Problem 58 - Chemistry Matter and Change

Question

Hot-Air Balloons A sample of air occupies 2.50 L at a temperature of 22.0°C. What volume will this sample occupy inside a hot-air balloon at a temperature of 43.0°C? Assume that the pressure inside the balloon remains constant.

Step-by-Step Solution

Answer

The volume of the air sample inside the hot-air balloon at a temperature of 43.0°C is approximately 2.68 L, assuming the pressure remains constant.

Step 1: Convert temperatures to Kelvin

First, we need to convert the Celsius temperatures to Kelvin. Recall that to convert Celsius to Kelvin, we can use the formula K = C + 273.15, where K is the temperature in Kelvin, and C is the temperature in Celsius. Initial temperature (T1) = 22.0°C + 273.15 Final temperature (T2) = 43.0°C + 273.15

Step 2: Perform calculations

Now we can calculate the initial and final temperatures in Kelvin: T1 = 22.0°C + 273.15 = 295.15 K T2 = 43.0°C + 273.15 = 316.15 K We also have the initial volume (V1) = 2.50 L. We need to find the final volume (V2) of the air sample when it goes from 295.15 K to 316.15 K. Using Charles's Law formula: (V1/T1) = (V2/T2) Plug in the values: (2.50 L / 295.15 K) = (V2 / 316.15 K)

Step 3: Solve for the final volume

To solve for the final volume (V2), we can multiply both sides of the equation by the final temperature (T2): V2 = (2.50 L / 295.15 K) * 316.15 K V2 = 2.50 L * (316.15 K / 295.15 K) Now, calculate the final volume (V2): V2 = 2.50 L * (1.071) V2 ≈ 2.68 L

Step 4: State the answer

The volume of the air sample inside the hot-air balloon at a temperature of 43.0°C is approximately 2.68 L, assuming the pressure remains constant.

Key Concepts

Temperature Conversion

Understanding temperature conversion is essential when working with gas laws. The task here involves converting Celsius to Kelvin, which is crucial because gas laws require temperature to be in the absolute temperature scale, Kelvin. The formula to convert Celsius to Kelvin is: \
\( K = C + 273.15 \)
\
where K is the Kelvin temperature and C is the Celsius temperature. Remember that Kelvin does not include negative numbers because it starts from absolute zero, where all molecular motion stops. By converting temperatures to Kelvin, we ensure more accurate and standard calculations in gas laws.

Gas Laws

Gas laws are the governing principles that describe the behavior of gases under various conditions of temperature, volume, and pressure. Charles's Law, in particular, is one of these gas laws and it states the volume of a gas is directly proportional to its temperature when the pressure is held constant. Other important gas laws include Boyle's Law, which deals with pressure and volume at constant temperature, and the Ideal Gas Law, which combines several individual laws into one universal equation. These laws are foundational for understanding how gases will react when subjected to different conditions, and they are used extensively in the fields of chemistry and physics.

Kelvin Temperature Scale

The Kelvin temperature scale is integral to the study of gas laws. It is an absolute scale based on absolute zero, unlike Celsius which is based on the properties of water. The Kelvin scale is used in scientific calculations because it provides consistency and accuracy. Zero Kelvin is the point at which particles theoretically stop moving entirely, making it a natural starting point for temperature in the study of gases. It's important to always use Kelvin when applying gas laws, as doing otherwise may result in incorrect answers.

Volume-Temperature Relationship

The volume-temperature relationship is beautifully demonstrated in Charles's Law. Charles's Law asserts that for a given mass of gas at constant pressure, the volume is directly proportional to its Kelvin temperature. This can be represented by the formula: \
\( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)
\
where \( V_1 \) and \( T_1 \) are the initial volume and temperature, respectively, and \( V_2 \) and \( T_2 \) are the final volume and temperature. When you know any three of these values, the fourth can be calculated. This relationship is particularly useful when predicting the behavior of gases in different thermal conditions, such as predicting the expansion of air in a hot air balloon as it is heated.