A balloon initially contains 40 m^3 of helium gas at atmospheric conditions of 100 kPa and 17^∘

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A balloon initially contains 40 m^3 of helium gas at...

A balloon initially contains $40 \mathrm{~m}^{3}$ of helium gas at atmospheric conditions of $100 \mathrm{kPa}$ and $17^{\circ} \mathrm{C} .$ The balloon is connected by a valve to a large reservoir that supplies helium gas at $125 \mathrm{kPa}$ and $25^{\circ} \mathrm{C}$. Now the valve is opened, and helium is allowed to enter the balloon until pressure equilibrium with the helium at the supply line is reached. The material of the balloon is such that its volume increases linearly with pressure. If no heat transfer takes place during this process, determine the final temperature in the balloon.

Key Concepts

Adiabatic Process

An adiabatic process is one in which no heat is transferred into or out of the system. In thermodynamics, this means that any change in the internal energy of the system is entirely due to work done on or by the system or changes in its composition. This concept is crucial when analyzing processes where rapid changes occur or where the system is insulated, such that heat transfer is negligible, as in the scenario of the gas entering the balloon without heat exchange.

Ideal Gas Law

The ideal gas law provides a relationship between pressure, volume, temperature, and the number of moles of a gas. It is expressed as PV = nRT, where R is the universal gas constant. This law is fundamental for predicting how a gas will behave under different conditions and is particularly useful in problems involving changing states of the gas, as it connects the measurable quantities in both the initial and final states of the system.

Energy Balance in Open Systems

When dealing with systems where mass crosses the boundary, such as a balloon that is being filled with additional gas, an energy balance must account for the energy carried by the mass entering or leaving the system. In an adiabatic process for an open system, the change in the system’s internal energy arises from both the work done and the enthalpy (energy content) of the entering mass. This approach helps determine the final state variables like temperature, as the incoming gas has its own state and energy that affects the overall energy balance.

Pressure-Volume Relationship in Flexible Containers

In flexible systems like balloons, the material’s elasticity can dictate how the volume changes with pressure. A linear relationship between volume and pressure implies that as the internal pressure increases, the volume expands in a predictable manner, modifying the work done by or on the gas. Recognizing this relationship is essential when applying conservation laws because it affects how energy is partitioned between internal energy and work.

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