Second law of thermodynamics. II

One of the basic laws of physics, the law of non-decreasing entropy in an isolated system.
For a constant temperature system there is specific function state S - entropy, which is defined in such a way that
1. An adiabatic transition from equilibrium state A to equilibrium state B is possible only when

2. The increase in entropy in a slow quasi-static process is equal to

Where T is temperature.
The above formulation is very formal. There are many alternative formulations of the second law of thermodynamics. For example, Planck proposed the following formulation:
It is impossible to build a machine that cycles, cools a heat source, or lifts loads without causing however, no changes in nature.

Constantine Carathéodory gave an axiomatically strict formulation
Near state 1, such states 2 exist; adiabatic transitions from state 1 to state 2 are impossible.

Boltzmann formulated the second law of thermodynamics from the point of view of statistical physics:
Nature tends to move from states with a lower probability of implementation to states with a higher probability of implementation.

Such formulations are common.
It is impossible to be an eternal mover of another kind.

It is impossible to transfer heat from a cold body to a hot one without expending energy.

Every system tends to move from order to disorder.

The second law of thermodynamics was formulated in the mid-19th century, at the time when theoretical basis for the design and construction of heat engines. The experiments of Mayer and Joule established the equivalence between thermal and mechanical energies (the first law of thermodynamics). The question arose about the efficiency of heat engines. Experimental studies have shown that some heat is necessarily lost during the operation of any machine.
In the 1850s and 1860s, Clausius developed the concept of entropy in a number of publications. In 1865, he finally chose a name for the new concept. These publications also proved that heat cannot be completely converted into useful work, thus formulating the second law of thermodynamics.
Boltzmann gave a statistical interpretation to the second law of thermodynamics, introducing a new definition for entropy, which was based on microscopic atomistic concepts.
Statistical physics introduces a new definition of entropy, which at first glance is very different from the definition of thermodynamics. It is given by the Boltzmann formula:

Where? - the number of microscopic states corresponding to a given macroscopic state, k B- Boltzmann constant.
From the statistical definition of entropy it is obvious that an increase in entropy corresponds to a transition to a macroscopic state that is characterized highest value microscopic states.
If the initial state of a thermodynamic system is nonequilibrium, then over time it moves to an equilibrium state, increasing its entropy. This process occurs only in one direction. The reverse process - the transition from an equilibrium state to an initial nonequilibrium state - is not realized. That is, the flow of time receives direction.
The laws of physics that describe the microscopic world are invariant under the replacement of t by -t. This statement is true both for the laws of classical mechanics and the laws of quantum mechanics. In the microscopic world, conservative forces act; there is no friction, which is the dissipation of energy, i.e. the transformation of other types of energy into the energy of thermal motion, and this in turn is associated with the law of non-decreasing entropy.
Imagine, for example, a gas in a reservoir placed in a larger reservoir. If you open the valve of the smaller tank, the gas will after some time fill the larger tank so that its density is equalized. According to the laws of the microscopic world, there is also a reverse process, when gas from a larger reservoir is collected in a smaller container. But in the macroscopic world this never happens.
If the entropy of each isolated system only increases with time, and the Universe is an isolated system, then someday the entropy will reach a maximum, after which any changes in it will become impossible.
Such considerations that appeared after the establishment of the second law of thermodynamics, called heat death. This hypothesis was widely debated in the 19th century.
Every process in the world leads to the dissipation of part of the energy and its conversion into heat, leading to greater disorder. Of course, our Universe is still quite young. Thermonuclear processes in stars lead to a steady flow of energy to Earth, for example. The Earth is and will remain an open system for a long time, which receives energy from various sources: from the Sun, from processes radioactive decay in the core, i.e. In open systems, entropy can decrease, which leads to the emergence of a variety of comfortable structures.

Second law of thermodynamics. Entropy.

The second law is associated with the concept of entropy, which is a measure of chaos (or a measure of order). The second law of thermodynamics states that for the universe as a whole, entropy increases.

There are two classical definitions of the second law of thermodynamics:

• Kelvin and Planck

There is no cyclic process that extracts a quantity of heat from a reservoir at a certain temperature and completely converts that heat into work. (It is impossible to build a periodically operating machine that does nothing other than lift a load and cool a heat reservoir)

• Clausius

There is no process whose only result is the transfer of heat from a less heated body to a more heated one. (A circular process is impossible, the only result of which would be the production of work by cooling the heat reservoir)

Both definitions of the second law of thermodynamics rely on the first law of thermodynamics, which states that energy decreases.

The second law is related to the concept entropy (S).

Entropy generated by all processes, it is associated with the loss of the system’s ability to do work. The growth of entropy is a spontaneous process. If the volume and energy of a system are constant, then any change in the system increases entropy. If the volume or energy of the system changes, the entropy of the system decreases. However, the entropy of the universe does not decrease.

In order for energy to be used, there must be areas of high and low energy levels in the system. Useful work is produced as a result of the transfer of energy from the region with high level energy to an area with low energy levels.

• 100% of energy cannot be converted into work
• Entropy can be generated, but cannot be destroyed

Heat Engine Efficiency

The efficiency of a heat engine operating between two energy levels is determined in terms of absolute temperatures

η = (T h - T c) / T h = 1 - T c / T h

η = efficiency

T c = lower temperature limit (K)

In order to achieve maximum efficiency, T c should be kept as low as possible. For the effect to be 100%, T c must be equal to 0 on the Kelvin scale. In practice this is impossible, so the efficiency is always less than 1 (less than 100%).

