Heat Engines and the Second Law of Thermodynamics: Videos & Practice Problems

A heat engine converts heat energy into useful work, exemplified by car engines that burn fuel to generate motion. The efficiency of an engine, denoted as e, is calculated using the equation e = wqₕ or e = 1−qₗₕ. The second law of thermodynamics states that 100% efficiency is impossible, as engines must expel waste heat to a cold reservoir, preventing perpetual motion machines.

Introduction to Heat Engines

The second law of thermodynamics introduces the concept of heat engines, which are devices that convert heat energy into useful work. A practical analogy for understanding heat engines is a car engine, which ignites gasoline to produce heat and subsequently converts that heat into mechanical work to turn the wheels of the car. This process is not perfectly efficient, as some energy is lost as exhaust heat.

In the context of heat engines, an energy flow diagram is often used to illustrate the transfer of heat. This diagram includes three key components: the hot reservoir, the work output, and the cold reservoir. The hot reservoir is the source of heat energy, such as the burning gasoline in a car engine. The work output represents the useful energy produced by the engine, while the cold reservoir is where the waste heat is expelled, analogous to the exhaust pipe of a car.

To understand the operation of heat engines mathematically, we can refer to the first law of thermodynamics, expressed as:

$$\Delta E = q - w$$

In cyclic processes, where the internal energy change (\(\Delta E\)) is zero, the equation simplifies to:

$$q = w$$

This indicates that the work done by the engine is equal to the heat added during the cycle. In heat engines, the net heat (\(q_{net}\)) is the difference between the heat input (\(q_h\)) and the heat expelled (\(q_c\)), leading to the equation:

$$w = q_h - q_c$$

For example, if a heat engine takes in 500 joules of heat and does 300 joules of work, we can calculate the waste heat expelled. Here, \(q_h = 500 \, \text{J}\) and \(w = 300 \, \text{J}\). To find the waste heat (\(q_c\)), we rearrange the equation:

$$q_c = q_h - w$$

Substituting the values gives:

$$q_c = 500 \, \text{J} - 300 \, \text{J} = 200 \, \text{J}$$

This means that out of the 500 joules of heat energy, the engine converts 300 joules into work and expels 200 joules as waste heat. Understanding these principles is crucial for analyzing the efficiency and operation of various heat engines across different applications.

An aircraft engine takes in 9 kJ of heat and expels 6.4 kJ of heat each cycle. How much mechanical work does the engine do each cycle?

• A. 15.4 kJ
• B. 2.6 kJ
• C. 2.6 J
• D. 6.4 kJ

Answer: (B)

A heat engine uses a tank of ice water as a cold reservoir. The engine takes in 8 kJ of heat from the hot reservoir, and the heat expelled melts 18g of ice in the tank. How much work does this engine do?

• A. 14,012 J
• B. 6012 J
• C. 1,988 J
• D. 8000 J

Answer: (C)

Thermal Efficiency & The Second Law of Thermodynamics

In the study of heat engines, understanding thermal efficiency is crucial. Thermal efficiency measures how effectively an engine converts heat energy into work. It is defined as the ratio of useful work output (W) to the heat input (Q_h), expressed mathematically as:

$$ e = \frac{W}{Q_h} $$

Alternatively, thermal efficiency can also be calculated using the heat expelled (Q_c) from the engine, leading to the equation:

$$ e = 1 - \frac{Q_c}{Q_h} $$

In this context, Q_c represents the heat that is not converted into work and is released to a cold reservoir. The efficiency is often expressed as a percentage by multiplying the ratio by 100.

For example, if an engine receives 1,000 joules of heat and produces 0 joules of work, the efficiency would be:

$$ e = \frac{0}{1000} = 0\% $$

This indicates a poorly functioning engine. Conversely, if an engine takes in 1,000 joules and expels 600 joules, the efficiency can be calculated as:

$$ e = 1 - \frac{600}{1000} = 0.4 $$

Multiplying by 100 gives an efficiency of 40%. If the work output is known, and the efficiency is given, one can rearrange the efficiency formula to find the heat input:

$$ Q_h = \frac{W}{e} $$

For instance, if 600 joules of work is produced with an efficiency of 30%, the heat input would be:

$$ Q_h = \frac{600}{0.3} = 2000 \text{ joules} $$

Subsequently, the waste heat can be calculated as:

$$ Q_c = Q_h - W = 2000 - 600 = 1400 \text{ joules} $$

It is important to note that achieving 100% efficiency, where all heat energy is converted into work, is theoretically impossible. This principle is encapsulated in the second law of thermodynamics, which states that some energy must always be expelled as waste heat to a cold reservoir. This prevents the creation of a perpetual motion machine, which would violate the laws of physics by continuously converting heat into work without any energy loss.

In summary, thermal efficiency is a key concept in evaluating the performance of heat engines, and understanding the limitations imposed by the second law of thermodynamics is essential for grasping the fundamental principles of energy conversion.

A steam turbine takes in 75g of water and boils it as heat energy to run a 40% efficient engine. How much work does this engine do per cycle?

• A. 67,800 J
• B. 1.695×10⁵ J
• C. 10,020 J
• D. 4.24×10⁵ J

Answer: (A)

Here’s what students ask on this topic:

What is a heat engine and how does it work?

A heat engine is a device that converts heat energy into useful work. It operates by taking in heat from a high-temperature source (hot reservoir), using part of this energy to perform work, and then expelling the remaining heat to a low-temperature sink (cold reservoir). A common example is a car engine, which burns fuel to generate heat, converting some of this heat into mechanical work to move the car, while the rest is expelled as exhaust. The efficiency of a heat engine is never 100% due to the second law of thermodynamics, which mandates that some energy must always be lost as waste heat.

What is the second law of thermodynamics in relation to heat engines?

The second law of thermodynamics states that it is impossible for any heat engine to convert all the heat energy it absorbs into work. This means that no heat engine can be 100% efficient. Some of the input heat must always be expelled as waste heat to a cold reservoir. This principle prevents the creation of a perpetual motion machine, which would otherwise be able to sustain its own operation indefinitely without any external energy input. The second law is often expressed in terms of entropy, indicating that the total entropy of a closed system can never decrease over time.

How do you calculate the thermal efficiency of a heat engine?

The thermal efficiency (e) of a heat engine is a measure of how well it converts heat into work. It is calculated using the equation:

$e=\frac{W}{Q}$

where W is the work output and Q_H is the heat input from the hot reservoir. Alternatively, it can be expressed as:

$e=1-\frac{Q}{C}$

where Q_C is the heat expelled to the cold reservoir. This ratio indicates the fraction of heat converted to work, with the remainder being waste heat.

What is the significance of the hot and cold reservoirs in a heat engine?

The hot and cold reservoirs are crucial components of a heat engine. The hot reservoir provides the heat energy that the engine converts into work. This is typically a high-temperature source, such as burning fuel in a car engine. The cold reservoir, on the other hand, is where the waste heat is expelled. It is a low-temperature sink that absorbs the unused heat energy. The temperature difference between these two reservoirs drives the heat flow, enabling the engine to perform work. The efficiency of the engine depends on the temperatures of these reservoirs, with greater differences generally leading to higher efficiencies.

Why can't a heat engine achieve 100% efficiency?

A heat engine cannot achieve 100% efficiency due to the second law of thermodynamics. This law states that some energy must always be lost as waste heat to a cold reservoir. This unavoidable loss is due to the increase in entropy, which means that not all the input heat can be converted into work. As a result, every real heat engine must expel some heat to the environment, preventing it from being perfectly efficient. This principle ensures that perpetual motion machines, which would operate indefinitely without external energy, are impossible.