A weather balloon is inflated to a volume of 28.5 L at a pressure of 748 mmHg and a temperature of 28.0 °C. The balloon rises in the atmosphere to an altitude of approximately 25,000 ft, where the pressure is 385 mmHg and the temperature is -15.0 °C. Assuming the balloon can freely expand, calculate the volume of the balloon at this altitude.

Verified step by step guidance

1. Identify the initial and final conditions of the balloon: initial volume \( V_1 = 28.5 \text{ L} \), initial pressure \( P_1 = 748 \text{ mmHg} \), initial temperature \( T_1 = 28.0 \degree \text{C} \), final pressure \( P_2 = 385 \text{ mmHg} \), and final temperature \( T_2 = -15.0 \degree \text{C} \).
2. Convert the temperatures from Celsius to Kelvin using the formula \( T(\text{K}) = T(\degree \text{C}) + 273.15 \).
3. Use the combined gas law \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \) to relate the initial and final states of the gas.
4. Rearrange the combined gas law to solve for the final volume \( V_2 \): \( V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \).
5. Substitute the known values into the equation and solve for \( V_2 \).

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is fundamental in predicting how gases behave under varying conditions. In this scenario, it allows us to understand how the volume of the balloon changes as it ascends and experiences different pressure and temperature.

Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its temperature (in Kelvin) when pressure is held constant. This concept is crucial for understanding how the volume of the balloon will change as it rises and the temperature decreases. It emphasizes the relationship between temperature and volume in gas behavior.

Boyle's Law

Boyle's Law states that the pressure of a gas is inversely proportional to its volume when temperature is held constant. This principle is essential for analyzing the changes in pressure and volume of the balloon as it ascends to higher altitudes, where the pressure decreases significantly, affecting the balloon's volume.