ET-third law

Third Law of Thermodynamics

The Third Law of Thermodynamics extends the definition of Entropy:
Entropy is zero only in a perfect crystal at absolute zero ( 0 kelvin [- 273.15 degree Celsius ]).
The Third Law of Thermodynamics can mathematically be expressed as
lim ST→0 = 0 (1)
where
S = entropy (J/K)
T = absolute temperature (K)
At a temperature of absolute zero there is no thermal energy or heat. At a temperature of zero Kelvin the atoms in a pure crystalline substance are aligned perfectly and do not move. There is no entropy of mixing since the substance is pure. By -VBM

ET-zeroth law

The Zeroth law of Thermodynamics can be stated as:
If two thermodynamic systems A and B are in thermal equilibrium, and B is also in thermal equilibrium with another system C, then A and C are in thermal equilibrium.

OR

"" If a body isolated from the other environment is in thermal equilibrium with one body & is separately in thermal equilibrium with the another body then three bodies are said to be in the thermal equilibrium with each other.""
This may seem obvious as we are quite familiar with this experiment. When we place in a cup of water (System A) a thermometer (System B) we wait a period of time until they reach equilibrium then read the measurement on the thermometer.
It is called the Zeroth Law as it is not derivable from the other laws and is often useful to understand the concept before presenting the other laws of thermodynamics.
It also states that at absolute zero temperature( i.e. zero Kelvin ) all molecular motion inside a crystal ceases.
The application of zeroth law is mainly seen in the thermodynamic properties.

The zeroth law of thermodynamics is a generalization about the thermal equilibrium among bodies, or thermodynamic systems, in contact. It results from the definition and properties of temperature.

Contents 1 Zeroth law as equivalence relation 2 Thermal equilibrium between many systems 3 Temperature and the zeroth law 4 History 5 Notes 6 References

Zeroth law as equivalence relation

A system is said to be in thermal equilibrium when its temperature does not change over time. Let A, B, and C be distinct thermodynamic systems or bodies. The zeroth law of thermodynamics can then be expressed as:

"If A and C are each in thermal equilibrium with B, A is also in thermal equilibrium with C."

The preceding sentence asserts that thermal equilibrium is a Euclidean relation between thermodynamic systems. If we also grant that all thermodynamic systems are (trivially) in thermal equilibrium with themselves, then thermal equilibrium is also a reflexive relation. Relations that are both reflexive and Euclidean are equivalence relations. One consequence of this reasoning is that thermal equilibrium is a transitive relation between the temperature T of A, B, and C:

If T (A) = T(B)

And T (B) = T(C)

Then T (A) = T(C).

Thermal equilibrium between many systems

Main article: Thermal equilibrium

Many systems are said to be in equilibrium if the small, random exchanges (due to Brownian motion, for example) between them do not lead to a net change in the total energy summed over all systems. A simple example illustrates why the zeroth law is necessary to complete the equilibrium description.
Consider N systems in adiabatic isolation from the rest of the universe (i.e., no heat exchange is possible outside of these N systems), all of which have a constant volume and composition, and which can only exchange heat with one another.
The combined First and Second Laws relate the fluctuations in total energy, δU, to the temperature of the ith system, and the entropy fluctuation in the ith system, as follows:

.

The adiabatic isolation of the system from the remaining universe requires that the total sum of the entropy fluctuations vanishes, or:

That is, entropy can only be exchanged between the N systems. This constraint can be used to rearrange the expression for the total energy fluctuation and obtain:

where is the temperature of any system j we may choose to single out among the N systems. Finally, equilibrium requires the total fluctuation in energy to vanish, in which case:

which can be thought of as the vanishing of the product of an antisymmetric matrix and a vector of entropy fluctuations . In order for a non-trivial solution to exist,

That is, the determinant of the matrix formed by must vanish for all choices of N. However, according to Jacobi's theorem, the determinant of a NxN antisymmetric matrix is always zero if N is odd, although for N even we find that all of the entries must vanish, , in order to obtain a vanishing determinant. Hence at equilibrium. This non-intuitive result means that an odd number of systems are always in equilibrium regardless of their temperatures and entropy fluctuations, while equality of temperatures is only required between an even number of systems to achieve equilibrium in the presence of entropy fluctuations.
The zeroth law solves this odd vs. even paradox, because it can readily be used to reduce an odd-numbered system to an even number by considering any three of the N systems and eliminating one by application of its principle, and hence reduce the problem to even N which subsequently leads to the same equilibrium condition that we expect in every case, i.e., . The same result applies to fluctuations in any extensive quantity, such as volume (yielding the equal pressure condition), or fluctuations in mass (leading to equality of chemical potentials). Hence the zeroth law has implications for a great deal more than temperature alone. In general, we see that the zeroth law breaks a certain kind of asymmetry present in the First and Second Laws.

