Question

A thermoelectric device is used to build a small refrigerator that can hold two pop cans. When the device is operating, the cold side of the device is at \(T=10{ }^{\circ} \mathrm{C}\) while the hot side of the device, outside the refrigerator, is at \(T=42^{\circ} \mathrm{C}\). What is the maximum possible efficiency of this device?

Short answer

The maximum possible efficiency is 10.14%.

Step-by-step solution

Understand the Concept of Maximum Efficiency

The maximum possible efficiency of a thermoelectric device is determined by the Carnot efficiency, which is the maximum efficiency any heat engine can achieve operating between two reservoirs at different temperatures.

Convert Temperatures to Kelvin

To use the Carnot efficiency formula, convert temperatures from Celsius to Kelvin. Add 273.15 to each temperature. Thus, the cold side temperature is \(10 + 273.15 = 283.15\,\text{K}\) and the hot side temperature is \(42 + 273.15 = 315.15\,\text{K}\).

Apply the Carnot Efficiency Formula

The Carnot efficiency \( \eta_c \) is given by the formula: \( \eta_c = 1 - \frac{T_{cold}}{T_{hot}} \), where \( T_{cold} \) and \( T_{hot} \) are in Kelvin.

Compute the Efficiency

Substitute \( T_{cold} = 283.15\,\text{K} \) and \( T_{hot} = 315.15\,\text{K} \) into the formula: \( \eta_c = 1 - \frac{283.15}{315.15} \approx 0.1014 \). Convert this to a percentage by multiplying by 100, giving \( 10.14\% \).

Key concepts

Carnot Efficiency

The concept of Carnot efficiency is central to understanding how thermal systems perform. It defines the upper limit of efficiency for any heat engine operating between two heat reservoirs at different temperatures. This principle, formulated by Sadi Carnot in 1824, is theoretical and represents the most efficient possible conversion of heat into work. To calculate the Carnot efficiency \( \eta_c \), use the equation: \( \eta_c = 1 - \frac{T_{\text{cold}}}{T_{\text{hot}}} \). Here, \( T_{\text{cold}} \) and \( T_{\text{hot}} \) are the absolute temperatures of the cold and hot reservoirs, respectively. These temperatures need to be in Kelvin to ensure accuracy. This efficiency highlights the importance of operating temperature ranges, as a greater temperature difference between the reservoirs leads to higher efficiency.

Temperature Conversion

Temperature conversion is essential in thermodynamics as it standardizes temperature units, enabling accurate calculations and comparisons. In scientific contexts, temperatures are often converted to the Kelvin scale from Celsius or Fahrenheit. For Celsius to Kelvin conversion, add 273.15 to the Celsius temperature. For instance, a temperature of \(10 \degree \text{C} \) converts to \(10 + 273.15 = 283.15 \, \text{K}\). This conversion is crucial when calculating efficiencies, such as Carnot efficiency, where temperatures must be in Kelvin to apply the appropriate formulas correctly.

Kelvin Scale

The Kelvin scale is the fundamental scale for thermodynamic temperature measurement and is based on absolute zero, the point at which particles have minimal vibrational motion. Unlike Celsius and Fahrenheit, the Kelvin scale begins at absolute zero \( 0 \text{K} \), equivalent to \(-273.15 \degree \text{C} \). Thus, it does not have negative values. This feature makes Kelvin especially useful in scientific equations and calculations. When performing conversions for thermodynamic calculations, such as finding a system’s efficiency, using the Kelvin scale ensures accuracies in results since it directly relates to the absolute thermodynamic principles.

Thermodynamics

Thermodynamics is the area of physics that examines how heat and temperature relate to energy, work, and the physical properties of matter. It is based on four laws that describe the behavior of these elements in various systems. The key components include energy conservation, the direction of energy transfer, and the concept of entropy, among others. It helps us understand how to improve system efficiencies, such as maximizing the efficiency of refrigeration systems using the principles of heat exchange and energy conservation, as depicted by the Carnot cycle. Considering energy transformations, thermodynamics thus enables the maximization of energy usage in devices like thermoelectric refrigerators.