• Entropy change > 0
  Irreversible process
• Entropy change= 0
  Bilateral process (reversible)
• Entropy change< 0
  Impossible process (not feasible)

Entropy determines the relative ability of one system to influence another. As energy moves to a lower energy level, where the potential for impact on the environment decreases, entropy increases.

Definition of entropy

Entropy is defined as:

T = absolute temperature (K)

A change in the entropy of a system is caused by a change in the temperature content in it. The change in entropy is equal to the change in the temperature of the system divided by the average absolute temperature(Ta):

The sum of the values ​​(H/T) for each complete Carnot cycle is 0. This is because every positive H is opposed negative meaning H.

• Carnot thermal cycle

The Carnot cycle is an ideal thermodynamic cycle.

In a heat engine, gas is (reversibly) heated and then cooled. The cycle model is as follows: Position 1 --() --> Position 2 --() --> Position 3 --(isothermal compression) --> Position 4 --(adiabatic compression) --> Position 1

Position 1 - Position 2: Isothermal expansion
Isothermal expansion. At the beginning of the process, the working fluid has a temperature T h, that is, the temperature of the heater. The body is then brought into contact with a heater, which isothermally (at a constant temperature) transfers to it an amount of heat QH. At the same time, the volume of the working fluid increases. Q H =∫Tds=T h (S 2 -S 1) =T h ΔS
Position 2 - Position 3: Adiabatic expansion
Adiabatic (isentropic) expansion. The working fluid is disconnected from the heater and continues to expand without heat exchange with the environment. At the same time, its temperature decreases to the temperature of the refrigerator.
Position 3 - Position 4: Isothermal compression
Isothermal compression. The working fluid, which by that time has a temperature Tc, is brought into contact with the refrigerator and begins to compress isothermally, giving the amount of heat Qc to the refrigerator. Q c =T c (S 2 -S 1)=T c ΔS
Position 4 - Position 1: Adiabatic compression
Adiabatic (isentropic) compression. The working fluid is disconnected from the refrigerator and compressed without heat exchange with the environment. At the same time, its temperature increases to the temperature of the heater.

During isothermal processes, the temperature remains constant; during adiabatic processes, there is no heat exchange, which means entropy is conserved.

Therefore, it is convenient to represent the Carnot cycle in T and S coordinates (temperature and entropy).

The laws of thermodynamics were determined empirically (experimentally). The second law of thermodynamics is a generalization of experiments related to entropy. It is known that dS of the system plus dS of the environment is equal to or greater than 0.

• The entropy of an adiabatically isolated system does not change!

Example - Entropy when heating water

The process of heating 1 kg of water from 0 to 100 o C (273 to 373 K)

At 0 o C = 0 kJ/kg (specific - per unit mass)

At 100 o C = 419 kJ/kg

Change in specific entropy:

dS = dH / T a

= ((419 kJ/kg) - (0 kJ/kg)) / ((273 K + 373 K)/2)

= 1.297 kJ/kg*K

Example - Entropy during evaporation of water

The process of converting 1 kg of water at 100 o C (373 K) into saturated steam at 100 o C (373 K) under normal conditions.

Specific enthalpy of steam at 100 o C (373 K) before evaporation = 0 kJ/kg

100 o C (373 K) at evaporation = 2,258 kJ/kg

Change in specific entropy:

dS = dH / T a

= (2 258 - 0) / ((373 + 373)/2)

= 6.054 kJ/kg*K

The total change in the specific entropy of water evaporation is the sum of the specific entropy of water (at 0 o C) plus the specific entropy of steam (at a temperature of 100 o C).

How is energy generated, how is it converted from one form to another, and what happens to energy in a closed system? The laws of thermodynamics will help answer all these questions. The second law of thermodynamics will be discussed in more detail today.

Laws in everyday life

Laws govern everyday life. Traffic laws say you must stop at stop signs. Government workers are required to provide a portion of their salaries to the state and federal government. Even scientific ones are applicable to Everyday life. For example, the law of gravity predicts a pretty bad outcome for those who try to fly. Another set of scientific laws that affect everyday life are the laws of thermodynamics. So, a number of examples can be given to see how they affect everyday life.

First law of thermodynamics

The first law of thermodynamics states that energy cannot be created or destroyed, but it can be converted from one form to another. This is also sometimes called the law of conservation of energy. So how does this relate to everyday life? Well, take for example the computer you are using now. It feeds on energy, but where does this energy come from? The first law of thermodynamics tells us that this energy couldn't come from out of thin air, so it came from somewhere.

You can track this energy. The computer is powered by electricity, but where does this electricity come from? That's right, from a power plant or hydroelectric power station. If we consider the second, it will be associated with a dam that holds back the river. A river has a connection with kinetic energy, which means that the river flows. The dam converts this kinetic energy into potential energy.

How does a hydroelectric power plant work? Water is used to rotate the turbine. When the turbine rotates, a generator is driven, which will create electricity. This electricity can be carried entirely in wires from the power plant to your home so that when you plug the power cord into an electrical outlet, electricity flows into your computer so it can operate.