Temperature and the zeroth law

It is often claimed, for instance by Max Planck in his influential textbook on thermodynamics, that the Zeroth law implies that we can define a "temperature function" or more informally, that we can "construct a thermometer."
In the space of thermodynamic parameters, zones of constant temperature will form a surface, which provides a natural order of nearby surfaces. It is then simple to construct a global temperature function that provides a continuous ordering of states. Note that the dimensionality of a surface of constant temperature is one less than the number of thermodynamic parameters (thus, for an ideal gas described with 3 thermodynamic parameters P, V and n, it is a 2 dimensional surface). The temperature so defined may indeed not look like the Celsius temperature scale, but it is a temperature function nonetheless.
For example, if two systems of ideal gas are in equilibrium, then P₁V₁/N₁ = P₂V₂/N₂ where P_i is the pressure in the ith system, V_i is the volume, and N_i is the "amount" (in moles, or simply the number of atoms) of gas.
The surface PV/N = const defines surfaces of equal temperature, and the obvious (but not only) way to label them is to define T so that PV/N = RT, where R is some constant. These systems can now be used as a thermometer to calibrate other systems.

ET-5.thermodynamics application

Contents

• 1 One Component Systems
  
  
  • 1.1 Gibbs Phase Rule
• 2 Psychrometry
  
  
  • 2.1 Adiabatic Saturation
  • 2.2 Wet Bulb Temperature
  • 2.3 Psychrometric Chart
  • 2.4 Air Conditioning
• 3 Common Thermodynamic Cycles
  
  
  • 3.1 Rankine Cycle
  • 3.2 Otto Cycle
    
    
    • 3.2.1 Analysis
  • 3.3 Diesel Cycle
    
    
    • 3.3.1 Analysis
  • 3.4 Dual Cycle
  • 3.5 Gas Turbine Cycle (or Joule-Brayton Cycle)
    
    
    • 3.5.1 Analysis
  • 3.6 Refrigeration Cycles

One Component Systems

All materials can exist in three phases: solid, liquid, and gas. All one component systems share certain characteristics, so that a study of a typical one component system will be quite useful.

For this analysis, we consider heat transferred to the substance at constant pressure. The above chart shows temperature vs. specific volume (1/density) curves for at three different constant pressures. The three line-curves labeled p₁, p₂, and p_c above are isobars, showing conditions at constant pressure. When the liquid and vapor coexist, it is called a saturated state. There is no change in temperature or pressure when liquid and vapor are in equilibrium, so that the temperature is called saturation temperature and the pressure is called saturation pressure. Saturated states are represented by the horizontal lines in the chart. In the temperature range where both liquid and vapor of a pure substance can coexist in equilibrium, for every value of saturated temperature, there is only one corresponding value of saturation pressure. If the temperature of the liquid is lower than the saturation temperature, it is called subcooled liquid. If the temperature of the vapor or gas is greater than the saturation temperature it is called superheated vapor.
The amount of liquid and vapor in a saturated mixture is specified by its quality x, which is the fraction of vapor in the mixture. Thus, the horizontal line representing the vaporization of the fluid has a quality of x=0 at the left endpoint where it is 100% liquid and a quality of x=1 at the right endpoint where it is 100% vapor. The blue curve in the preceding diagram shows saturation temperatures for saturated liquid i. e. where x=0. The green curve in the diagram shows saturation temperatures for saturated vapor i. e. where x=1. These curves are not isobars.


