Question

Since the sun drives energy processes on Earth, we could explore the maximum possible thermodynamic efficiency of a process operating between the surface temperature of the sun \((5,800 \mathrm{~K})\) and Earth's surface temperature \((288 \mathrm{~K}) .\) What is this maximum efficiency? \(^{69}\)

Short answer

Answer: The maximum thermodynamic efficiency is approximately 95.03%.

Step-by-step solution

Write down the given temperatures

We are given the surface temperature of the sun, \(T_H = 5,800 \mathrm{~K}\), and Earth's surface temperature, \(T_C = 288 \mathrm{~K}\).

Write down the formula for Carnot efficiency

The formula for Carnot efficiency is given by: Carnot efficiency = \(1 - \frac{T_C}{T_H}\) where \(T_C\) is the temperature of the cold reservoir and \(T_H\) is the temperature of the hot reservoir.

Substitute the given temperature values into the formula

Replacing \(T_C\) with \(288 \mathrm{~K}\) and \(T_H\) with \(5,800 \mathrm{~K}\), we get: Carnot efficiency = \(1 - \frac{288}{5,800}\)

Calculate the maximum efficiency

Now, we can calculate the maximum thermodynamic efficiency: Carnot efficiency = \(1 - \frac{288}{5,800} \approx 1 - 0.0497 \approx 0.9503\) So, the maximum thermodynamic efficiency of a process operating between the surface temperature of the sun and Earth's surface temperature is approximately \(95.03 \%\).

Key concepts

Carnot Efficiency

Carnot efficiency is a critical concept in thermodynamics, providing a theoretical limit for the efficiency of heat engines. Named after the French physicist Sadi Carnot, it sets the maximum possible efficiency that any heat engine can achieve while operating between two reservoirs of heat.

The formula for Carnot efficiency is:

• Carnot Efficiency = \(1 - \frac{T_C}{T_H}\)

where \(T_C\) is the absolute temperature of the cold reservoir and \(T_H\) is the absolute temperature of the hot reservoir. This formula illustrates that:

• Increasing the temperature difference between the hot and cold reservoirs increases efficiency.
• If the temperatures of the reservoirs are equal, the efficiency drops to zero.

This equation highlights the inherent limitations posed by the Second Law of Thermodynamics, implying that no heat engine can be 100% efficient as some energy is always lost to the cold reservoir.

Temperature in Thermodynamics

Temperature plays a pivotal role in thermodynamics, influencing how energy and heat transfer occur within a system. Measured in Kelvin (K) when dealing with thermodynamic equations, temperature provides a measure of the thermal energy present.

• Absolute temperature is crucial as it allows consistent calculations without negative values.
• It helps determine the direction of heat flow—from high to low temperature—according to the Second Law of Thermodynamics.

In thermodynamic cycles, such as those analyzed using the Carnot efficiency, understanding temperature differences between hot and cold reservoirs is essential for determining potential energy conversion efficiency.

Larger differences usually lead to higher potential efficiencies, encouraging technologies to maximize temperature gradients where feasible. However, physical constraints often limit these temperature extremes.

Solar Energy

Solar energy is the radiant light and heat harnessed from the sun using a range of ever-evolving technologies. It is the primary source of energy for Earth's climate system and provides a sustainable alternative to fossil fuels.

• Solar panels convert sunlight directly into electricity using photovoltaic cells.
• Besides electricity, solar energy is used for heating water and air in solar thermal applications.

By understanding solar energy within the context of thermodynamics, especially efficiency, we can better appreciate how harnessing it effectively can reduce reliance on diminishing fossil fuels.

Given the sun's surface temperature of about 5,800 K, significant potential exists for energy conversion processes, like solar concentrators and photovoltaic cells, which often focus on optimizing their design to approach closer to the theoretical limits provided by Carnot efficiency.

Earth's Surface Temperature

Earth's surface temperature varies broadly across different regions and times of the year, but the average surface temperature is about 288 K (15°C). This temperature is pivotal for sustaining life and also plays a crucial part in energy exchange and thermodynamic processes.

• It acts as the cold reservoir in many thermodynamic cycles, including those capturing solar energy.
• Changes in Earth's surface temperature can affect the efficiency of energy systems, e.g., a warmer Earth could potentially increase efficiency in capturing solar energy.

In the context of the Carnot cycle, understanding Earth's surface temperature is essential, as it helps determine the efficiency limits of engines and systems converting solar or other thermal energy forms into work.

Accurate modeling and prediction of such temperatures allow better planning and optimization of energy harnessing systems worldwide, aiding both technological advancement and environmental preservation.