Question

A heat engine pulls $100\mathrm{J}$ out of a hot bath at $800\mathrm{K}$, and transfers $80\mathrm{J}$ of heat into the cold bath at $300\mathrm{K}$. What efficiency does this heat engine achieve in producing useful work, and how does it compare to the theoretical maximum?

Short answer

Answer: The actual efficiency of the heat engine in this problem is 20%, while the theoretical maximum efficiency based on the Carnot engine is 62.5%. This means that the heat engine's efficiency is less than the theoretical maximum efficiency and there is room for improvement in its design to increase its efficiency.

Step-by-step solution

Calculate the work output

To determine the work output of the heat engine, we need to find the difference between the heat extracted from the hot bath and the heat transferred to the cold bath. Using the formula: Work output = Heat extracted - Heat transferred Substitute the given values: Work output = $100\mathrm{J}-80\mathrm{J}=20\mathrm{J}$

Calculate the actual efficiency of the heat engine

The actual efficiency of the heat engine can be calculated using the formula: Actual efficiency = Work output / Heat extracted Substitute the values from Step 1: Actual efficiency = $20\mathrm{J}/100\mathrm{J}=0.2$ Multiply by 100 to express it in percentage: Actual efficiency = $0.2\times 100=20\mathrm{\%}$

Calculate the theoretical maximum efficiency

The theoretical maximum efficiency of a heat engine is determined by the efficiency of an ideal Carnot engine. The efficiency of a Carnot engine is given by: Carnot efficiency = $1-\frac{T_{cold}}{T_{hot}}$ Substitute the given temperatures: Carnot efficiency = $1-\frac{300\mathrm{K}}{800\mathrm{K}}=1-\frac{3}{8}=\frac{5}{8}$ Multiply by 100 to express it in percentage: Theoretical maximum efficiency = $\frac{5}{8}\times 100=62.5\mathrm{\%}$

Compare the actual efficiency to the theoretical maximum efficiency

Now that we have calculated both the actual efficiency and the theoretical maximum efficiency, let's compare them: Actual efficiency: $20\mathrm{\%}$ Theoretical maximum efficiency: $62.5\mathrm{\%}$ The efficiency of this heat engine is less than the theoretical maximum efficiency of an ideal Carnot engine. This means there is room for improvement in the heat engine design to increase its efficiency.

Key concepts

Heat Engine Efficiency

When we talk about heat engine efficiency, we're referring to how well the engine converts heat into useful work. In simpler terms, efficiency tells us what portion of the input energy is turned into work rather than wasted heat. For our given problem, the engine extracts 100 J of heat from the hot source and transfers 80 J of it to a cold sink. This means it performs 20 J of useful work. The formula used to calculate efficiency is:

• Efficiency = (Work output) / (Heat extracted)

In this scenario, the work output is 20 J, and the heat extracted is 100 J, leading to an efficiency of 20%. This efficiency is not very high, indicating that a significant amount of energy is lost as waste heat.

Carnot Engine

The Carnot engine is a theoretical construct that represents the most efficient possible heat engine cycle. Named after the French physicist Sadi Carnot, it defines the upper limit of efficiency that any heat engine operating between two temperature reservoirs can achieve. The Carnot efficiency is determined by the temperatures of the hot and cold reservoirs.

• Carnot Efficiency = 1 - (T_cold/T_hot)

This is derived from the absolute temperatures of the two reservoirs. For our problem, T_cold is 300 K and T_hot is 800 K, giving us a Carnot efficiency of 62.5%. No real engine can ever reach this efficiency due to inevitable energy losses in practical systems.

Work Output

Work output is the amount of useful work derived from the engine after accounting for the heat transfer. In the context of heat engines, it is the net energy available for use after a portion of the heat input is rejected to the cold sink. From the solved problem, the work output is determined by subtracting the heat transferred to the cold sink (80 J) from the heat extracted from the hot source (100 J). Hence, the work output is:

• Work Output = Heat Extracted - Heat Transferred = 100 J - 80 J = 20 J

This work output is crucial because it represents the actual reason we use heat engines—the ability to perform work using the thermal energy absorbed from the hot reservoir.

Heat Transfer

Heat transfer is the movement of thermal energy from one place or object to another. In the case of heat engines, understanding heat transfer is vital to figuring out how much energy can be turned into work. When a heat engine operates, it takes in energy from a high-temperature reservoir (hot bath) and expels some of it to a low-temperature reservoir (cold bath). The amount of heat taken in minus the heat expelled gives the engine's work output. In practice, two main types of heat transfer occur in engines:

• Conduction: Heat transfer through materials touching each other.
• Convection: Heat transfer through fluids (liquids or gases) moving between areas of different temperature.

It's essential to understand these processes because they determine the efficiency of energy conversion from heat to work in a real-world engine. In the completed exercise, 100 J is absorbed, and 80 J are transferred out, affecting the overall efficiency significantly.