v_fg = v_g - v_f
If you also consider the solid state, then we get the phase diagram for the material. The point where the solid, liquid, and the vapor state exist in equilibrium is called the
triple point. Note that as the saturation temperatures increase, the liquid and vapor specific volumes approach each other until the blue and green curves come together and meet at point C on the p_c isobar. At that point C, called the critical point, the liquid and vapor states merge together and all their thermodynamic properties become the same. The critical point has a certain temperature T_c, and pressure p_c, which depend on the substance in question. At temperatures above the critical point, the substance is considered a super-heated gas.
This diagram is based on the diagram for water. Other pure (one-component) substances have corresponding temperature vs. specific volume diagrams which are fairly similarly shaped, but the temperatures, pressures, and specific volumes will vary.
The thermodynamic properties of materials are given in charts. One commonly used chart is the Mollier Chart, which is the plot of enthalpy versus entropy. The pressure enthalpy chart is frequently used in refrigeration applications. Charts such as these are useful because many processes are isenthalpic, so obtaining values would be as simple as drawing a straight line on the chart and reading off the data.
Steam tables give the values of specific volume, enthalpy, entropy, and internal energy for different temperatures for water. They are of great use to an engineer, with applications in steam turbines, steam engines, and air conditioning, among others.

Gas tables give the same equations for common gases like air. Although most gases roughly obey the ideal gas equation, gas tables note the actual values which are more accurate in many cases. They are not as important as steam tables, but in many cases it is much easier to lookup from a table rather than compute answers.

Gibbs Phase Rule

Gibbs phase rule states that for a heterogeneous system in equilibrium with C components in P phases, the degree of freedom F = C - P + 2. Thus, for a one component system with two phases, there is only one degree of freedom. F=1-2+2 F =1 That is, if you are given either the pressure or temperature of wet steam, you can obtain all the properties, while for superheated steam, which has just one phase, you will need both the pressure and the temperature.

Psychrometry

Psychrometry is the study of air and water vapor mixtures for air conditioning. For this application, air is taken to be a mixture of nitrogen and oxygen with the other gases being small enough so that they can be approximated by more of nitrogen and oxygen without much error. In this psychrometry section, vapor refers to water vapor. For air at normal (atmospheric) pressure, the saturation pressure of vapor is very low. Also, air is far away from its critical point in those conditions. Thus, the air vapor mixture behaves as an ideal gas mixture. If the partial pressure of the vapor is smaller than the saturation pressure for water for that temperature, the mixture is called unsaturated. The amount of moisture in the air vapor mixture is quantified by its humidity.
The absolute humidity ω is the ratio of masses of the vapor and air, i.e., ω = m_v/m_a. Now, applying ideal gas equation, pV = mRT for water vapor and for air, we have, since the volume and temperature are the same, ω = 0.622 p_v/p_a. The ratio of specific gas constants (R in preceding equation) of water vapor to air equals 0.622 .
The relative humidity φ is the ratio of the vapor pressure to the saturation vapor pressure at that temperature, i.e., φ = p_v/p_v,sat.
The saturation ratio is the ratio of the absolute humidity to the absolute humidity at saturation, or, ψ = ω/ω_sat. It is easy to see that the saturation ratio is very close to the value of relative humidity.

The above plot shows the value of absolute humidity versus the temperature. The initial state of the mixture is 1, and it is cooled isobarically, and at constant absolute humidity. When it reaches 2, it is saturated, and its absolute humidity is ω_a. Further cooling causes condensation and the system moves to point 3, where its absolute humidity is ω_b. The temperature at 2 is called the dew point.
It is customary to state all quantities in psychrometry per unit mass of dry air. Thus, the amount of air condensed in the above chart when moving from 2 to 3 is ω_b − ω_a.

Adiabatic Saturation

Consider an unsaturated mixture entering a chamber. Suppose water was sprayed into the stream, so that the humidity increases and it leaves as a saturated mixture. This is accompanied by a loss of temperature due to heat being removed from the air which is used for vaporization. If the water supplied is at the temperature of exit of the stream, then there is no heat transfer from the water to the mixture. The final temperature of the mixture is called adiabatic saturation temperature.

Wet Bulb Temperature

The relative humidity of air vapor mixtures is measured by using dry and wet bulb thermometers. The dry bulb thermometer is an ordinary thermometer, while the wet bulb thermometer has its bulb covered by a moist wick. When the mixture flows past the two thermometers, the dry bulb thermometer shows the temperature of the stream, while water evaporates from the wick and its temperature falls. This temperature is very close to the adiabatic saturation temperature if we neglect the heat transfer due to convection.

Psychrometric Chart

This chart gives the value of absolute humidity versus temperature, along with the enthalpy. From this chart you can determine the relative humidity given the dry and wet bulb temperatures. We have, from the first law, that for a flow system with no heat transfer, the enthalpy is a constant. Now, for the adiabatic saturation process, there is no heat transfer taking place, so that the adiabatic saturation lines are the same as the wet bulb temperature and the constant enthalpy lines.
Questions
1. The temperature at Phoenix is 35°C with a relative humidity of 40%. Can a room be cooled using a conventional air cooler?

