Carnot Engine and PV work

[vicinihil]

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Can anyone shed some light on this? I'm thoroughly confused and I tend to miss most of these questions in the EK 1001

 

[Foghorn]

Basics

Heat Engine Cycle
A heat engine typically uses energy provided in the form of heat to do work and then exhausts the heat which cannot be used to do work. Thermodynamics is the study of the relationships between heat and work. The first law and second law of thermodynamics constrain the operation of a heat engine. The first law is the application of conservation of energy to the system, and the second sets limits on the possible efficiency of the machine and determines the direction of energy flow.

Heat engines such as automobile engines operate in a cyclic manner, adding energy in the form of heat in one part of the cycle and using that energy to do useful work in another part of the cycle.

PV Diagrams
Pressure-Volume (PV) diagrams are a primary visualization tool for the study of heat engines. Since the engines usually involve a gas as a working substance, the ideal gas law relates the PV diagram to the temperature so that the three essential state variables for the gas can be tracked through the engine cycle. Since work is done only when the volume of the gas changes, the diagram gives a visual interpretation of work done. Since the internal energy of an ideal gas depends upon its temperature, the PV diagram along with the temperatures calculated from the ideal gas law determine the changes in the internal energy of the gas so that the amount of heat added can be evaluated from the first law of thermodynamics. In summary, the PV diagram provides the framework for the analysis of any heat engine which uses a gas as a working substance.

For a cyclic heat engine process, the PV diagram will be closed loop. The area inside the loop is a representation of the amount of work done during a cycle. Some idea of the relative efficiency of an engine cycle can be obtained by comparing its PV diagram with that of a Carnot cycle, the most efficient kind of heat engine cycle.

Heat Engines
A heat engine typically uses energy provided in the form of heat to do work and then exhausts the heat which cannot be used to do work. Thermodynamics is the study of the relationships between heat and work. The first law and second law of thermodynamics constrain the operation of a heat engine. The first law is the application of conservation of energy to the system, and the second sets limits on the possible efficiency of the machine and determines the direction of energy

General heat engines can be described by the reservoir model (left) or by a PV diagram (right)

Energy Reservoir Model
One of the general ways to illustrate a heat engine is the energy reservoir model. The engine takes energy from a hot reservoir and uses part of it to do work, but is constrained by the second law of thermodynamics to exhaust part of the energy to a cold reservoir. In the case of the automobile engine, the hot reservoir is the burning fuel and the cold reservoir is the environment to which the combustion products are exhausted.

The efficiency expression given is a general one, but the maximum efficiency is limited to that of the Carnot cycle. This limitation is often called the thermal bottleneck.

 

[Foghorn]

The cycle and entropy

Carnot Cycle
The most efficient heat engine cycle is the Carnot cycle, consisting of two isothermal processes and two adiabatic processes. The Carnot cycle can be thought of as the most efficient heat engine cycle allowed by physical laws. When the second law of thermodynamics states that not all the supplied heat in a heat engine can be used to do work, the Carnot efficiency sets the limiting value on the fraction of the heat which can be so used.

In order to approach the Carnot efficiency, the processes involved in the heat engine cycle must be reversible and involve no change in entropy. This means that the Carnot cycle is an idealization, since no real engine processes are reversible and all real physical processes involve some increase in entropy.

The conceptual value of the Carnot cycle is that it establishes the maximum possible efficiency for an engine cycle operating between Th and Tc.

Entropy and the Carnot Cycle

The efficiency of a heat engine cycle is given by

For the ideal case of the Carnot cycle, this efficiency can be written

Using these two expressions together

If we take Q to represent heat added to the system, then heat taken from the system will have a negative value. For the Carnot cycle

which can be generalized as an integral around a reversible cycle

(Clausius Theorem)

For any part of the heat engine cycle, this can be used to define a change in entropy S for the system

or in differential form at any point in the cycle

For any irreversible process, the efficiency is less than that of the Carnot cycle. This can be associated with less heat flow to the system and/or more heat flow out of the system. The inevitable result is

(Clausius Inequality)

Any real engine cycle will result in more entropy given to the environment than was taken from it, leading to an overall net increase in entropy.

 

[michigator04]

Is that stuff even on the list of topics? I've never seen that before (and I already took the MCAT).

 

[vicinihil]

yea I didn't think so, but I'm working through EK 1001 for Chem and this part is rocking my booty

 

[majik1213]

if it shows up on the MCAT, it should appear as a passage. I don't think any questions would directly test your knowledge of the carnot engine. Some of the EK questions ask for more discreet knowledge than required by the MCAT for the purposes of making you stronger. I know for sure that no calculus-dependent questions shall appear on the MCAT, whether passage-based or discreet.

 

[BrokenGlass]

yea I didn't think so, but I'm working through EK 1001 for Chem and this part is rocking my booty

The idea here is to convert heat to work (heat engines) or work to heat (heat pumps or refrigerators).

 

[BerkReviewTeach]

The idea here is to convert heat to work (heat engines) or work to heat (heat pumps or refrigerators).

Excellent!!! Keep it simple.

Complex concepts such as the Carnot cycle are simplified at the MCAT level. While the previous posts with the calculus-based derivations are accurate and pertinent to a physical chemistry course, be careful not to go beyond the scope of the exam you are preparing for. Keep it simple.

Engines convert heat into work by way of expanding a gas within a cylinder-piston system. If heat is not converted to work (wasted as exhaust) or some of the work is converted back into heat (via friction), then the engine is not as efficient. Efficiency is simply what you get out divided by what you put in.

Think about it conceptually and not mathematically. The passages may present the daunting graphs and perhaps even some ugly math and unfamiliar terms, but it's all about questions. Keep things simple and you'll be fine on the questions.