Question

A refrigerator does \(153 \mathrm{~J}\) of work to transfer \(568 \mathrm{~J}\) of heat from its cold compartment. (a) Calculate the refrigerator's coefficient of performance. ( \(b\) ) How much heat is exhausted to the kitchen?

Short answer

The refrigerator's coefficient of performance is approximately 3.71 and the amount of heat exhausted to the kitchen is \(721 J\).

Step-by-step solution

Identify given values

From the problem statement, the work done by the refrigerator \(W = 153 J\) and the heat removed from the cold compartment \(Q_c = 568 J\).

Calculating the Coefficient of Performance

To calculate the coefficient of performance (COP) use the formula \(COP = \frac{Q_c}{W}\). Substituting the given values: \(COP = \frac{568 J}{153 J}\). Calculating this gives \(COP \approx 3.71\).

Calculating the Heat Exhausted to the Kitchen

According to the first law of thermodynamics, the heat exhausted to the kitchen \(Q_h\) is the sum of the heat removed from the cold compartment and the work done by the refrigerator. This can be written as: \(Q_h = Q_c + W\). Substituting the given values: \(Q_h = 568 J + 153 J\). Calculating this gives \(Q_h = 721 J\).

Key concepts

First Law of Thermodynamics

The first law of thermodynamics, also known as the law of energy conservation, is a cornerstone in the study of energy systems. It states that energy can neither be created nor destroyed, only transformed from one form to another within a closed system. Mathematically, it's expressed as:

\[ \text{Change in internal energy} = \text{Heat added to the system} - \text{Work done by the system} \]
In practical terms, for any refrigeration process, it means that the energy (in the form of heat) removed from the refrigerated space plus any work input must equal the heat expelled to the surroundings. When solving problems involving refrigerators or heat pumps, this principle ensures that the total amount of energy remains constant. Understanding this law aids in the calculation of the heat expelled to the environment, as demonstrated in the textbook example, where the heat exhausted to the kitchen was found by adding up the heat removed from the cold compartment and the work done by the refrigerator.

Refrigerator Thermodynamics

Delving into refrigerator thermodynamics, the performance of a refrigerator is often quantified by its Coefficient of Performance (COP). This is a measure of how effectively a refrigerator transfers heat from a colder area to a warmer one, compared to the work it consumes to carry out the process. The formal definition of COP for a refrigerator is:

\[ COP = \frac{Q_c}{W} \]
where \(Q_c\) is the amount of heat removed from the cold compartment (inside the refrigerator), and \(W\) is the work done by the refrigerator. A higher COP signifies a more efficient refrigerator, meaning it requires less work to transfer a certain amount of heat. In the example from the textbook, a COP of approximately 3.71 was calculated, indicating that for every unit of energy expended in work, the refrigerator moves about 3.71 units of heat energy out of the cold space.

Heat Transfer

Heat transfer is the movement of thermal energy due to the temperature difference between two objects or systems. In the context of refrigerators, heat transfer is a fundamental process that occurs when the refrigerant absorbs heat from the refrigerator's interior and then releases it to the exterior environment, usually a kitchen. There are three modes of heat transfer: conduction, convection, and radiation. Most refrigerators primarily utilize conduction and convection. Conduction occurs within the refrigerant as it circulates inside the coils, absorbing heat from the interior space. Convection takes place as the external coils dissipate heat into the surrounding air. This expulsion of heat is necessary to maintain the cooling effect inside the refrigerator and is quantified when you investigate how much heat is exhausted to the kitchen, which was 721 J according to the textbook solution.