We need to find the wet bulb temperature for the point T = 35°C and φ = 40%. We have, from the psychrometric chart, the wet bulb temperature is between 20 and 25°C. Thus, you can cool the room down to a comfortable temperature using an evaporative cooler.

2. The temperature of Los Angeles is 37°C with relative humidity of 83%. To what temperature can a room be cooled using a conventional air cooler?

The wet bulb temperature is about 34.2°C for this situation. Thus, you cannot use an ordinary cooler to reduce room temperature in this situation. You will need to use an air conditioner.

Air Conditioning

The human body can work efficiently only in a narrow range of conditions. Further, it rejects about 60 W of heat continuously into the surroundings, and more during heavy exercise. The temperature of the body is maintained by the evaporation of sweat from the body. Thus, for comfort, both the temperature and the relative humidity should be low.
Conventional air conditioning consists of setting the humidity at an acceptable level, while reducing the temperature. Reducing the humidity to zero is not the ideal objective. For instance, low humidity leads to issues like high chances of static electricity building up, leading to damage of sensitive electronic equipment. A humidity level of 50% is more acceptable in this case.
The most common method of reducing humidity is to cool the air using a conventional air conditioner working on a reversed Carnot cycle. The vapor that condenses is removed. Now, the air that is produced is very cold, and needs to be heated back up to room temperature before it is released back to the air conditioned area.

Common Thermodynamic Cycles

Several thermodynamic cycles used in machines can be approximated with idealized cycles. It was shown previously that a Carnot engine was the most efficient engine operating between two thermal reservoirs. However, due to practical difficulties, Carnot cycle cannot be implemented in all situations. The following sections deal with idealized (non Carnot) systems found in practice.

Rankine Cycle

In the Rankine cycle, also called the standard vapor power cycle, the working fluid follows a closed cycle. We will consider water as a working substance. In the Rankine cycle, water is pumped from a low pressure to a high pressure using a liquid pump. This water is then heated in the boiler at constant pressure where its temperature increases and it is converted to superheated vapor. This vapor is then expanded in an expander to generate work. This expander can be a turbine or a reciprocating (i.e. piston) machine such as those used in older steam locomotive or ship. The output of the expander is then cooled in a condenser to the liquid state and fed to the pump. The Rankine cycle differs from the Carnot cycle in that the input to the pump is a liquid (it is cooled more in the condenser). This allows the use of a small, low power pump due to the lower specific volume of liquid compared to steam. Also, the heat transfer in the boiler takes place mainly as a result of a phase change, compared to the isothermal heating of the ideal gas in the Carnot cycle, so that the efficiency is quite good (even though it is still lower than the Carnot efficiency). The amount of heat transferred as the liquid is heated to its boiling point is very small compared to the heat transfer during phase change. The steam is superheated so that no liquid state exists inside the turbine. Condensation in the turbine can be devastating as it can cause corrosion and erosion of the blades.
There are several modifications to the Rankine cycle leading to even better practical designs. In the reheat cycle there are two expanders working in series, and the steam from the high pressure stage is heated again in the boiler before it enters the low pressure expander. This avoids the problem of moisture in the turbine and also increases the efficiency. The regenerative cycle is another modification to increase the efficiency of the Rankine cycle. In many Rankine cycle implementations, the water enters the boiler in the subcooled state, and also, the large difference in temperature between the one at which heat is supplied to the boiler and the fluid temperature will give rise to irreversibilities which will cause the efficiency to drop. In the regenerative cycle, the output of the condenser is heated by some steam tapped from the expander. This causes the overall efficiency to increase, due to the reasons noted above.

Otto Cycle

The Otto Cycle is the idealization for the process found in the reciprocating internal combustion engines which are used by most automobiles. While in an actual engine the gas is released as exhaust, this is found to be a good way to analyze the process. There are, of course, other losses too in the actual engine. For instance, partial combustion and aspiration problems for a high speed engine. The working material in the idealized cycle is an ideal gas, as opposed to the air fuel mixture in an engine.

