A portion of helium gas in a vertical cylindrical container is in thermodynamic equilibrium with the surroundings. The gas is confined by a movable heavy piston. The piston is slowly elevated by a distance \(H\) from its equilibrium position and then kept in the elevated position long enough for the thermodynamic equilibrium to be re-established. After that, the container is insulated and then the piston is released. After the piston comes to rest, what is the new equilibrium position of the piston with respect to initial position? (a) The piston ends up \(0.4 H\) above its initial position (b) The piston ends up \(0.6 H\) above its initial position (c) The piston ends at its initial position (d) The piston ends up \(0.4 H\) below its initial position

At the core of understanding thermal systems, such as a helium gas contained by a piston, is the First Law of Thermodynamics. This principle is the conservation law of energy for thermodynamic systems. It essentially states that the change in the internal energy of a closed system is equal to the amount of heat energy transferred to the system minus the work done by the system on its surroundings. In formula terms, it is expressed as \(\Delta U = Q - W\), where \(\Delta U\) is the change in internal energy, \(Q\) the heat added to the system, and \(W\) the work done by the system.

In context with the exercise, when the gas in the cylinder reaches a new equilibrium after the piston is elevated, and no heat is exchanged due to insulation, the First Law implies that the internal energy remains constant if no work is done. This is why, when the piston is released and eventually comes to rest, it should be at the same internal energy level as initially, which further leads to the conclusion that the piston returns to its original position.

Thermodynamics entails the study of energy transformations, and thermodynamic processes are the paths or sequences of events that a system undergoes from one state of equilibrium to another.

The exercise at hand illustrates a sequence of thermodynamic processes involving an initial state of equilibrium, followed by a change (piston elevation), a re-establishment of equilibrium, insulation, and the subsequent release of the piston. This sequence demonstrates a quasi-static process, where each intermediate state of the system is near equilibrium, ensuring that the thermodynamic properties of the system are well-defined throughout the process.

Understanding each step of this sequence is critical for predicting the behavior of the gas and the piston. Notably, in the final step, as the system is insulated (an adiabatic process since \(Q=0\)), and no work is done (since the piston comes to rest), these conditions strongly influence the final equilibrium state according to the First Law of Thermodynamics.

The behavior of gases under various conditions is described by the gas laws in physics. These laws, such as Boyle's Law, Charles's Law, and Avogadro's Law, simplify the prediction of how a gas will respond to changes in pressure, temperature, and volume.

In the case of the exercise problem, we refer to the Ideal Gas Law, which combines these principles into the equation \(PV=nRT\). Here, \(P\) represents the pressure of the gas, \(V\) its volume, \(n\) the number of moles, \(R\) the ideal gas constant, and \(T\) the temperature. Assuming the number of moles and the gas constant are fixed, this relationship implies that temperature and pressure determine a gas's volume in equilibrium.

Since the helium gas returns to thermal equilibrium with its surroundings and the number of moles and temperature are constant, according to the Ideal Gas Law, the piston must return to its initial position to maintain the same volume and fulfill the equilibrium conditions.

The piston and cylinder system is a foundational concept in thermodynamics, often used to illustrate various thermodynamic processes. A piston is a movable barrier that can change the volume of the gas within the cylinder.

In the exercise provided, the heavy piston is initially in equilibrium, and when it is slowly elevated, it does work on the gas, compressing it and changing the volume of the container. After a new equilibrium state is reached and the system is insulated, the system becomes adiabatic—meaning that heat cannot enter or leave.

If no further work is done, the piston will settle at a position where the conditions of pressure and volume reflect the original equilibrated state. This demonstrates how piston and cylinder systems can vividly model processes such as adiabatic expansions or compressions, under the rule set by thermodynamic principles. It's also a prime example of how work done on or by the gas can result in volume changes while maintaining thermodynamic balance.