius is standard in most of the world and education, while Fahrenheit still dominates in a few regions, guiding everyday activities like weather reports and cooking.