1. Heat is transferred at constant volume during 1-2.
2. The gas expands reversibly and adiabatically during 2-3, where work is done.
3. Heat is rejected at constant volume at low temperature during 3-4.
4. The gas is compressed reversibly and adiabatically in 4-1.

Analysis

Heat is transferred at constant volume in 1-2, so that Q_1-2 = m c_v(T₂ − T₁). Similarly, the heat rejected in 3-4 is Q_3-4 = m c_v (T₃ − T₄). The thermal efficiency of the Otto cycle is thus
η_th = (Q_1-2 − Q_3-4)/Q_1-2
η_th = 1 − Q_3-4/Q_1-2
η_th = 1 − (T₃ − T₄)/(T₂ − T₁)
Since 2-3 and 4-1 are reversible adiabatic processes involving an ideal gas, we have,
T₂/T₃ = (V₃/V₂)^{γ − 1}
and
T₄/T₁ = (V₁/V₄)^{γ − 1}
But,
V₁ = V₂
and
V₃ = V₄
So, we have
T₂/T₃ = T₁/T₄
Thus,
η_th = 1 − (T₃/T₂)(1 − T₄/T₃)/(1 − T₁/T₂)
Or
η_th = 1 − T₃/T₂
If we introduce the term r_c = V₃/V₂ for the compression ratio, then we have,
η_th = 1 − r_c^{1 − γ}
As can be seen, increasing the compression ratio will improve thermal efficiency. However, increasing the compression ratio causes the peak temperature to go up, which may cause spontaneous, uncontrolled ignition of the fuel, which leads to a shock wave traveling through the cylinder, and is called knocking.

Diesel Cycle

The Diesel cycle is the idealized cycle for compression ignition engines (ones that don't use a spark plug). The difference between the Diesel cycle and the Otto cycle is that heat is supplied at constant pressure.

1. Heat is supplied reversibly at constant pressure in 1-2.
2. Reversible adiabatic expansion during which work is done in 2-3.
3. Heat is rejected reversibly at constant volume in 3-4.
4. Gas is compressed reversibly and adiabatically in 4-1.

Analysis

Heat is transferred to the system at constant pressure during 1-2 so that
Q_in = m c_p (T₂ − T₁)
Heat is rejected by the system at constant volume during 3-4:
Q_out = m c_v (T₃ − T₄)
Thus, the efficiency of the Diesel cycle is
η_th = (Q_in − Q_out)/Q_in
η_th = 1 − Q_out/Q_in
η_th = 1 − (c_v (T₃ − T₄))/(c_p (T₂ − T₁))
η_th = 1 − (1/γ) (T₃ − T₄)/(T₂ − T₁)

We define the cutoff ratio as r_t = V₂/V₁, and since the pressures at 1 and 2 are equal, we have, applying the ideal gas equation, T₂/T₁ = r_t. Now, for the adiabatic processes 2-3 and 4-1 we have,


Since V₃ = V₄, we have

Dual Cycle

The dual cycle is sometimes used to approximate actual cycles as the time taken for heat transfer in the engine is not zero for the Otto cycle (so not constant volume). In the Diesel cycle, due to the nature of the combustion process, the heat input does not occur at constant pressure.

Gas Turbine Cycle (or Joule-Brayton Cycle)

Gas turbines are rotary internal combustion engines. As the first stage air is drawn in from outside and compressed using a compressor. Then the fuel is introduced and the mixture is ignited in the combustion chamber. The hot gases are expanded using a turbine which produces work. The output of the turbine is vented outside as exhaust.

The ideal gas turbine cycle is shown above. The four stages are

1. Heat input at constant pressure during 1-2.
2. Reversible adiabatic expansion during 2-3, where work is done.
3. Heat rejection at constant pressure during 3-4.
4. Reversible adiabatic compression during 4-1 where work is consumed.

Large amount of work is consumed in process 4-1 for a gas turbine cycle as the working material (gas) is very compressible. The compressor needs to handle a large volume and achieve large compression ratios.

Analysis

The heat input in a gas turbine cycle is given by Q_in = m c_p (T₂ - T₁) and the heat rejected Q_out = m c_p (T₃ - T₄). Thus the thermal efficiency is given by



Since the adiabatic processes take place between the same pressures, the temperature ratios are the same

Or

Where r_p is the pressure ratio and is a fundamental quantity for the gas turbine cycle.

