A balloon behaves such that the pressure inside is...

A balloon behaves such that the pressure inside is proportional to the diameter squared. It contains $2 \mathrm{kg}$ of ammonia at $0^{\circ} \mathrm{C},$ with $60 \%$ quality. The balloon and ammonia are now heated so that a final pressure of $600 \mathrm{kPa}$ is reached. Considering the ammonia as a control mass, find the amount of work done in the process.

Key Concepts

Geometric Relationships in Expanding Systems

When a system like a balloon expands, its geometric properties, such as volume and diameter, are interrelated. Recognizing the relationship between these variables (for instance, using the volume formula for a sphere) is important to accurately express how changes in one (like diameter) affect the overall volume and, consequently, the work done by the system.

Control Mass Process

A control mass process in thermodynamics involves analyzing a system with a fixed amount of mass where energy is exchanged with the surroundings via heat and work. Understanding this concept is essential for applying conservation of energy principles and tracking changes in state variables as the system undergoes a process such as heating.

First Law of Thermodynamics

The first law for a closed system (control mass) relates the change in internal energy to the heat added to the system minus the work done by the system. This principle is fundamental when accounting for energy transfers during processes that involve changes in temperature, phase, and volume.

Boundary Work and Pressure–Volume Integration

In thermodynamic processes involving expansion or compression, work is done by or on the system through the movement of its boundaries. When pressure is not constant but varies with a geometric parameter (such as the diameter of a balloon), calculating the work requires integrating the pressure with respect to the differential change in volume, taking into account the specific relationship between pressure and the changing dimensions.

Phase Quality in Two-Phase Mixtures

The concept of quality represents the mass fraction of vapor in a two-phase liquid–vapor mixture. It is critical in determining the internal energy, enthalpy, and other properties of a substance at conditions where both phases coexist, thus influencing the state and energy analysis during processes such as heating.

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