Question

A Carnot engine is operated between two heat reservoirs at temperatures of 520 $K$ and 300 $K$. (a) If the engine receives 6.45 $kJ$ of heat energy from the reservoir at 520 $K$ in each cycle, how many joules per cycle does it discard to the reservoir at 300 $K$? (b) How much mechanical work is performed by the engine during each cycle? (c) What is the thermal efficiency of the engine?

Short answer

The engine discards 3721.65 J, performs 2728.35 J work, and has 42.3% efficiency.

Step-by-step solution

Understand the Carnot Engine

A Carnot engine is a theoretical engine that operates on the reversible Carnot cycle. It serves between two thermal reservoirs at different temperatures. The efficiency of a Carnot engine is determined by the temperatures of these two reservoirs.

Apply the Carnot Efficiency Formula

The thermal efficiency of the Carnot engine is given by the formula: $\eta =1-\frac{T_C}{T_H}$, where $T_H$ is the temperature of the hot reservoir (520 K) and $T_C$ is the temperature of the cold reservoir (300 K).

Calculate Efficiency

Plug in the values into the efficiency formula: $\eta =1-\frac{300}{520}$. This calculates to $\eta \approx 0.423$ or 42.3%.

Find the Work Done per Cycle

The work done by the engine, $W$, is the difference in the heat absorbed, $Q_H=6450\text{ J}$, and the heat rejected, $Q_C$. The relation is $W=Q_H-Q_C$.

Heat Rejection Formula

Another way is to use the relationship $Q_C=Q_H(1-\eta )$. We already know $Q_H=6450\text{ J}$ and $\eta =0.423$. Substitute these values to find $Q_C=6450\times (1-0.423)=3721.65\text{ J}$.

Calculate Work Using Efficiency

Using the efficiency formula $\eta =\frac{W}{Q_H}$, rearrange to find $W=\eta \times Q_H=0.423\times 6450=2728.35\text{ J}$.

Compile Results

In each cycle, 3721.65 J of heat energy is discarded, 2728.35 J of mechanical work is performed, and the thermal efficiency is 42.3%.

Key concepts

Carnot cycle

The Carnot cycle is a fundamental concept in thermodynamics, which is essential to understand the operations of a Carnot engine. It represents a theoretical cycle made up of four reversible processes: two isothermal (constant temperature) and two adiabatic (no heat exchange) transformations. The cycle helps illustrate the maximum possible efficiency a heat engine can achieve.
In its complete form:

• First, the engine absorbs heat from a hot reservoir through an isothermal process.
• Then, it expands adiabatically, doing work on its surroundings.
• Third, it discharges heat to a cold reservoir through an isothermal process.
• Finally, it compresses adiabatically back to its initial state.

This cycle is ideal and perfectly reversible, meaning no energy is lost to the environment as waste. The Carnot cycle not only sets a benchmark for efficiency but also provides a framework to study real-life engines.

thermal efficiency

Thermal efficiency is a key metric in evaluating the performance of heat engines, including the Carnot engine. It is defined as the ratio of the work done by the engine to the heat energy absorbed from the hot reservoir. This efficiency represents how effectively an engine converts heat into useful work.
For a Carnot engine, thermal efficiency can be calculated using the formula: $$\eta =1-\frac{T_C}{T_H}$$ where $T_H$ and $T_C$ are the absolute temperatures of the hot and cold reservoirs, respectively. The closer the efficiency of an engine is to 1 (or 100%), the more efficient it is.
Since real engines suffer from practical limitations, their efficiencies are always less than the maximum possible efficiency represented by the Carnot cycle. However, by understanding thermal efficiency, engineers can design engines that operate as efficiently as possible given real-world constraints.

heat reservoirs

In the context of a Carnot engine, heat reservoirs are essential components. They are large thermal systems from which the engine absorbs or to which it rejects heat. These reservoirs maintain a constant temperature even as heat energy is exchanged.
The two reservoirs in a Carnot engine serve very specific purposes:

• The hot reservoir supplies heat energy to the engine, enabling it to do mechanical work.
• The cold reservoir absorbs the residual heat energy which is not converted into work.

The temperatures of these reservoirs directly affect the engine's thermal efficiency and overall performance. The larger the temperature differential between the reservoirs, the greater potential for efficiency, as outlined by the Carnot efficiency formula. A thorough understanding of heat reservoirs is crucial for analyzing how heat engines function and how they can be optimized for better performance.

mechanical work

Mechanical work in a Carnot engine is the useful output obtained when heat energy is converted. It is essentially the energy used to perform tasks such as driving machinery or generating electricity.
For a Carnot cycle, the work done per cycle, $W$, can be calculated using the equation:$$W=Q_H-Q_C$$ where $Q_H$ is the heat absorbed from the hot reservoir, and $Q_C$ is the heat discarded to the cold reservoir. This formula illustrates the transformation of received heat into work and rejected heat.
Another approach to calculate mechanical work is by using the efficiency of the engine: $$W=\eta \times Q_H$$Here, $\eta$ represents the thermal efficiency, showing how effectively the engine utilizes the absorbed heat to do work. Understanding how mechanical work is derived from thermal processes is fundamental in evaluating the practicality and limitations of engines.

temperature differential

Temperature differential in a Carnot engine refers to the difference in temperatures between the hot and cold reservoirs, denoted as $T_H-T_C$. This differential is critical in determining the engine's maximum efficiency and performance.
A larger temperature differential means a greater potential efficiency for the engine, as suggested by the formula for Carnot efficiency:$$\eta =1-\frac{T_C}{T_H}$$This equation shows that efficiency approaches nearer to its theoretical maximum as the cold reservoir temperature $T_C$ decreases, or the hot reservoir temperature $T_H$ increases.
It's important to note that in practical applications, the temperature differential is often limited by material constraints and environmental considerations. Nevertheless, optimizing this differential is a key focus in the design and improvement of heat engines, highlighting its significance in thermodynamics research and engineering.