Question

Show that processes that use work for mixing are irreversible by considering an adiabatic system whose contents are stirred by turning a paddle wheel inside the system (e.g., stirring a cake mix with an electric mixer).

Short answer

Answer: The process of mixing in an adiabatic system using a paddle wheel is considered irreversible because the internal energy and entropy of the system increase during the process, and there is no way to return the system to its initial state without external changes, as no heat is exchanged with the surroundings in an adiabatic system.

Step-by-step solution

Identify important terms and concepts

First, let's clarify some terms and concepts that are important for this exercise: 1. Adiabatic system: A system that does not exchange heat with its surroundings. 2. Paddle wheel: A device inside the system that stirs its contents by using work (in this case, operated by an electric mixer). 3. Irreversible process: A process that cannot be reversed, following its natural course. In other words, the system cannot return to its initial state without any change in the surroundings.

Describe the initial state of the adiabatic system

At the beginning of the process, the system is in equilibrium and the internal energy is constant. No work has been done on the system yet, and since it is adiabatic, no heat has been transferred either.

Apply work to the system using the paddle wheel

When we turn on the electric mixer, the paddle wheel begins to stir the contents of the system. In doing so, it does work on the system. Since the system is adiabatic, this work does not transfer any heat to the surroundings, but it increases the internal energy of the system.

Observe the entropy change of the system

As the paddle wheel stirs the contents and increases the internal energy, the entropy of the system also increases. Entropy is a measure of the disorder or chaos of a system. In this case, the mixing process is creating more disorder in the system, which leads to an increase in entropy. Entropy can only increase or remain the same in a natural process, so the process is irreversible.

Check if the process can return to the initial state without external changes

To determine if the process is irreversible, we need to check whether the system can return to its initial state without any external changes. To do this, we would need to remove the work done by the paddle wheel. However, since no heat was exchanged with the surroundings (adiabatic system), we cannot remove this work and return the system to its initial state. Therefore, the process is irreversible In conclusion, the process of mixing in an adiabatic system using a paddle wheel is an irreversible process. The increase in internal energy and entropy, along with the inability to return the system to its initial state without external changes, show that the process is inherently irreversible.

Key concepts

Adiabatic System

Imagine a system that remains untouched by the temperature of its environment. This is what an adiabatic system is - a sealed off scenario where no heat enters or leaves. It's akin to putting a cake mix in an oven without turning it on and expecting it not to bake. In our particular exercise, the adiabatic system signifies a container in which a substance, say our theoretical cake mix, is being stirred. The unique attribute of such a system is that all the work done by the stirring paddle wheel gets converted directly into the internal energy of the substance.

Due to this isolation from the environment, we're forced to look within the system to understand changes, such as in temperature or energy, purely as results of work done inside it, not as a consequence of heat exchange with the surroundings. This plays a significant role in defining the nature of the thermodynamic process occurring within it.

Entropy Increase

Entropy is quite the character - often described using terms like 'disorder' and 'randomness'. In everyday language, entropy is what happens to your bedroom over a week of not tidying it up. In thermodynamics, entropy quantifies the level of molecular chaos. When an adiabatic system is stirred, and the particles inside begin moving more vigorously and randomly, the entropy jumps up.

An important note is that entropy isn't just about messiness; it reflects the number of ways a system can be arranged. Stirring, as it happens in our scenario, creates more arrangements for the particles, hence the entropy increases. This jump is a telltale sign of irreversibility - once increased, entropy doesn't just go back down on its own, just like your messy room won't tidy itself up.

Internal Energy

Think of internal energy as the stash of energy that every substance has locked within it due to the movements and vibrations of its atoms and molecules. It's somewhat like the energy you have before starting an exercise - ready and available. In thermodynamics, when work is done on or by the system, we see a shift in this internal energy. For our paddle wheel stirring the cake mix, all the work done by the paddle wheel is directly transformed into this internal energy.

If you couldn't feel the heat or see the batter mixing, you'd know something is happening just by checking the internal energy of the system. Just as a hiker's energy diminishes with every step, the energy of our system increases with each stir - since no heat is lost in our adiabatic system, every joule of work is a joule of internal energy gained.

Paddle Wheel Stirring

By bringing in the 'paddle wheel', we're essentially introducing a device that enacts work on a substance. It's a perfect example of work input that doesn't involve heat exchange with the environment, which can be hard to visualize. Imagine using a hand mixer in the kitchen, but instead of creaming butter and sugar, you're increasing the kinetic energy of the molecules in our adiabatic system's contents.

Every turn of the paddle wheel injects energy into the system; the particles move faster, bash into each other more frequently, and in a sense, create an unseen dance of chaos that adds to the entropy. While it may seem trivial, the action of the paddle wheel illustrates a foundational concept of thermodynamics: that work can alter a system's state in ways not just tied to heating or cooling.