Consider a thermally isolated system consisting of two...

Consider a thermally isolated system consisting of two volumes, $V$ and 2 V of an ideal gas separated by a thermally conducting and movable partition. The temperatures and pressures are as shown. The partition is now allowed to move without the gases mixing. When equilibrium is established what is the change in the total internal energy? The total entropy? What is the equilibrium temperature? Pressure? FIGURE CAN'T COPY.

Consider a thermally isolated system consisting of two volumes, $V$ and 2 V of an ideal gas separated by a thermally conducting and movable partition.
The temperatures and pressures are as shown. The partition is now allowed to move without the gases mixing.

When equilibrium is established what is the change in the total internal energy? The total entropy?
What is the equilibrium temperature? Pressure?
FIGURE CAN'T COPY.

Key Concepts

Movable Partition in Equilibrium

A movable and thermally conducting partition allows the two sections of the system to exchange energy and perform work on each other until equilibrium is reached. This setup is essential in many thermodynamic problems because it facilitates the simultaneous achievement of uniform temperature and pressure across the system while conserving overall energy.

Thermal Isolation

Thermal isolation means that the system does not exchange heat with its surroundings, making it an adiabatic system. This concept is critical because it ensures that the total energy of the system remains constant, regardless of the internal redistribution of energy. It also implies that any changes in the system’s energy (such as work done when a partition moves) are fully accounted for internally.

Ideal Gas Properties

For an ideal gas, the internal energy is a function solely of its temperature and degrees of freedom, independent of pressure or volume. This key property allows simplification of energy calculations, because knowing the temperature is sufficient to determine the internal energy, which is crucial when assessing energy changes during rearrangements within the system.

Conservation of Internal Energy

In a thermally isolated system, the first law of thermodynamics (conservation of energy) applies. The total internal energy remains constant despite internal re-distributions. This concept is used to determine that any work or energy transfer occurring internally (for example, due to the movement of a partition) does not change the total energy of the system.

Entropy and the Second Law of Thermodynamics

Entropy is a measure of the number of microstates or the disorder within a system. The second law states that for an isolated system undergoing any process, the total entropy will tend to increase or, in the case of a reversible process, remain constant. This concept is essential to understand how the system evolves towards equilibrium and why the final state typically has a higher entropy than the initial state.

Equilibrium Conditions (Thermal and Mechanical)

When the system reaches equilibrium, both the thermal and mechanical conditions are balanced; this means that the temperature becomes uniform throughout the system (thermal equilibrium) and the pressures equalize (mechanical equilibrium). Achieving these conditions is necessary for the system to be in its maximum entropy state under the given constraints.

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