What happened here? There was already a certain amount of energy that was associated with the water in the river like kinetic energy. Then it turned into potential energy. The dam then took that potential energy and turned it into electricity, which could then go into your home and power your computer.

Second law of thermodynamics

By studying this law, one can understand how energy works and why everything moves towards possible chaos and disorder. The second law of thermodynamics is also called the law of entropy. Have you ever wondered how the Universe came into being? According to Theory Big Bang, before everything around was born, a huge amount of energy gathered together. After the Big Bang, the Universe appeared. All this is good, but what kind of energy was that? At the beginning of time, all the energy in the Universe was contained in one relatively small place. This intense concentration represented a huge amount of what is called potential energy. Over time, it spread across the vast expanse of our Universe.

On a much smaller scale, the reservoir of water held by a dam contains potential energy because its location allows it to flow through the dam. In each case, the stored energy, once released, spreads out and does so without any effort being exerted. In other words, release potential energy is a spontaneous process that occurs without the need for additional resources. As energy spreads, some of it is converted into useful energy and does some work. The rest is converted into unusable energy, simply called heat.

As the Universe continues to expand, it contains less and less useful energy. If a less useful one is available, less work can be done. As the water flows through the dam, it also contains less useful energy. This decrease in useful energy over time is called entropy, where entropy is the amount of unused energy in a system, and a system is simply the collection of objects that make up the whole.

Entropy can also be referred to as the amount of randomness or chaos in an organization without organization. As useful energy decreases over time, disorganization and chaos increase. Thus, as the accumulated potential energy is released, not all of it is converted into useful energy. All systems experience this increase in entropy over time. This is very important to understand, and this phenomenon is called the second law of thermodynamics.

Entropy: randomness or defect

As you may have guessed, the second law follows the first, which is commonly called the law of conservation of energy, and it states that energy cannot be created and cannot be destroyed. In other words, the amount of energy in the Universe or any system is constant. The second law of thermodynamics is commonly called the law of entropy, and it holds that as time passes, energy becomes less useful and its quality decreases over time. Entropy is the degree of randomness or defects a system has. If a system is very disordered, then it has high entropy. If there are many faults in the system, then the entropy is low.

Speaking in simple words, the second law of thermodynamics states that the entropy of a system cannot decrease over time. This means that in nature things move from a state of order to a state of disorder. And this is irreversible. The system will never become more orderly on its own. In other words, in nature the entropy of a system always increases. One way to think about this is your home. If you never clean and vacuum it, then pretty soon you will have a terrible mess. Entropy has increased! To reduce it, you need to use energy to use a vacuum cleaner and mop to clear the dust from the surface. The house won't clean itself.

What is the second law of thermodynamics? The formulation in simple words states that when energy changes from one form to another form, matter either moves freely or entropy (disorder) in a closed system increases. Differences in temperature, pressure and density tend to level out horizontally after a while. Due to gravity, density and pressure do not equalize vertically. The density and pressure at the bottom will be greater than at the top. Entropy is a measure of the spread of matter and energy wherever it has access. The most common formulation of the second law of thermodynamics is mainly associated with Rudolf Clausius, who said:

It is impossible to construct a device which produces no other effect than the transfer of heat from a body at a lower temperature to a body at a higher temperature. high temperature.

In other words, everything tries to maintain the same temperature over time. There are many formulations of the second law of thermodynamics that use different terms, but they all mean the same thing. Another Clausius statement:

Heat does not itself occur from a colder body to a hotter one.

The second law applies only to large systems. It concerns the likely behavior of a system in which there is no energy or matter. The larger the system, the more likely the second law is.

Another wording of the law:

Total entropy always increases in a spontaneous process.

The increase in entropy ΔS during the process must exceed or be equal to the ratio of the amount of heat Q transferred to the system to the temperature T at which the heat is transferred.

Thermodynamic system

IN in a general sense The statement of the second law of thermodynamics in simple terms states that temperature differences between systems in contact with each other tend to equalize and that work can be obtained from these nonequilibrium differences. But at the same time there is a loss of thermal energy, and entropy increases. Differences in pressure, density and temperature tend to equalize if given the chance; Density and pressure, but not temperature, depend on gravity. A heat engine is a mechanical device that produces useful work due to the difference in temperature of two bodies.

A thermodynamic system is one that interacts and exchanges energy with the region around it. Exchange and transfer must occur in at least two ways. One way must be heat transfer. If a thermodynamic system is "in equilibrium", it cannot change its state or status without interacting with its environment. Simply put, if you are in equilibrium, you are a "happy system", there is nothing you can do. If you want to do something, you must interact with the world around you.

Second law of thermodynamics: irreversibility of processes

It is impossible to have a cyclic (repeating) process that completely converts heat into work. It is also impossible to have a process that transfers heat from cold objects to warm objects without using work. Some energy in a reaction is always lost to heating. In addition, the system cannot convert all its energy into work energy. The second part of the law is more obvious.

A cold body cannot heat a warm body. Warm naturally tends to flow from warmer to cooler areas. If heat moves from cooler to warmer temperatures, it is contrary to what is "natural", so the system must do some work to make this happen. in nature - the second law of thermodynamics. This is perhaps the most famous (at least among scientists) and important law in all of science. One of his formulations:

The entropy of the Universe tends to its maximum.

