Question

Consider a room that is initially at the outdoor temperature of \(20^{\circ} \mathrm{C}\). The room contains a \(40-\mathrm{W}\) lightbulb, a \(110-\mathrm{W}\) TV set, a \(300-\mathrm{W}\) refrigerator, and a \(1200-\mathrm{W}\) iron. Assuming no heat transfer through the walls, determine the rate of increase of the energy content of the room when all of these electric devices are on.

Short answer

Answer: The rate of increase of the energy content of the room is 1650 W or 1650 J/s.

Step-by-step solution

Determine Power Provided by Each Device

The power of each device is given: - Lightbulb: \(40\,\text{W}\) - TV set: \(110\,\text{W}\) - Refrigerator: \(300\,\text{W}\) - Iron: \(1200\,\text{W}\)

Add Power of All Devices

Add the power supplied by each device: \(P_\text{total} = P_\text{lightbulb} + P_\text{TV set} + P_\text{refrigerator} + P_\text{iron}\) \(P_\text{total} = 40\,\text{W} + 110\,\text{W} + 300\,\text{W} + 1200\,\text{W}\) \(P_\text{total} = 1650\,\text{W}\)

Calculate Rate of Increase of Energy Content of the Room

The energy content of the room will increase at a rate that is equal to the total power added by all the devices. As there is no heat transfer through the walls, all of this energy remains within the room. So, the rate of increase of the energy content of the room is \(1650\,\text{W}\) or \(1650\,\text{J/s}\).

Key concepts

Thermodynamics

The exploration of the problem at hand touches upon a fundamental concept in physics known as thermodynamics. In simple terms, thermodynamics is the study of heat, energy, and the work they do within physical systems. It operates under four laws that predict how different forms of energy will behave under varying circumstances. In the scenario provided, we consider a closed system — the room — where the energy supplied by electrical devices is transformed into heat. This transformation and accumulation of heat within the room is a superb demonstration of the first law of thermodynamics, sometimes referred to as the law of energy conservation. This law states that energy cannot be created or destroyed, only changed in form. Here, the electrical energy consumed by devices is fully converted to heat energy because of no heat exchange with the environment, which upholds the law perfectly.

When dealing with similar problems, recognizing that it is a closed system is key to understanding the absence of energy loss. It's imperative when learning thermodynamics to grasp the relationship between heat, work, and internal energy. In instances where the energy output of appliances is considered, students should note that this energy contributes to the internal energy of the room's air, thus increasing its temperature over time.

Power Calculation

Power calculation plays an essential role in the problem we are reviewing. Power, in a physical sense, refers to the rate at which work is done or energy is transferred. In this context, it is measured in watts (W), where one watt is equal to one joule of energy transferred per second. By considering each electrical device as an energy source, the problem requires us to calculate the total power output. It is analogous to calculating the total output rate of energy transfer into the room's air.

The simplicity of adding the wattage of each individual device to find the total power input into the system is a basic yet integral example of power calculation in practice. Students often mistake power for total energy, whereas power is actually the rate at which energy is used or generated. It is beneficial to emphasize the difference: energy is the capacity to do work, while power is how quickly work is done. Remembering this distinction can vastly improve the understanding of power-related calculations in thermodynamics.

Heat Transfer

Heat transfer is pivotal for solving the issue outlined in the exercise, even though in this particular problem it is characterized by its absence. Heat transfer is the physical act of thermal energy being exchanged between different areas or systems. There are three methods of heat transfer: conduction, convection, and radiation. However, in the stated scenario, the crucial point is that the room has been described as insulated from its environment, suggesting no heat exchange through the walls.

This means that heat generated by the appliances cannot escape, leading to a steadily increasing internal energy and thus a rise in temperature. Normally, heat transfer would play a natural, modulating role in balancing the room's temperature, but it’s the prevention of this process that leads us to calculate the accumulation of energy inside the room. Understanding heat transfer concepts aids students in discerning conditions where energy would stabilize (through dissipating heat) versus conditions where it would accumulate, such as in the completely insulated room of the exercise.