A thermally insulated cylinder, closed at both ends, is...

A thermally insulated cylinder, closed at both ends, is fitted with a frictionless heat-conducting piston which divides the cylinder into two parts. Initially, the piston is clamped in the center, with 1 litre of air at 200 K and 2 atm pressure on one side and 1 litre of air at 300 K and 1 atm on the other side. The piston is released and the system reaches equilibrium in pressure and temperature, with the piston at a new position. (a) Compute the final pressure and temperature. (b) Compute the total increase in entropy. Be sure to give all your reasoning.

A thermally insulated cylinder, closed at both ends, is fitted with a frictionless heat-conducting piston which divides the cylinder into two parts. Initially, the piston is clamped in the center, with 1 litre of air at 200 K and 2 atm pressure on one side and 1 litre of air at 300 K and 1 atm on the other side. The piston is released and the system reaches equilibrium in pressure and temperature, with the piston at a new position.
(a) Compute the final pressure and temperature.
(b) Compute the total increase in entropy.

Be sure to give all your reasoning.

Key Concepts

Reversible vs. Irreversible Processes

Reversible processes are idealized transformations that occur infinitely slowly so that the system remains in equilibrium and no net entropy is produced, whereas irreversible processes are characterized by finite differences and net increases in system entropy. Recognizing this distinction is essential for analyzing the efficiency and entropy production in real-world thermodynamic processes.

Entropy and the Second Law of Thermodynamics

Entropy is a measure of the degree of disorder or randomness within a system, and according to the second law of thermodynamics, the entropy of an isolated system tends to increase during irreversible processes. Calculating entropy changes helps to quantify the irreversibility and feasibility of thermodynamic processes.

Internal Energy and Specific Heat Capacities

For ideal gases, the internal energy depends directly on temperature and the number of moles, with specific heats (like Cv for constant volume) linking temperature changes to changes in internal energy. This understanding is key when applying energy conservation principles to compute final states in thermodynamic processes.

Energy Conservation in Isolated Systems

In a thermally insulated or isolated system, the total internal energy remains constant since there is no energy transfer across the system’s boundary. This principle is crucial for analyzing processes where internal energy changes due to work and heat redistribution occur without external influences.

Mechanical Equilibrium

Mechanical equilibrium is achieved when there is no net force acting on any movable barrier or diaphragm in a system, resulting in equal pressures on either side. This concept is essential when dealing with systems containing a movable interface, such as a piston that shifts until the pressures balance.

Thermal Equilibrium

Thermal equilibrium refers to the state reached when all parts of a system that can exchange heat come to have the same temperature. In processes where initial temperature differences exist, heat conduction ensures that eventually no temperature gradients remain throughout the system.

Ideal Gas Law

The ideal gas law, expressed by the equation pV = nRT, establishes a relationship between pressure, volume, and temperature for an ideal gas. It is indispensable for determining state variables during thermodynamic changes and is fundamental in evaluating the behavior of gases in confined volumes.

Transcript

Please give Ace some feedback