In other words, entropy either remains the same or becomes greater; the entropy of the universe can never decrease. The problem is that this is always true. If you take a bottle of perfume and spray it in a room, soon fragrant atoms will fill the entire space, and this process is irreversible.

Relationships in thermodynamics

The laws of thermodynamics describe the relationships between thermal energy, or heat, and other forms of energy, and how energy affects matter. The first law of thermodynamics states that energy cannot be created or destroyed; total energy in the Universe remains unchanged. The second law of thermodynamics deals with the quality of energy. It states that as energy is transferred or converted, more and more useful energy is lost. The Second Law also states that there is a natural tendency for any isolated system to become more disordered.

Even when the order increases in certain place when you take into account the entire system, including the environment, there is always an increase in entropy. In another example, crystals may form from a salt solution when water is evaporated. Crystals are more ordered than salt molecules in solution; however, evaporated water is much more messy than liquid water. The process taken as a whole results in a net increase in disorder.

Work and Energy

The second law explains that it is impossible to convert thermal energy into mechanical energy with 100 percent efficiency. You can give an example with a car. After the process of heating the gas to increase its pressure to drive the piston, there is always some heat left in the gas, which cannot be used to perform any additional work. This waste heat must be rejected by transferring it to the radiator. In the case of a car engine, this is done by extracting the spent fuel and air mixture into the atmosphere.

Additionally, any device with moving parts creates friction, which converts mechanical energy into heat, which is usually unusable and must be removed from the system by transferring it to a heat sink. When a hot and cold body come into contact with each other, thermal energy will flow from a hot body to a cold body until they reach thermal equilibrium. However, the heat will never return the other way; the temperature difference between two bodies will never increase spontaneously. Moving heat from a cold body to a hot body requires work, which must be performed by an external energy source such as a heat pump.

Fate of the Universe

The Second Law also predicts the end of the universe. This is the ultimate level of disorder, if there is constant thermal equilibrium everywhere, no work can be done and all energy will end up as random motion of atoms and molecules. According to modern data, the Metagalaxy is an expanding non-stationary system; there can be no talk of the thermal death of the Universe. Heat death is a state of thermal equilibrium in which all processes cease.

This position is erroneous, since the second law of thermodynamics applies only to closed systems. And the Universe, as we know, is limitless. However, the term “heat death of the Universe” itself is sometimes used to designate a scenario for the future development of the Universe, according to which it will continue to expand indefinitely into the darkness of space until it turns into scattered cold dust.

As is known, the first law of thermodynamics reflects the law of conservation of energy in thermodynamic processes, but it does not give an idea of ​​​​the direction of the processes. In addition, you can come up with many thermodynamic processes that will not contradict the first law, but in reality such processes do not exist. The existence of the second law (law) of thermodynamics is caused by the need to establish the possibility of a particular process. This law determines the direction of flow of thermodynamic processes. When formulating the second law of thermodynamics, they use the concepts of entropy and the Clausius inequality. In this case, the second law of thermodynamics is formulated as the law of growth of entropy of a closed system if the process is irreversible.

Statements of the second law of thermodynamics

If a process occurs in a closed system, then the entropy of this system does not decrease. In the form of a formula, the second law of thermodynamics is written as:

where S is entropy; L is the path along which the system moves from one state to another.

In this formulation of the second law of thermodynamics, attention should be paid to the fact that the system under consideration must be closed. In an open system, entropy can behave in any way (it can decrease, increase, or remain constant). Note that entropy does not change in a closed system during reversible processes.

An increase in entropy in a closed system during irreversible processes is the transition of a thermodynamic system from states with a lower probability to states with a higher probability. The famous Boltzmann formula gives a statistical interpretation of the second law of thermodynamics:

where k is Boltzmann's constant; w - thermodynamic probability (the number of ways in which the macrostate of the system under consideration can be realized). Thus, the second law of thermodynamics is a statistical law that is associated with the description of the patterns of thermal (chaotic) movement of molecules that make up a thermodynamic system.

Other formulations of the second law of thermodynamics

There are a number of other formulations of the second law of thermodynamics:

1) Kelvin's formulation: It is impossible to create a circular process, the result of which will be exclusively the conversion of the heat received from the heater into work. From this formulation of the second law of thermodynamics they conclude that it is impossible to create perpetual motion machine second kind. This means that periodically acting heat engine must have a heater, working fluid and refrigerator. In this case, the efficiency of an ideal heat engine cannot be greater than the efficiency of the Carnot cycle:

where is the heater temperature; — refrigerator temperature; ( title="Rendered by QuickLaTeX.com" height="15" width="65" style="vertical-align: -3px;">).!}

2) Clausius' formulation: It is impossible to create a circular process as a result of which only heat will be transferred from a body with a lower temperature to a body with a higher temperature.

The second law of thermodynamics notes the essential difference between the two forms of energy transfer (work and heat). From this law it follows that the transition of the ordered movement of the body as a whole into the chaotic movement of the molecules of the body and external environment- is an irreversible process. In this case, ordered movement can turn into chaotic without additional (compensatory) processes. Whereas the transition from disordered motion to ordered motion must be accompanied by a compensating process.

Examples of problem solving

EXAMPLE 1

Exercise

What is the essence of the “Heat Death of the Universe” problem? Why is this problem untenable?

