Consider a system consisting of the following apparatus, in which gas is confined in one flask and there is a vacuum in the other flask. The flasks are separated by a valve. Assume that the flasks are perfectly insulated and will not allow the flow of heat into or out of the flasks to the surroundings. When the valve is opened, gas flows from the filled flask to the evacuated one. (a) Is work performed during the expansion of the gas? (b) Why or why not? (c) Can you determine the value of \(\Delta E\) for the process?

To know if work is performed, we should understand the conditions under which work is done. In thermodynamics, work is done when there is a transfer of energy due to a force acting through a displacement. In the case of gas expansion, work is performed when the gas applies a force on a movable object (e.g., a piston) while being displaced. In this exercise, the gas expands from one flask to another with no movable objects in the system. Thus, no force is acting through any displacement, and no work is performed during the gas expansion.

In thermodynamics, when a gas expands against a vacuum, the process is known as "free expansion." As mentioned in Step 1, during the free expansion of the gas, there are no external forces acting on the gas and no movable objects involved in the system. Since there is no force being exerted by the gas through a displacement, no work is done in the process.

The change in internal energy, \(\Delta E\), of a system is related to the work done (\(W\)) and the heat exchanged with the surroundings (\(Q\)) through the first law of thermodynamics:\[\Delta E = Q - W\]However, this problem states that the flasks are perfectly insulated, meaning that there is no heat transfer between the system and the surroundings: \(Q=0\). Furthermore, as established in Steps 1 and 2, no work is done during the process, so \(W=0\). Hence, we can rewrite the first law of thermodynamics equation for this case as:\[\Delta E = 0 - 0\]This equation implies that the change in internal energy, \(\Delta E\), is zero. In conclusion, during the free expansion of the gas (a) no work is performed, because there is no force acting through a displacement, (b) the reason for the absence of work done is the lack of external forces and movable objects during the expansion, and (c) the value of \(\Delta E\) for the process is zero, as no heat is exchanged with the surroundings and no work is done.