Question

A certain refrigerator is rated as being \(32.0 \%\) as ef ficient as a Carnot refrigerator. To remove \(100 .\) J of heat from the interior at \(0^{\circ} \mathrm{C}\) and eject it to the outside at \(22^{\circ} \mathrm{C}\), how much work must the refrigerator motor do?

Short answer

Answer: The work done by the refrigerator motor is approximately 4184.10 J.

Step-by-step solution

Convert the temperatures to Kelvin

We need to work with temperatures in Kelvin. To convert the given temperatures from Celsius to Kelvin, we will use the equation: \(K = °C + 273.15\) For the interior temperature (\(T_{cold}\)): \(T_{cold} = 0°C + 273.15 = 273.15 K\) For the exterior temperature (\(T_{hot}\)): \(T_{hot} = 22°C + 273.15 = 295.15 K\)

Calculate the efficiency of the Carnot refrigerator

The efficiency of a Carnot refrigerator (or heat engine) is given by the equation: \(Efficiency_{Carnot} = 1 - \frac{T_{cold}}{T_{hot}}\) Substitute the calculated temperatures from Step 1: \(Efficiency_{Carnot} = 1 - \frac{273.15}{295.15} \approx 0.0746\)

Calculate the efficiency of the given refrigerator

We know that the given refrigerator is \(32.0 \%\) as efficient as a Carnot refrigerator. To find the efficiency of the given refrigerator, we will multiply the efficiency of the Carnot refrigerator by \(0.32\). \(Efficiency_{given} = 0.32 \times Efficiency_{Carnot} = 0.32 \times 0.0746 \approx 0.0239\)

Calculate the work done by the refrigerator motor

We are given that the refrigerator removes \(100 J\) of heat from the interior. Using the efficiency of the given refrigerator, we can find the work done by the motor with the equation: \(Work = \frac{Q_{removed}}{Efficiency_{given}}\) Substitute the given heat removed (\(100 J\)) and the calculated efficiency of the given refrigerator: \(Work = \frac{100}{0.0239} \approx 4184.10 J\) So, the work done by the refrigerator motor is approximately \(4184.10 J\).

Key concepts

Thermodynamic Temperature Conversion

Understanding the conversion between different temperature scales is crucial when working with thermodynamic systems, as these often require temperatures to be expressed in an absolute scale such as Kelvin. The Kelvin scale is an absolute temperature scale, starting at absolute zero, the theoretical point where all molecular motion ceases.

To convert Celsius to Kelvin, the operation is straightforward: you simply add 273.15 to the Celsius temperature. This is key in thermodynamics because calculations involving gas laws, heat cycles, or any thermodynamic process rely on absolute temperatures to ensure accuracy.

For example, if we have a temperature of 20°C and we want to convert it to Kelvin, the equation would look like this:
\(K = 20°C + 273.15 = 293.15 K\). This process of temperature conversion is foundational when analyzing refrigeration systems like the Carnot refrigerator.

Carnot Cycle Efficiency

The Carnot cycle efficiency is vital in understanding the theoretical limits of a heat engine or refrigerator’s performance. For a Carnot refrigerator, the efficiency is defined by the difference in temperature between the cold reservoir (inside of the fridge) and the hot reservoir (the environment), both expressed in Kelvin.

The Carnot efficiency equation is:
\(Efficiency_{Carnot} = 1 - \frac{T_{cold}}{T_{hot}}\).

This formula allows us to understand that as the temperature difference between the two reservoirs decreases, the Carnot refrigerator becomes more efficient. However, actual refrigerators can never be 100% efficient due to practical constraints and the Second Law of Thermodynamics.

Refrigeration Work Calculation

When calculating the work required for a refrigeration process, it is essential to understand the concept of efficiency. For a given refrigerator, we express efficiency as a fraction of the Carnot efficiency. To calculate the work, \(W\), done by the refrigerator motor, one must divide the amount of heat, \(Q_{removed}\), extracted from the cold reservoir by the efficiency of the refrigerator, \(Efficiency_{given}\):

\(Work = \frac{Q_{removed}}{Efficiency_{given}}\).

This formula demonstrates how the refrigerator’s work relates directly to its efficiency. Lower efficiency requires more work to remove the same amount of heat. In practice, this corresponds to more significant electrical energy consumption, essential for environmental and economic reasons.

Kelvin Temperature Scale

The Kelvin scale is an absolute temperature scale used predominantly in the scientific community because it is directly related to the energy of particles. Zero Kelvin, or absolute zero, is the point where particles have the least possible energy and are not moving. Unlike Celsius or Fahrenheit, Kelvin does not use 'degrees'.

The Kelvin temperature scale is imperative in thermodynamics, including studies of heat engines and refrigerators, as calculations depend on an absolute temperature measurement. The conversion from Celsius to Kelvin helps students properly calculate heat transfer, work, and efficiency in thermodynamic systems.