Solution

This problem was formulated in the 19th century. If we consider the Universe to be a closed system and try to apply the second law of thermodynamics to it, then according to the Clausius hypothesis, the entropy of the Universe will reach a certain maximum. That is, after some time, all forms of motion will become thermal motion. All heat from bodies with a higher temperature will move to bodies with a higher temperature. low temperature, that is, the temperatures of all bodies in the Universe will become equal. The Universe will come to a state of thermal equilibrium, all processes will stop - this is called the thermal death of the Universe. Error this provision about the thermal death of the Universe lies in the fact that the second law of thermodynamics is not applicable to open systems, and the Universe should not be considered closed. Since it is limitless and consists of endless development.

EXAMPLE 2

Exercise

What is the efficiency of the cycle shown in Fig. 1? Consider that an ideal gas is involved in the process (the number of degrees of freedom is i) and its volume changes n times.

Solution

Coefficient useful action cycle, which is presented in Fig. 1, we find it as: where is the amount of heat that the working fluid receives from the heater in the presented cycle. In adiabatic processes there is no supply or removal of heat; it turns out that heat is supplied only in process 1-2. - the amount of heat that is removed from the gas in process 3-4. Using the first law of thermodynamics, we find the amount of heat received by the gas in process 1-2, which is isochoric: since there is no change in volume in this process. Change internal energy gas we define as: By analogy, for an isochoric process in which heat is removed, we have: Let us substitute the obtained result (2.2 - 2.5) into expression (2.1): We use the adiabatic equation to find temperature differences, and consider Fig. 1. For process 2-3 we write:

The first law of thermodynamics is one of the most general and fundamental laws of nature. There is not a single process known where at least

to some extent there would be a violation of it. If any process is prohibited by the first law, then you can be absolutely sure that it will never happen. However, this law does not give any indication of the direction in which processes that satisfy the principle of conservation of energy develop.

Let's explain this with examples.

Direction of thermal processes. The first law of thermodynamics does not say anything about the direction in which heat exchange occurs between bodies brought into thermal contact and located at different temperatures. As discussed above, heat exchange occurs in such a way that the temperatures are equalized and the entire system tends to a state of thermal equilibrium. But the first law would not be violated if, on the contrary, the transfer of heat occurred from a body with a low temperature to a body with a higher one, provided that the total supply of internal energy remained unchanged. However, everyday experience shows that this never happens by itself.

Another example: when a stone falls from a certain height, all the kinetic energy of its translational motion disappears when it hits the ground, but at the same time the internal energy of the stone itself and the bodies surrounding it increases, so that the law of conservation of energy, of course, is not violated. But the reverse process would not contradict the first law of thermodynamics, in which a certain amount of heat would be transferred from surrounding objects to a stone lying on the ground, as a result of which the stone would rise to a certain height. However, no one has ever observed such spontaneously jumping stones.

Inequality different types energy. Thinking about these and other similar examples, we come to the conclusion that the first law of thermodynamics does not impose any restrictions on the direction of transformations of energy from one type to another and on the direction of the transfer of heat between bodies, requiring only the conservation of the full reserve of energy in closed systems. Meanwhile, experience shows that different types of energy are not equivalent in terms of their ability to transform into other types.

Mechanical energy can be completely converted into internal energy of any body, regardless of what its temperature was. Indeed, any body can be heated by friction, increasing its internal energy by an amount equal to the work done. Similar Electric Energy can be completely converted into internal, for example, by passing an electric current through a resistance.

For the reverse transformations of internal energy into other types, there are certain restrictions, consisting in the fact that the stock of internal energy under no circumstances can be converted

entirely into other types of energy. The noted features of energy transformations are associated with the direction of processes in nature. The second law of thermodynamics, which reflects the direction of natural processes and imposes restrictions on the possible directions of energy transformations in macroscopic systems, is, like any fundamental law, a generalization large number experienced facts.

In order to more clearly imagine the physical content of the second law of thermodynamics, let us consider in more detail the issue of the reversibility of thermal processes.

Reversible and irreversible processes. If the conditions are changed slowly enough so that the speed of the process occurring in the system under consideration is significantly less speed relaxation, then such a process will physically represent a chain of equilibrium states close to each other. Therefore, such a process is described by the same macroscopic parameters as the equilibrium state. These slow processes are called equilibrium or quasi-static. In such processes, the system can be characterized by parameters such as pressure, temperature, etc. Real processes are nonequilibrium and can be considered equilibrium with greater or lesser accuracy.

Consider the following examples.

Let the gas be in a cylindrical vessel closed by a piston. If you extend the piston at a finite speed, then the expansion of the gas will be an irreversible process. Indeed, as soon as the piston is extended, the gas pressure directly at the piston will be less than in other parts of the cylinder. Such a process cannot be carried out reversibly through the same intermediate states, since when the piston is pushed back at a finite speed, not a rarefaction of the gas will occur near the piston, but its compression. Thus, the rapid expansion or compression of a gas provides an example of an irreversible process.

To expand the gas in a strictly reversible manner, the piston must be advanced infinitely slowly. In this case, the gas pressure will be the same throughout the entire volume at each moment, the state of the gas will depend on the position of the piston, and not on the direction of its movement, and the process will be reversible.

