Problem 14 - University Physics with Modern Physics

Question

A liquid is irregularly stirred in a well-insulated container and thereby undergoes a rise in temperature. Regard the liquid as the system. (a) Has heat been transferred? How can you tell? (b) Has work been done? How can you tell? Why is it important that the stirring is irregular?(c) What is the sign of \(\Delta U ?\) How can you tell?

Step-by-Step Solution

Answer

(a) No heat transfer; system is insulated. (b) Yes, work is done by stirring. (c) \( \Delta U > 0 \) ; temperature rise indicates increased internal energy.

Step 1: Understanding Heat Transfer

To determine if heat has been transferred, consider that the system (liquid) is well-insulated. In thermodynamics, an insulated system means that there is no heat exchange with the surroundings. Thus, no heat has been transferred in this scenario as the system is isolated.

Step 2: Analyzing Work Done

Work is done on a system when an external force is applied causing a process change. In the case of the liquid, stirring involves applying a mechanical force, which is considered work done on the system. Therefore, work has been done on the liquid.

Step 3: Evaluating the Role of Irregular Stirring

Irregular stirring is significant because it indicates that the mechanical energy input is random and not systematic. This randomness ensures that the energy distribution causes a rise in temperature uniformly throughout the liquid rather than a systematic change.

Step 4: Determining the Sign of ΔU

The sign of the change in internal energy (\(\Delta U\)) depends on whether the energy of the system has increased or decreased. Since the temperature of the liquid has risen due to work done (stirring), the internal energy has increased. Thus, \(\Delta U > 0\).

Key Concepts

Heat Transfer

Heat transfer occurs when thermal energy moves between a system and its surroundings due to a temperature difference. However, in the context of a thermodynamics problem where the system is insulated, understanding heat transfer becomes crucial. An insulated system refers to a scenario where no heat transfer can occur. In such systems, the energy exchange is isolated from the surrounding environment.

• The system being insulated means the container does not allow heat in or out.
• In scenarios with no heat exchange, the temperature change must be due to other factors like work done within the system.

When evaluating schematics or setups, one can easily recognize that in an insulated system, like the described container with the liquid, no heat transfer takes place.

Work Done

Work in thermodynamics refers to the process of energy transfer that results from an external force applied to a system. When you stir a liquid, you are exerting a force that causes motion in the liquid molecules. This constitutes work being done on the system.

• The mechanical action of stirring translates into energy imparted to the liquid molecules.


• The energy input due to stirring appears as thermal energy, raising the temperature of the liquid.


• This change is why stirring is an effective mechanism to increase energy without traditional heat transfer.

In this exercise, since the liquid is in an insulated system, stirring is the sole method of increasing internal energy by doing work.

Insulated System

An insulated system plays a pivotal role in controlling energy exchange. By definition, an insulated system guarantees that no heat can move in or out of the container. This is achieved through materials or designs that serve as thermal barriers.

• The main purpose is to prevent energy loss or gain from the surroundings.


• Such systems allow scientists and engineers to isolate variables to study specific energy exchanges, like mechanical work.


• In this scenario, it allows us to focus on how work done affects the system without the interference of external heat.

The insulated system property is thus essential to analyze the changes in internal energy solely due to the work done inside the container.

Internal Energy Change

The internal energy change (\(\Delta U\)) in a system indicates whether the system has absorbed or released energy. It is calculated as the difference between the initial and final state energy levels.

• If \(\Delta U > 0\), the system has absorbed energy, leading to an increase in temperature.


• A positive \(\Delta U\) suggests that work done on the system, in this case through stirring, has added energy to the system.


• \(\Delta U = 0\) would mean no net change, though it's rare in dynamic situations like stirring.

In this exercise, the rise in temperature points directly to an increase in internal energy, meaning \(\Delta U > 0\), primarily driven by the work done through stirring within the insulated system.