# Entropy and the Second Law of Thermodynamics | iCalculator™

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13.10Entropy and the Second Law of Thermodynamics

In this Physics tutorial, you will learn:

• Why the first law of thermodynamics is not sufficient in explaining all thermal-related phenomena?
• What are reversible and irreversible processes?
• What is entropy?
• What are the factors affecting the entropy of a thermodynamic system?
• What happens to the entropy in the universe during a thermal process?
• What does the Second Law of Thermodynamics say?
• What are some equivalent formulations of the Second Law of Thermodynamics?
• What is the operation principle of heat engines?
• How to calculate the efficiency of a heat engine?
• What are Carnot engines and what advantages do they offer compared to the traditional heat engines?
• Why Carnot engines have no practical applications in daily life?

## Introduction

Imagine you are living alone and one day you wake up late in the morning. You don't have time to clean you room, and you leave the house in total mess. After turning back in the afternoon, you find everything in order. What is the first thing that comes to your mind? Is it possible all thing have been fixed automatically without the human interference (for example, the wind blowing through the open window has put everything in order)?

Can you throw a number of pencils up and all pencils line up parallel to each other when they fall on the table? Why?

Which atomic structures do you think are more regular: those with high or low temperature? Why?

All these questions will get answer in this tutorial, which explain things related to the way universe works.

### Does the First Law of Thermodynamics Explain Everything about Thermal Processes?

Despite its importance in determining the values of heat, work and internal energy, the First law of Thermodynamics does not provide sufficient answers to everything that happens during a thermal process. In nature, not every thermodynamic event occurs spontaneously although at least theoretically the First Law allows this; rather, most processes have a well-defined order of occurrence. Let's consider a few examples to explain this point.

When we place in contact a hot and a cold object, the heat energy always flows from the hottest to the coldest object. We know this from practice but we are not able to determine the direction of heat flow from the First Law of Thermodynamics. It simply tells us that the heat released by one object equals the heat absorbed by the other.

Ice melts at room temperature absorbing heat energy from surroundings but it never freezes at room temperature by giving off heat to the surroundings and making the room hotter.

A drop of ink diffuses in the entire volume of a glass filled with water but it never concentrates back into a single drop when diffused in water.

If you shake a box with salt and pepper, they mix up but they cannot separate again if you continue mixing up the box.

If you line up 20 pencils in parallel on a table and then shake the table, the order of pencils will be distorted and you will never be able again to align the pencils so regularly.

It is clear that all these processes are one-directional; this means they never occur spontaneously. From here, we can deduce they must obey to any physical law unexplained so far.

## Reversible and Irreversible Processes

By definition, a reversible process is a thermodynamic process that can reverse without leaving any trace in the surroundings. A process can be either reversible or irreversible, i.e. a process that is not reversible, is irreversible.

All examples taken in the previous paragraph are irreversible processes as it is impossible to turn back naturally into the original state the components of the process. Some other examples of irreversible processes include the effects of friction (teeth of a saw cannot seem anymore as new after months of consumption), chemical reactions, expansion of gas in vacuum, etc.

The term "irreversible" does not mean the system cannot turn again to the original state. For example, if you slide a ball along a horizontal surface, it eventually stops due to friction. You can make the ball slide again at the initial velocity but this needs the involvement of an external factor (the human energy) which causes a permanent change in the surroundings (a human needs to consume fresh food to replace the energy lost). Thus, the ball returns to the original state but the environment not.

If the above process were reversible, it would mean the heat energy exchanged due to friction is sent back to the ball. Obviously, this is impossible. Hence, a reversible process is more an idealization than a reality. However, we use the concept of reversible process as a model to explain more realistic processes, similarly to ideal gas model, which is used to explain the behaviour of real gases.

Slow expansion of compression of gases by supplying or taking of heat in small amounts can be considered as an action that produces reversible processes. We can also obtain slow compression by adding small weights on the piston.

## The Meaning of Entropy

In simple words, Entropy represents the degree of disorder in a thermodynamic system. Entropy therefore is high when the particles in a thermodynamic system move in irregular way. We have explained in the previous tutorials that particles motion is related to temperature, i.e. higher the temperature, more irregular the particles motion. Therefore, entropy is related to the temperature of system.

