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 \((\mathrm{b}) \mathrm{C}^{\circ}\) ?

Step-by-Step Solution

Answer

The temperature change is 10.0°C and 18.0°F.

Step 1: Understanding the Change in Temperature in Kelvin

The temperature change in the given problem is 10.0 Kelvin. Kelvin and Celsius have the same increment size, meaning a change of 1 Kelvin is equivalent to a change of 1 degree Celsius.

Step 2: Converting Kelvin to Celsius

Since the increment size of Kelvin and Celsius is identical, the change in temperature is the same in Celsius. Therefore, the temperature change in Celsius is 10.0°C.

Step 3: Converting Celsius to Fahrenheit

To convert the temperature change from Celsius to Fahrenheit, use the formula: \[ \Delta T(\mathrm{F}^{\circ}) = \Delta T(\mathrm{C}^{\circ}) \times \frac{9}{5} \]Substituting the given value:\[ \Delta T(\mathrm{F}^{\circ}) = 10.0 \times \frac{9}{5} = 18.0 \mathrm{F}^{\circ} \]

Step 4: Summary of Conversions

The temperature change in Kelvin is equivalent to a 10.0°C change and an 18.0°F change.

Key Concepts

Kelvin to Celsius Conversion

Converting temperature between the Kelvin and Celsius scales is straightforward. This is because both scales use the same size for their increments. When we say increments, we're talking about each individual step or division on the thermometer. To convert a temperature change from Kelvin to Celsius, you don't actually need to make any adjustments!

This is due to a simple but fundamental idea:

• A change of 1 Kelvin is exactly equal to a change of 1 degree Celsius.
• This equivalence means that if a temperature drops by 10 Kelvin, it also drops by 10 degrees Celsius.
• When working with temperature changes, you apply no additional calculations. Just remember the unity of the increment sizes in these two scales.

In our example, the soft drink has a temperature change of 10.0 Kelvin, so it has a corresponding change of 10.0°C. Chemistry and physics often use Kelvin, but any time you're converting temperature changes, it's as easy as matching numbers.

Celsius to Fahrenheit Conversion

Sometimes we need to convert temperatures for use in regions that use different temperature systems. Specifically, converting from Celsius to Fahrenheit is commonly required in the United States. For converting a temperature change, the formula to remember is:

\[ \Delta T(\mathrm{F}^{\circ}) = \Delta T(\mathrm{C}^{\circ}) \times \frac{9}{5} \]
This formula reflects the fact that Fahrenheit increments are smaller than those of Celsius.

Let's apply the formula to a change of 10 degrees Celsius:

• Multiply the Celsius change by the fraction \(\frac{9}{5}\).
• This results in \( 10.0 \times \frac{9}{5} = 18.0 \) Fahrenheit degree change.
• The logic here is tied to the historical development of the Fahrenheit scale, which tracks differently than both Kelvin and Celsius.

Remembering this ratio (9/5) helps quickly assess any temperature conversion between these two scales. Once you know it, converting is like doing simple multiplication.

Temperature Change Calculations

Understanding temperature changes is key to interpreting many scientific and real-world processes. Temperature change can be confusing, especially when different scales are involved.

When you are calculating a temperature change, you're determining how much warmer or cooler a substance has become over a period of time. Here are some important points:

• A temperature change is the difference in temperature, not the absolute temperature itself.
• To find a change, you subtract the initial temperature from the final temperature.
• For comparative understanding, Kelvin changes are equivalent to Celsius, making this particularly straightforward when converting between the two.

All changes equate when adjusting between Kelvin and Celsius, but not between Celsius and Fahrenheit. Different formulas are required due to the old historical differences developed in each scale's construction. However, once you know the basics, calculating in any scale becomes easier and second nature. Temperature change is an essential idea not just in academics but in everyday life, from weather forecasts to cooking.