The irreversibility of the process of gas expansion is most clearly manifested when the expansion occurs into emptiness without completing mechanical work.

All processes accompanied by heat exchange between bodies having different temperatures. The irreversibility of such heat exchange is especially clearly visible in the example of equalizing the temperatures of bodies brought into contact.

Irreversible processes are those in which mechanical energy is converted into internal energy in the presence of friction, which is often referred to as the release of heat due to friction. In the absence of friction, all mechanical processes would proceed reversibly.

Thus, equilibrium reversible processes are an abstraction, and in practice, due to the existence of friction and heat transfer, they do not occur. However, the study of equilibrium processes in thermodynamics makes it possible to indicate how processes should be carried out in real systems in order to obtain the best results.

Various formulations of the second law of thermodynamics. Historically, the discovery of the second law of thermodynamics was associated with the study of the issue of the maximum efficiency of heat engines, carried out by the French scientist Sadi Carnot. Later, R. Clausius and W. Thomson (Lord Kelvin) proposed different but equivalent formulations of the second law of thermodynamics.

According to Clausius's formulation, a process is impossible whose only result would be the transfer of heat from a body with a lower temperature to a body with a higher temperature.

Thomson formulated the second law of thermodynamics as follows: a periodic process is impossible, the only final result of which would be the performance of work due to heat taken from one body.

The expression “sole result” in these formulations means that no changes other than those indicated occur either in the systems under consideration or in the bodies surrounding them. A conventional diagram of this kind of process, prohibited by the Clausius postulate, is shown in Fig. 56, and the process prohibited by Thomson’s postulate is shown in Fig. 57.

In Thomson's formulation, the second law of thermodynamics imposes restrictions on the conversion of internal energy into mechanical energy. From Thomson's formulation it follows that it is impossible to build a machine that would do work only by receiving heat from the environment. Such a hypothetical machine was called a perpetual motion machine of the second kind, since due to the unlimited reserves of internal energy in the earth, ocean, and atmosphere, such a machine would be equivalent for all practical purposes to a perpetual motion machine.

A perpetual motion machine of the second kind is not in conflict with the first law of thermodynamics, in contrast to a perpetual motion machine of the first kind, i.e., a device for doing work without using any energy source at all.

Equivalence of the formulations of Clausius and Thomson. Equivalence of the formulations of the second law of thermodynamics,

proposed by Clausius and Thomson, is established by simple reasoning.

Let us assume that Thomson's postulate is not true. Then it is possible to carry out such a process, the only result of which would be the performance of work due to heat taken from a single source with a temperature T. This work could, for example, by friction, be completely converted again into heat transferred to a body whose temperature is higher than T The only result of such a composite process would be the transfer of heat from a body with temperature T to a body with a higher temperature. But this would contradict Clausius's postulate. So, Clausius's postulate cannot be valid if Thomson's postulate is false.

Let us now assume that, on the contrary, Clausius’s postulate is invalid, and we will show that in this case Thomson’s postulate also cannot be satisfied. Let's build an ordinary heat engine that will work by receiving a certain amount of heat from the heater, giving it to the refrigerator and converting the difference into work (Fig. 58).

Since Clausius's postulate is assumed to be incorrect, it is possible to carry out a process whose only result is the transfer of an equal amount of heat from the refrigerator to the heater. This is shown schematically on the right side of Fig. 58.

Rice. 56. Schematic diagram of a hypothetical device in which the Clausius postulate is violated

Rice. 57. Schematic diagram of a hypothetical device in which Thomson’s postulate is violated

Rice. 58. Combining the device shown in Fig. with a heat engine. 56, in which the Clausius postulate is violated, we obtain a system in which the Thomson postulate is violated

As a result, the heater will give off an amount of heat to the working fluid of the heat engine, and receive, in a process that contradicts the Clausius postulate, an amount of heat so that in general it will give off an amount of heat equal to exactly this amount

The machine converts heat into work. In the refrigerator as a whole, no changes occur at all, because it gives and receives the same amount of heat. Now it is clear that by combining the action of a heat engine and a process that contradicts Clausius’s postulate, one can obtain a process that contradicts Thomson’s postulate.

Thus, the postulates of Clausius and Thomson are either both true or both false, and in this sense they are equivalent. Their validity for macroscopic systems is confirmed by all available experimental facts.

Carathéodory's principle. The physical content of the second law of thermodynamics in the formulations of Clausius and Thomson is expressed in the form of a statement about the impossibility of specific thermal processes. But it is also possible to give a formulation that does not specify the type of process, the impossibility of which is stated by this law. This formulation is called Carathéodory's principle. According to this principle, near each equilibrium state of any thermodynamic system there are other equilibrium states that are unattainable from the first one adiabatically.

Let us show the equivalence of Thomson's formulation and Carathéodory's principle. Let an arbitrary thermodynamic system quasistatically transition from some state 1 to a close state 2, receiving a certain amount of heat and doing work. Then, in accordance with the first law of thermodynamics

Let us return the system adiabatically from state 2 to state Then in such a reverse process there is no heat transfer, and the first law of thermodynamics gives

where is the work performed by the system. Adding (1) and (2), we get

Relationship (3) shows that in such a cyclic process, the system, having returned to its original state, converted all the heat received into work. But this is impossible according to the second law of thermodynamics as formulated by Thomson. This means that such a cyclic process is not feasible. Its first stage is always possible: at this stage, heat is simply supplied to the system, and no other conditions are imposed. Therefore, only the second stage is impossible here, when, according to the condition, the system must return to its original state adiabatically. In other words,

the state is adiabatically unreachable from a state close to it 2.