Also, we know that more heat supplied to a system, higher the kinetic energy of its particles and therefore, higher the degree of disorder in the system. This means entropy is related to the heat supplied to the system.

Combining these two factors together, we obtain the equation of entropy S. It is

S = Q/t

where Q is the heat of system and T its temperature.

However, just like in Kinematics, where we were not that interested in the object's actual position but in the change in position, as it gives a clearer idea about the object's motion, here we are more interested in the change in entropy ΔS rather than in the actual entropy of a thermodynamic system. Therefore, we can write:

∆S = ∆Q/t

The unit of entropy is joule per kelvin [J/K].

### Example 1

200 g ice at 00 is inserted inside a closed room containing water steam at 100°C. Given that the latent heat of fusion for ice is 334 000 J/kg, calculate:

1. The change in entropy for ice
2. The change in entropy for steam
3. The change in entropy in the universe

### Solution 1

a) Since the room is closed, the system is considered as isolated.

Given that 200 g = 0.2 kg and Tice = 0°C = 273 K, we obtain for the change in entropy for ice:

∆Sice = ∆Qice/Tice
= m × Lice/Tice
= 0.2 kg × 334 000 J/kg/273 K
= 66 800 J/273 K
= 244.7J/K

b) The steam is at 100°C = 373 K. Given that the heat absorbed by ice is equal to the heat released by steam (with negative sign), we obtain for the change in entropy of steam:

∆Ssteam = ∆Qsteam/Tsteam
= -66 800 J/373 K
= -179.1J/K

c) The change in entropy in the universe is

∆Suniverse = ∆Sice + ∆Ssteam
= 244.7J/K - 179.1J/K
= 65.6J/K

## The Second Law of Thermodynamics

From the example in the previous paragraph, it is obvious that the change in entropy in the universe is positive for all thermodynamic processes. This means every decrease in the local entropy of a thermodynamic system brings an increase at a higher extent of the entropy in the surroundings (we often say "in the universe" instead of "in the surroundings"). This conclusion forms the basis of the Second Law of Thermodynamics, which says:

The total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible.

The total entropy of a system remains constant only in ideal cases, i.e. when the process is completely reversible.

The Second Law of Thermodynamics is expressed in various forms that apparently seem as not related to each other but that are all equivalent. Some of them include:

Entropy of an isolated system either increases or at best, it remains constant during any change in the system.

It is impossible to convert entirely the heat energy supplied to a thermodynamic system into work. This means no heat engine can provide an efficiency of 100 percent, i.e. no perfect machine can exist.

It is impossible for heat to flow by itself from a colder object to a hotter one (Clausius Statement).

## Heat Engines

Heat engines operate by converting heat into mechanical energy, i.e. they produce motion from heat energy. Car motors, diesel engines, steam turbines and steam power plants are all examples of heat engines.

A heat engine uses a gas at high pressure to push against a piston. For this, it needs a source of thermal energy to heat up the gas inside the cylinder. These thermal energy sources originally are in another form, most commonly as chemical energy of fuels. Therefore, the chain of energy conversions in heat engines is In simple terms, a heat engine absorbs heat energy from a source (called "hot reservoir") at high temperature, then it converts part of this energy into useful work and expels the rest outside the system (in the surroundings). Such a medium is at lower temperature than the system and is known as "cold reservoir" or "heat sink".

A simplified scheme of a heat engine operation is shown in the figure below. QH represents the amount of heat supplied to the heat engine from a source at high temperature,

QC represents the amount of heat energy given off by the heat engine to a low temperature reservoir, usually the atmosphere,

TH is the absolute temperature of the hot reservoir,

TC is the absolute temperature of the cold reservoir, and

W is the mechanical work done by the heat engine.

The cold reservoir is necessary as otherwise, the heat engine would heat up continuously and eventually, it would melt down. Also, both QH and QC are taken as positive.

For continuous operation, a heat engine must operate in cycles, i.e. it must cool down to the initial temperature before starting the new cycle.

### Efficiency of Heat Engines

Efficiency of heat engines is conceptually similar to the efficiency of all mechanical devices, i.e. it represents the ratio of output and input energy of engine. Its formula is

e = W/Q × 100%

where W is the work done by the engine, which represents the output (or useful) energy and Q is the input (or total) heat energy supplied by the source.

