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Chapter 5: Problem 4

An inventor claims to have developed a device that undergoes a thermodynamic cycle while communicating thermally with two reservoirs. The system receives energy \(Q_{C}\) from the cold reservoir and discharges energy \(Q_{\mathrm{H}}\) to the hot reservoir while delivering a net amount of work to its surroundings. There are no other energy transfers between the device and its surroundings. Using the second law of thermodynamics, evaluate the inventor's claim.

Short Answer

Expert verified
The inventor's claim violates the second law of thermodynamics.

Step by step solution

01

Understand the Second Law of Thermodynamics

The second law of thermodynamics states that it is impossible for any device to operate on a cycle while transferring heat from a colder body to a hotter body without some form of work being done on the system. In the context of a heat engine, this means that heat must flow naturally from a hot reservoir to a cold reservoir, and work can be extracted only if there is a temperature difference and energy flows in this direction.
02

Identify Heat and Work Terms

Let the heat absorbed from the cold reservoir be denoted by \(Q_C\) and the heat discharged to the hot reservoir be \(Q_H\). The work delivered by the system is denoted as \(W\). According to the problem, the system absorbs energy \(Q_C\) from a cold reservoir and releases \(Q_H\) to a hot reservoir while delivering work \(W\) to its surroundings.
03

Apply the First Law of Thermodynamics

The first law of thermodynamics states that the energy added to the system (heat in) minus the energy leaving the system (work out and/or heat out) is equal to the change in internal energy. For a cycle, the change in internal energy over one complete cycle is zero. Thus, the principle can be written as: \[ Q_H - Q_C = W \]
04

Evaluate Energy Flow

Considering the system receives \(Q_C\) from the cold reservoir and discharges \(Q_H\) to the hot reservoir, with it delivering a net work output of \(W\) to its surroundings, using the first law of thermodynamics: \[ Q_H - Q_C = W \].
05

Check the Second Law of Thermodynamics Compliance

For the second law of thermodynamics to hold, a real device taking \(Q_C\) from the cold reservoir and discharging \(Q_H\) to the hot reservoir cannot have \(Q_C > Q_H\) because it implies moving energy from cold to hot naturally. Usually, the heat flows from the hot reservoir to the cold reservoir in a natural process.
06

Conclude the Evaluation

Since the invention claims receiving heat \(Q_C\) from the cold reservoir and delivering \(Q_H\) to the hot reservoir while performing work on its surroundings, it essentially contradicts the second law of thermodynamics. This means the inventor's claim is not valid according to the laws of physics.

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Key Concepts

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

thermodynamic cycle
A thermodynamic cycle consists of a series of processes that return a system to its initial state. This means that over a complete cycle, the internal energy of the system does not change. It's crucial for understanding devices like engines and refrigerators. Each state within the cycle has specific properties like temperature, pressure, and volume. These cycles can be represented on a P-V diagram (pressure-volume) or T-S diagram (temperature-entropy). Essentially, a cycle allows us to study energy transfer and conversion in a controlled manner.
heat engine
A heat engine is a device that converts thermal energy into mechanical work. It operates on the principle that energy flows from a hot reservoir to a cold reservoir. The engine absorbs heat from the hot reservoir, performs work, and then releases some heat to the cold reservoir. In an ideal case, such as the Carnot engine, the process is reversible and maximum efficiency is achieved. However, real engines are not 100% efficient. Instead, they follow the second law of thermodynamics, which dictates that some energy will always be lost as waste heat.
energy transfer
Energy transfer in a thermodynamic system can occur in the form of heat or work. Heat is the energy transferred due to a temperature gradient, and work is the energy transfer that results when a force moves an object. In our example, the system receives heat \(Q_C\) from a cold reservoir and discharges heat \(Q_H\) to a hot reservoir, while delivering work \(W\) to its surroundings. It is important to track how energy enter and leaves the system to ensure all principles, especially the first and second laws of thermodynamics, are respected.
first law of thermodynamics
The first law of thermodynamics is essentially the law of energy conservation. It states that the change in the internal energy of a system is equal to the heat added to the system minus the work done by the system on the surroundings. For a complete thermodynamic cycle, the change in the internal energy is zero, which simplifies to \[Q_H - Q_C = W\]. This means that the total heat added to the system (minus the heat lost) must equal the work done by the system. Applying this to our problem confirms that the work output equals the net heat transfer.
entropy
Entropy is a measure of the disorder or randomness in a system. According to the second law of thermodynamics, in any energy transfer or transformation, the total entropy of a system and its surroundings always increases for irreversible processes. This is because natural processes tend to move towards a state of greater disorder. For a heat engine working between two reservoirs, entropy will increase overall even though the engine may convert some thermal energy into work. When assessing the inventor's device, the second law helps us understand why the proposed energy flow (cold to hot) without external work being applied violates this fundamental principle.

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Most popular questions from this chapter

A reversible power cycle whose thermal efficiency is \(50 \%\) operates between a reservoir at \(1800 \mathrm{~K}\) and a reservoir at a lower temperature \(T\). Determine \(T\), in \(\mathrm{K}\).

A building for which the heat transfer rate through the walls and roof is \(400 \mathrm{~W}\) per degree temperature difference between the inside and outside is to be maintained at \(20^{\circ} \mathrm{C}\). For a day when the outside temperature is \(4^{\circ} \mathrm{C}\), determine the power required at steady state, \(\mathrm{kW}\), to heat the building using electrical resistance elements and compare with the minimum theoretical power that would be required by a heat pump. Repeat the comparison using typical manufacturer's data for the heat pump coefficient of performance.

Steam at a given state enters a turbine operating at steady state and expands adiabatically to a specified lower pressure. Would you expect the power output to be greater in an internally reversible expansion or an actual expansion?

A certain reversible power cycle has the same thermal efficiency for hot and cold reservoirs at 1000 and \(500 \mathrm{~K}\), respectively, as for hot and cold reservoirs at temperature \(T\) and \(1000 \mathrm{~K}\). Determine \(T\), in \(\mathrm{K}\).

A method for generating electricity using gravitational energy is described in U.S. Patent No. \(4,980,572\). The method employs massive spinning wheels located underground that serve as the prime mover of an alternator for generating electricity. Each wheel is kept in motion by torque pulses transmitted to it via a suitable mechanism from vehicles passing overhead. What practical difficulties might be encountered in implementing such a method for generating electricity? If the vehicles are trolleys on an existing urban transit system, might this be a cost-effective way to generate electricity? If the vehicle motion were sustained by the electricity generated, would this be an example of a perpetual motion machine? Discuss.
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