The principle of adiabatic unattainability means that almost all real physical processes occur with heat exchange: adiabatic processes are a rare exception. Next to each equilibrium state there are many others, the transition to which necessarily requires heat exchange, and only a few of them can be reached adiabatically.

Based on the above formulations of the second law of thermodynamics, it is possible to obtain Carnot's results for the maximum possible efficiency of heat engines. For a heat engine cycling between a heater with a fixed temperature and a refrigerator with a temperature, the efficiency coefficient cannot exceed the value

The highest value determined by formula (4) is achieved by a heat engine performing a reversible cycle, regardless of what is used as the working fluid. This statement, usually called Carnot's theorem, will be proven below.

A cycle is reversible if it consists of reversible processes, that is, those that can be carried out in any direction through the same chain of equilibrium states.

Rice. 59. Carnot cycle on the ideal gas diagram

The only reversible cyclic process that can be carried out between a heater and a refrigerator at fixed temperatures is the so-called Carnot cycle, consisting of two isotherms and two adiabats. For an ideal gas, such a cycle is shown in Fig. 59. In section 1-2, the gas has a temperature equal to the temperature of the heater and expands isothermally, receiving the amount of heat from the heater. In this case, the gas does positive work equal to the heat received. In section 2-3, the gas expands adiabatically, and at the same time its temperature decreases from to a value equal to the temperature of the refrigerator. The work performed by the gas in this section is equal to the decrease in its internal energy. In the next section 3-4, the gas is isothermally compressed. At the same time, it transfers to the refrigerator an amount of heat equal to the work done on it during compression. In section 4-1, the gas is adiabatically compressed until it is

the temperature will not increase to the value The increase in internal energy of the gas is equal to the work of external forces performed during compression of the gas.

The Carnot cycle is the only closed process that can be carried out in a reversible manner. In fact, adiabatic processes are reversible if they are carried out slowly enough, that is, quasi-statically. Isothermal processes are the only processes involving heat exchange that can be carried out in a reversible manner. In any other process, the temperature of the working fluid changes and, according to the second law of thermodynamics, heat exchange with a heater or refrigerator cannot be reversible: heat exchange in the presence of a finite temperature difference is in the nature of approaching thermal equilibrium and is not an equilibrium process.

Of course, heat exchange in the absence of a temperature difference occurs infinitely slowly. Therefore, the reversible Carnot cycle continues indefinitely and the power of the heat engine at the maximum possible efficiency, determined by formula (4), tends to zero. Processes in any real machine necessarily contain irreversible links, and, therefore, its efficiency is always less than the theoretical limit (4).

Conditions for obtaining maximum work. The transformation of internal energy into mechanical energy, as follows from the second law of thermodynamics, cannot be carried out completely. In order to convert the maximum possible part of internal energy into mechanical energy, it is necessary to use exclusively reversible processes. To illustrate, consider the following example. Let there be some body that is not in a state of thermal equilibrium with the environment, for example, an ideal gas in a cylinder with a piston, which has a temperature higher than the ambient temperature T (Fig. 60). How can you obtain the greatest amount of work, provided that in the final state the gas should occupy the same volume as in the initial state?

Rice. 60. Towards maximum performance

If the gas temperature were equal to the ambient temperature, i.e. the gas would be in thermal equilibrium with the environment, it would be impossible to get any work at all. The transformation of internal energy into mechanical energy can only occur if the initial state of the entire system is not equilibrium.

But in a non-equilibrium initial state, the transition of the system to an equilibrium state is not necessarily accompanied by the conversion of internal energy into mechanical energy. If you just bring the gas into

thermal contact with the environment, preventing it from expanding, the gas will cool down and no work will be done. Therefore, in order to be able to perform work, it is necessary to provide the gas with the opportunity to expand, bearing in mind that then it will have to be compressed, since according to the condition, in the final state the gas must occupy the same volume as in the initial state.

To obtain maximum work, the transition from the initial state to the final state must be made reversibly. And this can only be done using adiabatic and isothermal processes. So, the gas should be expanded adiabatically until its temperature is equal to the ambient temperature T, and then compressed isothermally at this temperature to its original volume (Fig. 61). The work done by the gas during adiabatic expansion 1-2, as can be seen from the figure, is greater than the work that would have to be done on the gas during isothermal compression 2-3. The maximum work that can be obtained during the transition of a gas from state 1 to state 3 is equal to the area shaded in Fig. 61 curved triangles 1-2-3.

The studied patterns of action of a reversible heat engine allow us to consider the principles of operation of the refrigeration machine and heat pump. In a refrigeration machine, all processes occur in the opposite direction (compared to a heat engine) (Fig. 62). Due to the performance of mechanical work A, a certain amount of heat is removed from a reservoir with a lower temperature. At the same time, an amount of heat is transferred to a reservoir with a higher temperature, the role of which is usually played by the environment equal to the sum Due to the reversibility of the machine under consideration, the relation is valid for it

which, in accordance with (4) can be considered as the efficiency of the corresponding heat engine.

