A particular balloon can be stretched to a maximum surface...

A particular balloon can be stretched to a maximum surface area of $1257 \mathrm{~cm}^{2}$. The balloon is filled with $3.0 \mathrm{~L}$ of helium gas at a pressure of 755 torr and a temperature of $298 \mathrm{~K}$. The balloon is then allowed to rise in the atmosphere. If the atmospheric temperature is $273 \mathrm{~K}$, at what pressure will the balloon burst? (Assume the balloon to be in the shape of a sphere.)

Key Concepts

Unit Conversion in Thermodynamics

Accurate calculation in thermodynamics often requires converting between different units of measurement, such as from torr to atmospheres or from liters to cubic meters, to ensure consistency. Correct unit conversion is necessary for applying the ideal gas law and other formulas accurately.

Spherical Geometry Relationships

For objects that are spherical, the relationships between surface area (A = 4?r²) and volume (V = (4/3)?r³) are key. These geometric formulas allow us to relate the maximum allowable surface area of a balloon to its volume, thereby helping to determine the point of structural failure (bursting) based on how much the balloon can expand.

Ideal Gas Law

The ideal gas law (PV = nRT) is a foundational principle in thermodynamics that relates the pressure, volume, and temperature of a gas with the number of moles present. It is used to determine how a gas will respond under changing conditions, such as variations in temperature or volume, which is essential when calculating the conditions at which the balloon will burst.

Gas Behavior Under Changing Conditions

Understanding how gases react to changes in temperature and volume is crucial. As the balloon ascends and the temperature changes, the gas expands or contracts according to the ideal gas law, while the balloon's elasticity sets a geometric limit on expansion. This interplay defines the burst conditions.

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