Refrigeration Cycles

The ideal refrigeration cycle is reverse of Carnot cycle, working as a heat pump instead of as a heat engine. However, there are practical difficulties in making such a system work.
The gas refrigeration cycle is used in aircraft to cool cabin air. The ambient air is compressed and then cooled using work from a turbine. The turbine itself uses work from the compressed air, further cooling it. The output of the turbine as well as the air which is used to cool the output of the compressor is mixed and sent to the cabin.
The Rankine vapor-compression cycle is a common alternative to the ideal Carnot cycle. A working material such as Freon or R-134a, called the refrigerant, is chosen based on its boiling point and heat of vaporization. The components of a vapor-compression refrigeration system are the compressor, condenser, the expansion (or throttling) valve, and the evaporator. The working material (in gaseous form) is compressed by the compressor, and its output is cooled to a liquid in the condenser. The output of the condenser is throttled to a lower pressure in the throttling valve, and sent to the evaporator which absorbs heat. The gas from the evaporator is sent to the compressor, completing the cycle.
Standard refrigeration units use the throttling valve instead of a turbine to expand the gas as the work output that would be produced is not significant to justify the cost of a turbine. There are irreversibilities associated with such an expansion, but it is cost effective when construction costs are considered.

ET-4. second law

Contents

• 1 Introduction
• 2 Statement of the Second Law of Thermodynamics
  
  • 2.1 Kelvin-Planck Statement
  • 2.2 Clausius Statement
  • 2.3 PMM2
  • 2.4 Equivalence of Clausius and Kelvin-Planck Statements
• 3 Carnot Cycle
• 4 Thermodynamic Temperature Scale
• 5 NOTE
• 6 Clausius Theorem
• 7 Entropy
  
  • 7.1 Entropy for an Ideal Gas
• 8 Availability
  
  • 8.1 Availability Function
  • 8.2 Irreversibility
  • 8.3 Helmholtz and Gibbs Free Energies

 Introduction

The first law is a statement of energy conservation. The rise in temperature of a substance when work is done is well known. Thus work can be completely converted to heat. However, we observe that in nature, we don't see the conversion in the other direction spontaneously.
The statement of the second law is facilitated by using the concept of heat engines. Heat engines work in a cycle and convert heat into work. A thermal reservoir is defined as a system which is in equilibrium and large enough so that heat transferred to and from it does not change its temperature appreciably.
Heat Engine
Heat engines usually work between two thermal reservoirs, the low temperature reservoir and the high temperature reservoir. The performance of a heat engine is measured by its thermal efficiency, which is defined as the ratio of work output to heat input, i.e., η = W/Q₁, where W is the net work done, and Q₁ is heat transferred from the high temperature reservoir.
Heat Pumps
Heat pumps transfer heat from a low temperature reservoir to a high temperature reservoir using external work, and can be considered as reversed heat engines.

Statement of the Second Law of Thermodynamics

Kelvin-Planck Statement

It is impossible to construct a heat engine which will operate continuously and convert all the heat it draws from a reservoir into work.

Clausius Statement

It is impossible to construct a heat pump which will transfer heat from a low temperature reservoir to a high temperature reservoir without using external work.

"OR"

it is impossible to flow heat from low temperature(sink) to high temperature(source)without using expenditures.

PMM2

A perpetual motion machine of the second kind, or PMM2 is one which converts all the heat input into work while working in a cycle. A PMM2 has an η_th of 1.

Equivalence of Clausius and Kelvin-Planck Statements

Kelvin-Planck from Clausius
Suppose we can construct a heat pump which transfers heat from a low temperature reservoir to a high temperature one without using external work. Then, we can couple it with a heat engine in such a way that the heat removed by the heat pump from the low temperature reservoir is the same as the heat rejected by the heat engine, so that the combined system is now a heat engine which converts heat to work without any external effect. This is thus in violation of the Kelvin-Planck statement of the second law.
Clausius from Kelvin-Planck
Now suppose we have a heat engine which can convert heat into work without rejecting heat anywhere else. We can combine it with a heat pump so that the work produced by the engine is used by the pump. Now the combined system is a heat pump which uses no external work, violating the Clausius statement of the second law.
Thus, we see that the Clausius and Kelvin-Planck statements are equivalent, and one necessarily implies the other.

