Problem 16 - College Physics

Question

\(\cdot\) A window air-conditioner unit absorbs \(9.80 \times 10^{4} \mathrm{J}\) of heat per minute from the room being cooled and in the same period deposits \(1.44 \times 10^{5} \mathrm{J}\) of heat into the outside air. What is the power consumption of the unit in watts?

Step-by-Step Solution

Answer

The power consumption is approximately 767 watts.

Step 1: Identify the Energy Transferred

The air-conditioner absorbs heat energy from the room and releases it outside. This energy transfer mechanism involves some work done by the air-conditioner. Let's denote the energy absorbed from the room as \(Q_\text{in} = 9.80 \times 10^{4} \mathrm{J}\) and the energy released outside as \(Q_\text{out} = 1.44 \times 10^{5} \mathrm{J}\) per minute.

Step 2: Determine the Work Done by the Air-Conditioner

The work done by the air-conditioner, \(W\), is equal to the difference between the energy transferred to the outside and the energy absorbed from the room: \(W = Q_\text{out} - Q_\text{in}\). Substitute the values: \(W = 1.44 \times 10^{5} \mathrm{J} - 9.80 \times 10^{4} \mathrm{J} = 4.60 \times 10^{4} \mathrm{J}\).

Step 3: Convert Energy Transferred to Power Consumption

Power consumption is the rate of doing work or transferring energy. It is given by \(P = \frac{W}{t}\), where \(W\) is the work done and \(t\) is the time in seconds. Since the energy transfer is per minute, and there are 60 seconds in a minute, \(P = \frac{4.60 \times 10^{4} \mathrm{J}}{60 \mathrm{s}}\).

Step 4: Calculate Power in Watts

Compute the power using the formula: \(P = \frac{4.60 \times 10^{4} \mathrm{J}}{60 \mathrm{s}} = 766.67 \mathrm{W}\). Thus, the power consumption of the air-conditioner is approximately \(767 \mathrm{W}\) when rounded to the nearest whole number.

Key Concepts

Heat Transfer

Heat transfer is a key principle in thermodynamics, which involves the movement of heat energy from one place to another. In our window air-conditioner scenario, the device absorbs heat from the room and expels it outside.
This process can be broken down into two main components:

• Heat Absorption: The air-conditioner takes in heat from the indoor space, which is denoted as \(Q_\text{in} = 9.80 \times 10^{4} \mathrm{J}\) per minute.
• Heat Rejection: At the same time, it releases heat into the outdoor environment, represented by \(Q_\text{out} = 1.44 \times 10^{5} \mathrm{J}\).

The difference in these heat quantities reflects the external work done by the machine, crucial for its cooling effect.

Power Consumption

Power consumption tells us how much energy a device uses over time. It is measured in watts, which is equivalent to joules per second. The air-conditioner performs the dual role of absorbing heat from inside and releasing it outside, consuming energy to do this.
To find the unit's power consumption, calculate how much energy it uses per second. First, determine the total work done, represented in joules, and then divide this by the duration in seconds:

• Use the formula: \(P = \frac{W}{t}\), where \(W\) is work and \(t\) is time in seconds.
• Given that \(W = 4.60 \times 10^{4} \mathrm{J} \) and \(t = 60 \mathrm{s}\), the power consumed is \(766.67 \mathrm{W}\).

Rounding gives us approximately \(767 \mathrm{W}\), capturing the efficient energy usage of the air-conditioner.

Work Done

Work done in thermodynamic systems like air conditioners is the energy required to transfer heat against natural temperature gradients. In our air-conditioner example:
The work done by the unit \(W\) is calculated as the difference between the energy rejected outside and the energy absorbed from inside:

• Total work done is \(W = Q_\text{out} - Q_\text{in} = 1.44 \times 10^{5} \mathrm{J} - 9.80 \times 10^{4} \mathrm{J} = 4.60 \times 10^{4} \mathrm{J}\).
• This value represents how much extra work the air-conditioner must perform to cool the room effectively.

By understanding the work done, we can appreciate the mechanical effort required to maintain comfortable indoor temperatures.

Energy Transfer

Energy transfer in a thermodynamic system refers to how energy in the form of heat is moved or transformed. With an air-conditioner, energy is actively transferred between indoors and outdoors to achieve desired cooling. This transfer involves:

• Uptake of heat energy inside the room, where it is less useful since we want it cooler.
• Release of that energy to the outside, making it effective in achieving cooler indoor temperatures.

The cycle of absorbing heat, performing work to move this heat, and expelling it is central to how these systems alter the indoor climate. It's how machines efficiently manage the energy balance, ensuring comfort.