Question

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 $\mathrm{\Delta }E$ for the process?

Short answer

In this problem, when the gas expands from one flask to the other, no work is performed during the expansion as there is no external force acting on the gas due to the vacuum in the evacuated flask. As the flasks are perfectly insulated, there is no exchange of heat between the system and its surroundings. Consequently, according to the first law of thermodynamics, the change in internal energy, $\mathrm{\Delta }E$, is 0 for this process.

Step-by-step solution

(a) Determine if work is performed during the expansion

We need to determine if work is performed during the expansion of the gas from one flask to the other. When the valve is opened, the gas expands into the evacuated flask. During this process, no external force is applied on the gas, since the expansion occurs into the vacuum. Consequently, no work is performed on the surroundings by the gas. Therefore, the work performed during the expansion is zero.

(b) Explain the reason for the conclusion in part (a)

The reason behind the conclusion that no work is performed during the expansion is that there is no external force acting upon the gas during the process. In thermodynamics, work is defined as force applied over a certain distance, which in this case is absent due to the perfect vacuum in the evacuated flask. Therefore, no force is exerted on the surroundings while the gas expands, and no work is performed as a result.

(c) Calculate the value of $\mathrm{\Delta }E$ for the process

Now, we need to determine the value of $\mathrm{\Delta }E$ for the process. In the given problem, the flasks are assumed to be perfectly insulated. This means that there is no exchange of heat between the system and its surroundings. Since there is no exchange of heat and no work done on the surroundings, the first law of thermodynamics applied to this process states that for an internally reversible process, $\mathrm{\Delta }E=q-W$. Since $q=0$ (no exchange of heat) and $W=0$ (work performed during expansion is zero), we can write the equation as follows: $\mathrm{\Delta }E=0-0$ $\mathrm{\Delta }E=0$ Thus, the change in internal energy of the process, $\mathrm{\Delta }E$, is 0.

Key concepts

Understanding Work in Thermodynamics

In thermodynamics, work is a concept that explains how energy is transferred by force. Simply put, work is done when a force moves something over a distance. For example, think of pushing a box across the floor. If we apply a force and the box moves, we've done work on it.

However, in the given exercise, the gas expands into a vacuum. Since there's no opposing force in a vacuum, no work is performed. This is because work in thermodynamics requires some sort of resistance or a force to be moved against. Without this, such as in a vacuum, the work done is zero. That's why the process of gas expanding into another flask in a vacuum involves no work.

Key points about work in an insulated system:

• No work is done if there's no force opposing the movement.
• Work requires energy transfer through force and distance.
• In a vacuum, expansion occurs without external resistance.

Exploring Energy and Its Conservation

Energy is a fundamental concept in physics and thermodynamics. It's the ability to do work or cause change. The first law of thermodynamics introduces the idea of energy conservation, stating that energy cannot be created or destroyed—only transformed from one form to another.

In the context of the exercise, when the gas expands into the evacuated flask, it is crucial to consider the transfer of energy. Since there is no heat transfer and no work done, the energy within the system remains constant. This is represented by the change in internal energy, $\mathrm{\Delta }E$, being zero. This means:

• Internal energy remains unchanged without heat exchange or work.
• The system conserves energy, leading to $\mathrm{\Delta }E=0$.
• The first law of thermodynamics upholds energy balance.

The Role of an Insulated System

An insulated system is crucial in thermodynamics because it prevents heat exchange with its surroundings. This isolation ensures that any process occurring inside the system has no energy exchange due to heat. Imagine placing the flasks in a perfectly insulated container: they neither gain nor lose heat.

In this scenario, insulation allows us to see what happens with the gas without the added complexity of heat transfer. Insulation keeps everything constant, ensuring that only internal changes affect the system. This is why the solutions' assumptions about the insulated flasks are so vital:

• No heat flows in or out of the system.
• The system's energy changes are solely due to internal processes.
• Understanding isolated systems helps in analyzing pure thermodynamic processes.