Question

A heat pump receives energy by heat transfer from the outside air at \(0^{\circ} \mathrm{C}\) and discharges energy by heat transfer to a dwelling at \(20^{\circ} \mathrm{C}\). Is this in violation of the Clausius statement of the second law of thermodynamics? Explain.

Short answer

No, the heat pump does not violate the Clausius statement because it uses external work to transfer heat from cold to hot.

Step-by-step solution

Understand the Clausius Statement

The Clausius statement of the second law of thermodynamics states that 'It is impossible for any system to operate in such a way that the sole result would be an energy transfer by heat from a cooler to a hotter body.' This means that heat cannot spontaneously flow from a colder region to a hotter region.

Identify the Heat Pump Function

A heat pump is a device that transfers heat from a cooler place to a warmer place using external work. In this case, the heat pump extracts heat from the outside air at 0°C (cold reservoir) and releases it inside the dwelling at 20°C (hot reservoir).

Apply the Clausius Statement

Since the heat pump uses external work to transfer heat from the cold reservoir to the hot reservoir, it does not violate the Clausius statement. The Clausius statement only prohibits spontaneous heat transfer from cold to hot, not when work is being inputted to achieve this transfer.

Conclusion

The operation of the heat pump, which moves heat from a colder area (0°C) to a warmer area (20°C) with the assistance of work, is in accordance with the Clausius statement of the second law of thermodynamics.

Key concepts

second law of thermodynamics

The second law of thermodynamics is a fundamental principle in physics. It states that the total entropy of an isolated system can never decrease over time. Instead, it can only remain constant or increase. This means that energy naturally disperses or spreads out if not hindered by an external force. One of the most famous statements of this law is the Clausius statement. It highlights that heat cannot spontaneously transfer from a cooler body to a hotter body. This makes natural processes irreversible and emphasizes the inherent directionality in heat transfer.
For instance, if you place an ice cube in a warm drink, heat flows from the drink to the ice. The second law ensures we don't see ice cubes spontaneously forming in warm drinks under natural conditions.

heat pump

A heat pump is a device designed to transfer heat from a colder area to a warmer one. It works against the natural heat flow direction, essentially doing the opposite of what a refrigerator does. To achieve this, it requires external work (usually in the form of electricity).
Heat pumps are used in heating systems to maintain a comfortable indoor environment, even when it's cold outside. They extract heat from the outside air, even at temperatures as low as 0°C, and release it indoors at a higher temperature, such as 20°C. This might seem like it violates the second law of thermodynamics, but it doesn't. The law's Clausius statement explicitly allows for such transfers when external work is involved.
The efficiency of a heat pump is often expressed as a Coefficient of Performance (COP), which is the ratio of heat transfer to the work input. A typical heat pump might have a COP ranging from 3 to 5, meaning it transfers 3 to 5 times more energy than it consumes in work.

thermodynamic principles

Thermodynamic principles govern the behavior and interaction of energy and matter. These principles include the laws of thermodynamics, which describe how energy flows and transforms. The first law is the law of energy conservation, stating that energy cannot be created or destroyed, only changed from one form to another.
The second law, as discussed, introduces the concept of entropy and dictates the direction of energy transfer. The third law, less commonly referred to but still significant, states that as the temperature of a system approaches absolute zero, the entropy approaches a constant minimum.
In practical terms, these principles ensure that when designing devices like heat pumps, engineers must account for these laws. They influence not only the feasibility but also the efficiency and potential impact of these devices. For example, when considering the second law, engineers must ensure that external work is applied to drive heat from a cooler to a warmer place, thus respecting the natural thermodynamic behavior.

heat transfer

Heat transfer is the movement of thermal energy from one object or substance to another. There are three primary modes of heat transfer: conduction, convection, and radiation.
Conduction occurs through direct contact between materials, allowing heat to flow from a region of higher temperature to one of lower temperature. An example is the heat you feel when touching a metal spoon that’s been sitting in a hot drink.
Convection involves the movement of fluid (liquid or gas) that carries heat with it. For example, warm air rising and cool air sinking plays a role in heating and cooling buildings.
Radiation is the transfer of heat through electromagnetic waves. The warmth you feel from sunlight is due to radiative heat transfer.
In a heat pump, these principles of heat transfer are critical. The device uses a refrigerant that absorbs heat from outside air (convection and conduction) and releases it indoors. By comprehending how these processes work together, one can better understand why a heat pump can efficiently heat a dwelling even in colder conditions.