A particular balloon is designed by its manufacturer to be inflated to a volume of no more than 2.5 L. If the balloon is filled with 2.0 L helium at sea level, is released, and rises to an altitude at which the atmospheric pressure is only \(500 . \mathrm{mm}\) Hg, will the balloon burst? (Assume temperature is constant.)

Atmospheric pressure is the force per unit area exerted on a surface by the weight of the air above that surface in the atmosphere of Earth. It is an important factor to consider when studying the behavior of gases, as it can affect their volume and pressure. At sea level, the standard atmospheric pressure is about 101,325 Pascals or 760 mm Hg (millimeters of mercury). As altitude increases, atmospheric pressure decreases because there are fewer air molecules above the surface.

This concept is crucial when we talk about gases expanding or contracting in different altitudes. The exercise shows a practical application of atmospheric pressure in predicting whether a helium balloon will burst when it ascends to higher altitudes. In the context of the balloon, as it rises, the external pressure around it decreases, which influences the gas inside to expand.

Gas laws are the physical laws that describe the properties of gases, including their volume, temperature, pressure, and quantity. The most prominent of the gas laws are Boyle's Law, Charles's Law, Gay-Lussac's Law, and the Combined Gas Law. These laws enable us to predict the behavior of a gas when one of the properties changes, assuming all other properties remain constant or are controlled.

Boyle's Law

Boyle's Law, in particular, explains the relationship between the pressure and volume of a gas at constant temperature. According to Boyle’s Law, for a given amount of gas at constant temperature, the volume of the gas is inversely proportional to its pressure. This means if the pressure increases, the volume decreases, and vice versa, which is the focus of the exercise provided.

Understanding the relationship between volume and pressure of a gas under constant temperature is key to solving problems that apply Boyle's Law. This inverse relationship is mathematically represented as:
\(P_1V_1 = P_2V_2\)
where \(P_1\) and \(V_1\) are the initial pressure and volume and \(P_2\) and \(V_2\) are the final pressure and volume, respectively. When volume increases, pressure decreases, and when volume decreases, pressure increases.

In the balloon exercise, this relationship helps to predict whether the balloon will burst or not. If the balloon’s volume at sea level is 2.0 L and the pressure decreases as it ascends, the volume will increase. We calculate its final volume under the reduced atmospheric pressure using Boyle’s Law formula, and by comparing it to the balloon’s maximum allowed volume, we conclude if it is going to burst. The exercise shows that the final volume of 3.04 L exceeds the safe limit, indicating the balloon would indeed burst.