For a refrigeration machine, the greatest interest is the amount of heat removed from the cooled reservoir. From (5) for we have

A graph of the dependence on ambient temperature (for a reversible process) is shown in Fig. 63. It can be seen that when the heat is removed, But with a small temperature difference, the ratio can take large values. In other words, the efficiency of the refrigeration machine at close

values ​​can be very large, since the amount of heat removed from the cooled bodies can significantly exceed the work A, which in real refrigeration machines is performed by a compressor driven by an electric motor.

In technical thermodynamics, to characterize a refrigeration machine, the so-called refrigeration coefficient is used, defined as the ratio of the amount of heat taken from the cooled bodies to the work of external forces

Unlike a heat engine (4), the coefficient of performance can take values ​​greater than unity.

Rice. 61. The process of obtaining maximum work on the -diagram

Rice. 62. Schematic diagram of a refrigeration machine

In real industrial and domestic installations and more. As can be seen from (7), the less the temperature of the environment and the cooled body differ, the greater the coefficient of refrigeration.

Let us now consider the operation of a heat pump, i.e., a refrigeration machine operating to heat a hot reservoir (heated room) due to the heat removed from the cold reservoir (environment). The circuit diagram of a heat pump is identical to that of a refrigeration machine (see Fig. 62). Unlike a refrigeration machine for a heat pump, the practical interest is not the amount of heat received by the heated body: For analogous to (6) we have

In technical thermodynamics, to characterize the efficiency of heat pumps, the so-called heating coefficient eotope is introduced, equal to

The given formulas (7) and (9) are valid for reversible machines. For real machines, where processes are completely or partially irreversible, these formulas provide an estimate of the cooling and heating coefficients.

So, when using a heat pump, the heated room receives more heat than with direct heating. W. Thomson drew attention to this circumstance when he proposed the idea of ​​so-called dynamic heating, which consists in the following. The heat obtained from burning fuel is not used to directly heat the room, but is sent to a heat engine to produce mechanical work. With this work, the heat pump is activated, which heats the room. When the temperature difference between the environment and the heated room is small, the latter receives noticeably more heat than is released when burning fuel. This may seem paradoxical.

In reality, there is no paradox in the heat pump and dynamic heating, which becomes completely clear if we use the concept of the quality of internal energy. The quality of internal energy refers to its ability to transform into other types. In this sense highest quality characterized by energy in mechanical or electromagnetic forms, since it can be completely converted into internal energy at any temperature. As for internal energy, its quality is higher, the higher the temperature of the body in which it is stored. Any naturally occurring irreversible process, for example the transfer of heat to a body with a lower temperature, leads to a devaluation of internal energy and a decrease in its quality. In reversible processes, a decrease in energy quality does not occur, since all energy transformations can go in the opposite direction.

With the usual heating method, all the heat released when burning fuel when heating the coil electric shock or received from a hot reservoir, etc., passes into the room in the form of the same amount of heat, but at a lower temperature, which represents a qualitative depreciation of internal energy. A heat pump or dynamic heating system eliminates the direct irreversible heat exchange between bodies at different temperatures.

When a heat pump or dynamic heating system operates, the quality of internal energy transferred to the heated room from the environment increases. At a small temperature difference, when the quality of this energy does not increase significantly, its quantity becomes greater, which explains the high efficiency of the heat pump and dynamic heating in general.

Give examples of phenomena that satisfy the law of conservation of energy, but nevertheless are never observed in nature.

What is the difference between different types of energy? Illustrate this disparity with examples.

What is a reversible thermal process? Give examples of reversible and irreversible processes.

What requirements must a physical system satisfy in order for mechanical processes in it to occur reversibly? Explain why friction and dissipation of mechanical energy make all processes irreversible.

Give different formulations of the second law of thermodynamics. Prove the equivalence of the Clausius and Thomson formulations.

What does Carathéodory's principle mean in relation to ideal gas? Explain your answer using a -diagram to depict its state.

Show that the physical meaning of the second law of thermodynamics is to establish an inextricable connection between the irreversibility of real processes in nature and heat transfer.

Formulate the conditions under which the efficiency of a heat engine operating in a reversible cycle would be close to unity.

Show that the Carnot cycle is the only reversible cyclic process for an engine using two heat reservoirs at fixed temperatures.

When discussing the conditions for obtaining maximum work, it was not taken into account Atmosphere pressure, acting on the piston from the outside. How will taking this pressure into account affect the above reasoning and the result?

The gas in a cylinder closed by a piston has the same temperature as the surrounding air, but a higher (or lower) pressure than the pressure in the atmosphere. What processes should be carried out with gas in order to obtain maximum useful work due to the non-equilibrium of the system? Draw these processes on a diagram, assuming the gas in the cylinder is ideal.

The gas in a cylinder closed by a piston has the same pressure as the surrounding air, but a higher (or lower) temperature. What processes should be carried out with gas in order to obtain maximum useful work due to the nonequilibrium of the system? Draw them on a diagram.

Consider two different dynamic heating schemes in which a heat engine either releases heat environment, or a heated room. Show that in the case where all processes are reversible, both schemes have the same efficiency. Which scheme will be more effective in a real system, when processes cannot be considered completely reversible?