Two insulated tanks A and B are connected by a valve. Tank A...

Two insulated tanks $A$ and $B$ are connected by a valve. Tank $A$ has a volume of $30 \mathrm{ft}^{3}$ and initially contains argon at $50 \mathrm{lbf} / \mathrm{in.}^{2}, 50 \mathrm{F}$. Tank $B$ has a volume of $60 \mathrm{ft}^{3}$ and initially contains ethane at 30 lbffin. 120 F. The valve is opened and remains open until the resulting gas mixture comes to a uniform state. Find the final pressure and temperature.

Key Concepts

Conservation of Mass

Conservation of mass is a key principle stating that the mass of the system remains constant during the mixing process. In problems involving multiple tanks with different gases, it ensures that the total mass (or number of moles) of each component is accounted for, serving as a necessary check when applying other thermodynamic relations.

Mixing of Gases in Closed Systems

The mixing of gases in a closed system involves combining distinct gases into a single uniform state where both temperature and pressure equilibrate throughout the system. This process requires the application of both mass and energy conservation principles, taking into account the properties of each gas to derive the final state variables.

Internal Energy Dependence on Temperature for Ideal Gases

For ideal gases, the internal energy is a function solely of temperature and is independent of pressure and volume. This concept allows one to establish a direct relationship between the temperature of a gas and its internal energy, enabling the use of energy conservation to solve for the final temperature when different gases are mixed.

Conservation of Energy in Adiabatic, Isolated Systems

In an adiabatic process, where no heat is exchanged with the surroundings, the total internal energy of the isolated system remains constant. This principle is essential when analyzing the mixing of gases within insulated tanks, as the final equilibrium condition is determined by balancing the internal energies of the individual gases before and after the mixing.

Ideal Gas Law

The Ideal Gas Law (PV = nRT) is crucial in relating pressure, volume, and temperature for ideal gases. It allows one to compute the state properties of gases based on their mass or number of moles and is fundamental when determining changes in state due to mixing or expansion, as seen in problems involving tanks and connected volumes.

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