Question

A refrigerator has a coefficient of performance of \(5.0 .\) If the refrigerator absorbs 40.0 cal of heat from the low temperature reservoir in each cycle, what is the amount of heat expelled into the high-temperature reservoir?

Short answer

Answer: The amount of heat expelled into the high-temperature reservoir is 48.0 cal.

Step-by-step solution

Write down the formula for the coefficient of performance.

The coefficient of performance (COP) of a refrigerator is given by the formula: COP = \(\frac{Q_{absorbed}}{W}\) where \(Q_{absorbed}\) is the heat absorbed from the low temperature reservoir and \(W\) is the work done on the system (refrigerator).

Calculate the work done on the system.

We are given the coefficient of performance (COP) as 5.0 and the heat absorbed from the low temperature reservoir (\(Q_{absorbed}\)) as 40.0 cal. We can rearrange the formula for the coefficient of performance to solve for the work done on the system: \(W = \frac{Q_{absorbed}}{COP}\) Substituting the given values, we have: \(W = \frac{40.0 \text{ cal}}{5.0} = 8.0 \text{ cal}\)

Apply energy conservation to find the heat expelled into the high-temperature reservoir.

According to the energy conservation principle, the total energy input into the system must equal the total energy output. So, the sum of the heat absorbed from the low temperature reservoir (\(Q_{absorbed}\)) and the work done on the system (\(W\)) must equal the heat expelled into the high-temperature reservoir (\(Q_{expelled}\)): \(Q_{expelled} = Q_{absorbed} + W\) Using the results from Step 2, we can now calculate the heat expelled into the high-temperature reservoir: \(Q_{expelled} = 40.0 \text{ cal} + 8.0 \text{ cal} = 48.0 \text{ cal}\) Therefore, the amount of heat expelled into the high-temperature reservoir is 48.0 cal.

Key concepts

Understanding Heat Transfer

In the context of thermodynamics, heat transfer refers to the movement of thermal energy from one object or system to another, driven by a temperature difference. It encompasses various modes such as conduction, convection, and radiation. When we consider a refrigerator like in our exercise, heat is transferred from the inside of the refrigerator (the low temperature reservoir) to the outside environment (the high-temperature reservoir), effectively removing heat from the items we wish to keep cold.

Refrigerator as a Heat Transfer Device

A refrigerator works by absorbing heat from its interior and expelling it to the surrounding air. The exercise demonstrates a practical application by calculating the amount of heat expelled based on the coefficient of performance (COP) and the heat absorbed. This is a clear instance where a thermodynamic device is used to control heat transfer, an essential process in refrigeration technology.

Thermodynamics and Coefficient of Performance

The thermodynamics behind a refrigerator is governed by principles that dictate how energy in the form of heat and work is transferred and transformed. The coefficient of performance (COP) is a key concept in thermodynamics that measures a refrigerator's efficiency. It is defined as the ratio of heat absorbed from the low temperature reservoir to the work input required to transfer that heat.

Calculating Efficiency

COP provides insight into the energy-efficiency of the refrigerator—higher COP values indicate a more efficient refrigerator that requires less work input to transfer a certain amount of heat. In our provided exercise, a COP of 5 means that for every unit of work to remove heat, five units of heat can be absorbed from the cooled space, highlighting the importance of this thermodynamic concept in evaluating appliance performance.

Energy Conservation in Refrigeration

The principle of energy conservation is a fundamental concept in physics and thermodynamics. It states that the total energy in an isolated system remains constant; it can neither be created nor destroyed, only transferred or transformed from one form to another.

Applying Energy Conservation

In our example involving the refrigerator, we use energy conservation to relate the heat absorbed (from the space being cooled), the work done by the refrigerator, and the heat expelled to the environment. The computation in the exercise illustrates this principle by showing that the energy expelled (heat) into the high-temperature reservoir is equal to the sum of the heat absorbed from the low temperature reservoir and the work input. These calculations ensure a better understanding of the process a refrigerator uses to maintain its cool interior and how it adheres to the laws of energy conservation.