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Chapter 19: Problem 14

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?

Short Answer

Expert verified
(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 by step solution

01

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.
02

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.
03

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.
04

Determining the Sign of 螖U

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, \(\Delta U > 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A cylinder contains 0.250 mol of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) gas at a temperature of \(27.0^{\circ} \mathrm{C}\) . The cylinder is provided with a frictionless piston, which maintains a constant pressure of 1.00 \(\mathrm{atm}\) on the gas. The gas is heated until its temperature increases to \(127.0^{\circ} \mathrm{C}\) . Assume that the \(\mathrm{CO}_{2}\) may be treated as an ideal gas. (a) Draw a \(p V\) -diagram for this process. (b) How much work is done by the gas in this process? (c) On what is this work done? (d) What is the change in internal energy of the gas?(e) How much heat was supplied to the gas? (f) How much work would have been done if the pressure had been 0.50 \(\mathrm{atm} ?\)

Engine Thrbochargers and Intercoolers. The power output of an automobile engine is directly proportional to the mass of air that can be forced into the volume of the engine's cylinders to react chemically with gasoline. Many cars have a turbochargers which compresses the air before it enters the engine, giving a greater mass of air per volume. This rapid, essentially adiabatic compression also heats the air. To compress it further, the air then passes through an intercooler in which the air exchanges heat with its surroundings at essentially constant pressure. The air is then drawn into the cylinders. In a typical installation, air is taken into the turbocharger at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{Pa}\right)\) , density \(p=1.23 \mathrm{kg} / \mathrm{m}^{3}\) , and temperature \(15.0^{\circ} \mathrm{C}\) . It is compressed adiabatically to \(1.45 \times 10^{5} \mathrm{Pa}\) . In the intercooler, the air is cooled to the original temperature of \(15.0^{\circ} \mathrm{C}\) at a constant pressure of \(1.45 \times 10^{5} \mathrm{Pa}\) (a) Draw a \(p V\) -diagram for this sequence of processes. b) If the volume of one of the engine's cylinders is 575 \(\mathrm{cm}^{3}\) , what mass of air exiting from the intercooler will fill the cylinder at \(1.45 \times 10^{5} \mathrm{Pa} 2\) Compared to the power output of an engine that takes in air at \(1.01 \times 10^{5} \mathrm{Pa}\) at \(15.0^{\circ} \mathrm{C}\) , what percentage increase in power is obtained by using the turbocharger and intercooler? (c) If the intercooler is not used, what mass of air exiting from the turbocharger will fill the cylinder at \(1.45 \times 10^{5} \mathrm{Pa} ?\) Compared to the power output of an engine that takes in air at obtained by using the turbocharger alone?

In a certain process, \(2.15 \times 10^{5} \mathrm{J}\) of heat is liberated by a system, and at the same time the system contracts under a constant external pressure of \(9.50 \times 10^{5} \mathrm{Pa}\) . The internal energy of the system is the same at the beginning and end of the process. Find the change in volume of the system. (The system is not an ideal gas.)

A monatomic ideal gas expands slowly to twice its original volume, doing 300 \(\mathrm{J}\) of work in the process. Find the heat added to the gas and the change in internal energy of the gas if the process is (a) isothermal; (b) adiabatic; (c) isobaric.

A cylinder contains 0.0100 mol of helium at \(T=27.0^{\circ} \mathrm{C}\) (a) How much heat is needed to raise the temperature to \(67.0^{\circ} \mathrm{C}\) while keeping the volume constant? Draw a \(p V\) -diagram for this process. (b) If instead the pressure of the helium is kept constant, how much heat is needed to raise the temperature from \(27.0^{\circ} \mathrm{C}\) to \(67.0^{\circ} \mathrm{C}\) ? Draw a pV-diagram for this process. (c) What accounts for the difference between your answers to parts (a) and (b)? In which case is more heat required? What becomes of the additional heat? d) If the gas is ideal, what is the change in its internal energy in part (a)? In part \((b) ?\) How do the two answers compare? Why?
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