Problem 101 - Chemistry The Central Science In Si Units

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 \(\Delta E\) for the process?

Step-by-Step Solution

Answer

(a) No work is performed. (b) There is no external pressure to work against. (c) \(\Delta E = 0\).

Step 1: Understand the System

We have two flasks, one with gas and the other evacuated, separated by a valve. The setup is perfectly insulated, meaning no heat exchange occurs with the surroundings.

Step 2: Is Work Performed?

Work in thermodynamics is defined as a process where energy is transferred by a system to its surroundings. For work to be performed by the system, there would have to be an opposing force exerted against which the gas could do work—like a piston moving or resisting external pressure. Since the evacuated flask has no such resistance (as it is a vacuum), no work is performed during the expansion of the gas.

Step 3: Why is No Work Performed?

In this situation, the expansion of gas into a vacuum is called free expansion. Because there's no external pressure to work against, the gas does not perform any work. In the equation for work, work is calculated as: \[ W = P_{ext} \Delta V \] where \( P_{ext} \) is the external pressure. Here, \( P_{ext} = 0 \), thus \( W = 0 \).

Step 4: Determine \( \Delta E \) for the Process

According to the first law of thermodynamics, \[ \Delta E = Q - W \]where \( \Delta E \) is the change in internal energy, \( Q \) is the heat exchanged, and \( W \) is the work done. Since the system is perfectly insulated, \( Q = 0 \) and we know \( W = 0 \), thus:\[ \Delta E = 0 - 0 = 0 \]Hence, the change in internal energy of the system, \( \Delta E \), is zero.

Key Concepts

First Law of Thermodynamics

The First Law of Thermodynamics is a fundamental principle of physics that governs how energy is conserved and transformed in a system. It's often summarized as: energy cannot be created or destroyed, only changed from one form to another. In equation form, it is expressed as:\[ \Delta E = Q - W \]where:

• \( \Delta E \) is the change in internal energy of the system.
• \( Q \) is the heat added to the system.
• \( W \) is the work done by the system.

In the context of our exercise, since the system is perfectly insulated, no heat \( (Q) \) is exchanged with the surroundings, making \( Q = 0 \). Similarly, we learned that no work is done \( (W = 0) \). As a result, the change in internal energy \( \Delta E \) is also zero. This means the energy content within the gas remains constant despite the expansion.

Free Expansion

Free expansion refers to a situation where a gas expands without doing work on the surroundings. This happens because there's no external pressure or opposing force involved during the expansion.
In our exercise, the gas initially confined to one flask expands into the vacuum of another flask as soon as the valve is opened.
Due to the lack of resistance in the evacuated flask, the gas does not perform any mechanical work, demonstrating free expansion.
Free expansion is a special type of expansion where:

• The external pressure \( (P_{ext}) \) is zero.
• There is no change in the environment.
• It's an irreversible process because it happens spontaneously and without control.

The key takeaway is that, during free expansion, the internal energy of the system doesn't change, which aligns with the First Law of Thermodynamics.

Internal Energy

Internal energy is a crucial concept in thermodynamics, representing the total energy contained within a system. It includes kinetic energy from particle motion and potential energy from interactions between particles.
In the setup described, the internal energy of the gas remains unchanged during free expansion. Why? Because no heat is exchanged with the surrounding environment (due to insulation), and no work is done (it’s a free expansion into a vacuum). Thus, according to the First Law of Thermodynamics:\[ \Delta E = 0 \]This tells us that all forms of energy within the gas are conserved.
For students, the concept of internal energy clarifies how energy states can remain unchanged, even when gases expand freely without external interference. Understanding internal energy helps us predict system behavior, such as temperature stability in an insulated container during expansion.

Work in Thermodynamics

In thermodynamics, work involves energy being transferred or transformed when a system exerts force to move something against an external pressure.
The work \( (W) \) done by a thermodynamic system can be calculated with the formula:\[ W = P_{ext} \Delta V \]where:

• \( P_{ext} \) is the external pressure the system is working against.
• \( \Delta V \) is the change in volume.

Returning to our example of free expansion into a vacuum, there is no external pressure \( (P_{ext} = 0) \). Therefore, the formula results in \( W = 0 \).
This means no work is performed.
In simpler terms, think of work as pushing against something. Without something to push against, no work can be done. This distinction is crucial in thermodynamics, as it defines the conditions under which energy transfers occur, helping us predict and analyze system changes.