 Carnot Cycle

Nicholas Sadi Carnot devised a reversible cycle in 1824 called the Carnot cycle for an engine working between two reservoirs at different temperatures. It consists of two reversible isothermal and two reversible adiabatic processes. For a cycle 1-2-3-4, the working material

1. Undergoes isothermal expansion in 1-2 while absorbing heat from high temperature reservoir
2. Undergoes adiabatic expansion in 2-3
3. Undergoes isothermal compression in 3-4, and
4. Undergoes adiabatic compression in 4-1.

Heat is transferred to the working material during 1-2 (Q₁) and heat is rejected during 3-4 (Q₂). The thermal efficiency is thus η_th = W/Q₁. Applying first law, we have, W = Q₁ − Q₂, so that η_th = 1 − Q₂/Q₁.
Carnot's principle states that

1. No heat engine working between two thermal reservoirs is more efficient than the Carnot engine, and
2. All Carnot engines working between reservoirs of the same temperature have the same efficiency.

The proof by contradiction of the above statements come from the second law, by considering cases where they are violated. For instance, if you had a Carnot engine which was more efficient than another one, we could use that as a heat pump (since processes in a Carnot cycle are reversible) and combine with the other engine to produce work without heat rejection, to violate the second law. A corollary of the Carnot principle is that Q₂/Q₁ is purely a function of t₂ and t₁, the reservoir temperatures. Or,

Thermodynamic Temperature Scale

Lord Kelvin used Carnot's principle to establish the thermodynamic temperature scale which is independent of the working material. He considered three temperatures, t₁, t₂, and t₃, such that t₁ > t₃ > t₂.
As shown in the previous section, the ratio of heat transferred only depends on the temperatures. Considering reservoirs 1 and 2:

Considering reservoirs 2 and 3:

Considering reservoirs 1 and 3:

Eliminating the heat transferred, we have the following condition for the function φ.

Now, it is possible to choose an arbitrary temperature for 3, so it is easy to show using elementary multivariate calculus that φ can be represented in terms of an increasing function of temperature ζ as follows:

Now, we can have a one to one association of the function ζ with a new temperature scale called the thermodynamic temperature scale, T, so that

Thus we have the thermal efficiency of a Carnot engine as

The thermodynamic temperature scale is also known as the Kelvin scale, and it needs only one fixed point, as the other one is absolute zero. The concept of absolute zero will be further refined during the statement of the third law of thermodynamics.

• 1st Law: Energy can neither be created or destroyed
• 2nd Law: All spontaneous events act to increase total entropy
• 3rd Law: Absolute zero is removal of all thermal molecular motion

NOTE

Reservoirs are systems of large quantity of matter which no temperature difference will occur when finite amount of heat is transferred or removed. Ex- Ocean,lake, air and etc....

Clausius Theorem

Clausius theorem states that any reversible process can be replaced by a combination of reversible isothermal and adiabatic processes.

Consider a reversible process a-b. A series of isothermal and adiabatic processes can replace this process if the heat and work interaction in those processes is the same as that in the process a-b. Let this process be replaced by the process a-c-d-b, where a-c and d-b are reversible adiabatic processes, while c-d is a reversible isothermal process. The isothermal line is chosen such that the area a-e-c is the same as the area b-e-d. Now, since the area under the p-V diagram is the work done for a reversible process, we have, the total work done in the cycle a-c-d-b-a is zero. Applying the first law, we have, the total heat transferred is also zero as the process is a cycle. Since a-c and d-b are adiabatic processes, the heat transferred in process c-d is the same as that in the process a-b. Now applying first law between the states a and b along a-b and a-c-d-b, we have, the work done is the same. Thus the heat and work in the process a-b and a-c-d-b are the same and any reversible process a-b can be replaced with a combination of isothermal and adiabatic processes, which is the Clausius theorem.
A corollary of this theorem is that any reversible cycle can be replaced by a series of Carnot cycles.
Suppose each of these Carnot cycles absorbs heat dQ₁ⁱ at temperature T₁ⁱ and rejects heat dQ₂ⁱ at T₂ⁱ. Then, for each of these engines, we have dQ₁ⁱ/dQ₂ⁱ = −T₁ⁱ/T₂ⁱ. The negative sign is included as the heat lost from the body has a negative value. Summing over a large number of these cycles, we have, in the limit,

This means that the quantity dQ/T is a property. It is given the name entropy.
Further, using Carnot's principle, for an irreversible cycle, the efficiency is less than that for the Carnot cycle, so that


As the heat is transferred out of the system in the second process, we have, assuming the normal conventions for heat transfer,

So that, in the limit we have,


The above inequality is called the inequality of Clausius. Here the equality holds in the reversible case.

