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The first law of thermodynamics is a crucial principle that governs energy transfer in various systems, including heat pumps. According to this law, energy cannot be created or destroyed; it can only change forms. This implies that the total energy of an isolated system remains constant. In the context of a heat pump, this law takes the form of the equation: \( Q_h = Q_c + W \).

Here, \( Q_h \) refers to the heat delivered inside the house, \( Q_c \) represents the heat extracted from the outside environment, and \( W \) is the work done by the heat pump. This equation shows how the work input is utilized to transfer heat, increasing the system's efficiency.

Understanding this energy balance helps in evaluating how effectively a heat pump can move energy against a temperature gradient and maintain comfortable indoor conditions.

The coefficient of performance (COP) is an essential metric for assessing the efficiency of heat pumps. It describes how effectively a heat pump transfers energy in the form of heat relative to the work it consumes. Mathematically, it is expressed as:

\[ \text{COP} = \frac{Q_h}{W} \]

Where \( Q_h \) is the heat delivered to the interior and \( W \) is the work done by the heat pump. A higher COP indicates a more efficient heat pump since more heat is delivered to the interior per unit of work done.

The COP is particularly useful because it helps consumers compare the performance of different heat pumps. By choosing a unit with a higher COP, one can achieve more economical heating, reflecting better energy usage. For the example exercise, the COP calculated is approximately \(2.99\), meaning that for every unit of work, the heat pump delivers nearly three units of heat.

The work done by a heat pump is an integral part of its operation, as it powers the entire process of moving heat from a lower to a higher temperature zone. In the given exercise, it is calculated using the relation from the first law of thermodynamics.

Given the equation \( Q_h = Q_c + W \), where \( Q_h = 3140 \text{ J} \) and \( Q_c = 2090 \text{ J} \), we can rearrange to solve for \( W \):

\[ W = Q_h - Q_c \]

Substituting the known values yields: \( W = 3140 \text{ J} - 2090 \text{ J} = 1050 \text{ J} \).

This means the heat pump requires 1050 joules of work to transfer 2090 joules from outside, delivering a total of 3140 joules inside. Understanding the work done by a heat pump helps in assessing its power requirements, thereby aiding in sizing the unit correctly for specific applications and ensuring its operational efficiency.