### Example 2

400 g of fuel are consumed by a 65 kW heat engine to make a car travel for 4 minutes. The fuel has a calorific value of 45 megajoules/kg. What is the efficiency of the engine? The formula of heat energy released by a burning fuel is

Q = m × q

where m is the mass of fuel and q is its calorific value.

### Solution 2

Work (or useful energy) is obtained by multiplying power and time. Since 65 kW = 65 000 W and 4 minutes = 240 seconds, we have

W = P × t
= 65 000 W × 240 s
= 15 600 000 J

The heat energy released by the burning fuel (which represents the total or input energy of the source) is

Q = m × q
= 0.4 kg × 45 000 000 J/kg
= 18 000 000 J

Therefore, the efficiency of this heat engine is

e = W/Q × 100%
= 15 600 000 J/18 000 000 J × 100%
= 86.7 %

This means 86.7% of the original heat energy supplied by the source is used to do work while the rest 13.3% is wasted energy that goes into the environment through the cold reservoir during cyclic processes.

Now, we can provide a more formal definition for entropy, which is based on the quantities involved in application of the Second Law of Thermodynamics in thermal engines:

Entropy is a thermodynamic quantity representing the unavailability of a system's thermal energy for conversion into mechanical work, often interpreted as the degree of disorder or randomness in the system.

## Carnot Engine

The Second Law of Thermodynamics implies that no perfect machine can exist i.e. the efficiency of heat engines cannot be 1. This means there is some heat lost during the process.

The question arisen here is: How to construct a machine with the maximum efficiency possible and what is the value of this maximum efficiency?

From experiments, it is proven that the type of engine that produces the maximum efficiency is the "Carnot Engine", named after the famous French scientist Sadi Carnot. A Carnot engine produces a definite reversible cycle which has a maximum possible efficiency between two given heat reservoirs with different temperatures. In other words, a Carnot cycle is the most efficient cycle that can theoretically exist.

A schematic-graphical representation of a Carnot cycle is shown in the figure below. From the graph above, you can see that a Carnot cycle consists in two adiabatic and two isothermal processes. All these processes are reversible.

Based on the figure above, we can say that:

• Processes 2-3 and 4-1 are adiabatic. This means no heat is transferred between various parts of the engine.
• Process 1-2 represents an isothermal expansion. During this process, the gas absorbs some heat (QH) from a hot source with temperature TH.
• During the process 3-4, gas is compressed isothermally, giving of some heat (QC) to a cold reservoir with temperature TC.

The figure below shows what happens to the gas produced by the heat source during a Carnot cycle. For the same difference in temperatures, the Carnot engine is the most efficient among all the other types of engines. However, a Carnot engine is more a theoretical model than a reality. As we have stated earlier, it is quite impossible to obtain reversible cycles during a thermal process. Carnot engine however, is very useful in determining the maximum efficiency a heat engine can have. Hence, we can say the efficiency of a real heat engine operating between the temperatures TH and TC is lower than that of a Carnot engine with the same range of temperatures. Thus, we can write

ereal < eCarnot

Two factors decrease the efficiency of a real engine. They are:

1. A real machine does not operate according a Carnot cycle
2. Friction between various parts of the engine decreases further the efficiency

The equation used to find the efficiency in a Carnot engine is

eCarnot = TH - TC/TH × 100%
= TH/TH - TC/TH × 100%
= 1 - TC/TH × 100%

### Example 3

A Carnot engine operates between the minimum and maximum temperature of water. What is its efficiency of this engine?

### Solution 3

Water exists in liquid state between 0°C and 100°C. The first represents the temperature of cold reservoir (TC) and the other, that of the hot reservoir (TH). When converted into Kelvin scale, we obtain TC = 0 + 273 = 273 K and TH = 100 + 273 = 373 K.

Therefore, we obtain for the efficiency of this Carnot engine:

eCarnot = TH - TC/TH × 100%
= 373 K - 273 K/373 K × 100%
= 100 K/373 K × 100%
= 26.8%

This is a small value, which in reality decreases further when we consider the above-mentioned factors. Therefore, most engines operate between higher ranges of temperatures, in order to increase the numerator of fraction, which brings in an increase in the engine's efficiency.

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