Entropy

Entropy is the quantitative statement of the second law of thermodynamics. It is represented by the symbol S, and is defined by

Thus, we can calculate the entropy change of a reversible process by evaluating the Note that as we have used the Carnot cycle, the temperature is the reservoir temperature. However, for a reversible process, the system temperature is the same as the reversible temperature.
Consider a system undergoing a cycle 1-2-1, where it returns to the original state along a different path. Since entropy of the system is a property, the change in entropy of the system in 1-2 and 2-1 are numerically equal. Suppose reversible heat transfer takes place in process 1-2 and irreversible heat transfer takes place in process 2-1. Applying Clausius's inequality, it is easy to see that the heat transfer in process 2-1 dQ_irr is less than T dS. That is, in an irreversible process the same change in entropy takes place with a lower heat transfer. As a corollary, the change in entropy in any process, dS, is related to the heat transfer dQ as
dS ≥ dQ/T
For an isolated system, dQ = 0, so that we have
dS_isolated ≥ 0
This is called the principle of increase of entropy and is an alternative statement of the second law.
Further, for the whole universe, we have
ΔS = ΔS_sys + ΔS_surr > 0
For a reversible process,
ΔS_sys = (Q/T)_rev = −ΔS_surr
So that
ΔS_universe = 0
for a reversible process.

Since T and S are properties, you can use a T-S graph instead of a p-V graph to describe the change in the system undergoing a reversible cycle. We have, from the first law, dQ + dW = 0. Thus the area under the T-S graph is the work done by the system. Further, the reversible adiabatic processes appear as vertical lines in the graph, while the reversible isothermal processes appear as horizontal lines.

Entropy for an Ideal Gas

An ideal gas obeys the equation pv = RT. According to the first law,
dQ + dW = dU
For a reversible process, according to the definition of entropy, we have
dQ = T dS
Also, the work done is the pressure volume work, so that
dW = -p dV
The change in internal energy:
dU = m c_v dT
T dS = p dV + m c_v dT
Taking per unit quantities and applying ideal gas equation,
ds = R dV/v + c_v dT/T

As a general rule, all things being equal, entropy increases as, temperature increases and as pressure and concentration decreases and energy stored as internal energy has higher entropy than energy which is stored as kinetic energy.

Availability

From the second law of thermodynamics, we see that we cannot convert all the heat energy to work. If we consider the aim of extracting useful work from heat, then only some of the heat energy is available to us. It was previously said that an engine working with a reversible cycle was more efficient than an irreversible engine. Now, we consider a system which interacts with a reservoir and generates work, i.e., we look for the maximum work that can be extracted from a system given that the surroundings are at a particular temperature.
Consider a system interacting with a reservoir and doing work in the process. Suppose the system changes state from 1 to 2 while it does work. We have, according to the first law,
dQ - dW = dE,
where dE is the change in the internal energy of the system. Since it is a property, it is the same for both the reversible and irreversible process. For an irreversible process, it was shown in a previous section that the heat transferred is less than the product of temperature and entropy change. Thus the work done in an irreversible process is lower, from first law.

Availability Function

The availability function is given by Φ, where
Φ ≡ E − T₀S
where T₀ is the temperature of the reservoir with which the system interacts. The availability function gives the effectiveness of a process in producing useful work. The above definition is useful for a non-flow process. For a flow process, it is given by
Ψ ≡ H − T₀S

Irreversibility

Maximum work can be obtained from a system by a reversible process. The work done in an actual process will be smaller due to the irreversibilities present. The difference is called the irreversibility and is defined as
I ≡ W_rev − W
From the first law, we have
W = ΔE − Q
I = ΔE - Q - (Φ₂ − Φ₁)
As the system interacts with surroundings of temperature T₀, we have
ΔS_surr = Q/T₀
Also, since
E − Φ = T₀ ΔS_sys
we have
I = T₀ (ΔS_sys + ΔS_surr)
Thus,
I ≥ 0
I represents increase in unavailable energy.

Helmholtz and Gibbs Free Energies

Helmholtz Free Energy is defined as
F ≡ U − TS
The Helmholtz free energy is relevant for a non-flow process. For a flow process, we define the Gibbs Free Energy
G ≡ H − TS
The Helmholtz and Gibbs free energies have applications in finding the conditions for equilibrium.