Formula of internal energy. Internal energy and ways to change it

It is convenient to consider a particular physical phenomenon or class of phenomena using models of varying degrees of approximation. For example, when describing the behavior of a gas, a physical model is used - an ideal gas.

Any model has limits of applicability, beyond which it requires clarification or the use of more complex options. Here we will consider a simple case of describing the internal energy of a physical system based on the most essential properties of gases within certain limits.

Ideal gas

For the convenience of describing some fundamental processes, this physical model simplifies a real gas as follows:

• Neglects the size of gas molecules. This means that there are phenomena for which this parameter is not essential for an adequate description.
• It neglects intermolecular interactions, that is, it accepts that in the processes of interest they manifest themselves in negligibly small periods of time and do not affect the state of the system. In this case, the interactions are of the nature of an absolutely elastic impact, in which there is no energy loss due to deformation.
• Neglects the interaction of molecules with the walls of the tank.
• It assumes that the gas-reservoir system is characterized by thermodynamic equilibrium.

This model is suitable for describing real gases if the pressures and temperatures are relatively low.

Energy state of a physical system

Any macroscopic physical system (body, gas or liquid in a vessel) has, in addition to its own kinetic and potential, another type of energy - internal. This value is obtained by summing the energies of all the subsystems that make up the physical system - molecules.

Each molecule in a gas also has its own potential and kinetic energy. The latter is due to the continuous chaotic thermal movement of molecules. The various interactions between them (electrical attraction, repulsion) are determined by potential energy.

It must be remembered that if the energy state of any parts of a physical system does not have any effect on the macroscopic state of the system, then it is not taken into account. For example, under normal conditions, nuclear energy does not manifest itself in changes in the state of a physical object, so it does not need to be taken into account. But at high temperatures and pressures this must be done.

Thus, the internal energy of a body reflects the nature of the movement and interaction of its particles. This means that this term is synonymous with the often used concept of “thermal energy”.

Monatomic gases, that is, those whose atoms are not combined into molecules, exist in nature - these are inert gases. Gases such as oxygen, nitrogen or hydrogen can exist in such a state only under conditions where energy is expended from the outside to constantly renew this state, since their atoms are chemically active and tend to combine into a molecule.

Let us consider the energy state of a monatomic ideal gas placed in a vessel of some volume. This is the simplest case. We remember that the electromagnetic interaction of atoms with each other and with the walls of the vessel, and, consequently, their potential energy is negligible. So the internal energy of a gas includes only the sum of the kinetic energies of its atoms.

It can be calculated by multiplying the average kinetic energy of the atoms in the gas by their number. The average energy is equal to E = 3/2 x R / N A x T, where R is the universal gas constant, N A is Avogadro's number, T is the absolute temperature of the gas. We calculate the number of atoms by multiplying the amount of substance by Avogadro's constant. The internal energy of a monatomic gas will be equal to U = N A x m / M x 3/2 x R/N A x T = 3/2 x m / M x RT. Here m is the mass and M is the molar mass of the gas.

Let us assume that the chemical composition of the gas and its mass always remain the same. In this case, as can be seen from the formula we obtained, the internal energy depends only on the temperature of the gas. For a real gas, it will be necessary to take into account, in addition to temperature, the change in volume, since this affects the potential energy of the atoms.

Molecular gases

In the above formula, the number 3 characterizes the number of degrees of freedom of movement of a monatomic particle - it is determined by the number of coordinates in space: x, y, z. For the state of a monatomic gas, it is generally indifferent whether its atoms rotate.

Molecules are spherically asymmetric, therefore, when determining the energy state of molecular gases, it is necessary to take into account the kinetic energy of their rotation. Diatomic molecules, in addition to the listed degrees of freedom associated with translational motion, have two more associated with rotation around two mutually perpendicular axes; Polyatomic molecules have three such independent rotation axes. Consequently, particles of diatomic gases are characterized by the number of degrees of freedom f = 5, while for polyatomic molecules f = 6.

Due to the randomness inherent in thermal motion, all directions of both rotational and translational motion are completely equally probable. The average kinetic energy contributed by each type of motion is the same. Therefore, we can substitute the value of f into the formula, which allows us to calculate the internal energy of an ideal gas of any molecular composition: U = f / 2 x m / M x RT.

Of course, we see from the formula that this value depends on the amount of the substance, that is, on how much and what kind of gas we took, as well as on the structure of the molecules of this gas. However, since we agreed not to change the mass and chemical composition, we only need to take into account the temperature.

Now let's look at how the U value is related to other gas characteristics - volume and pressure.

Internal energy and thermodynamic state

Temperature, as is known, is one of the states of the system (in this case, gas). In an ideal gas, it is related to pressure and volume by the relation PV = m / M x RT (the so-called Clapeyron-Mendeleev equation). Temperature determines thermal energy. So the latter can be expressed through a set of other state parameters. She is indifferent to the previous state, as well as to the method of changing it.

Let's see how the internal energy changes when the system moves from one thermodynamic state to another. Its change during any such transition is determined by the difference between the initial and final values. If the system returns to its original state after some intermediate state, then this difference will be equal to zero.

Let's say we heated the gas in the reservoir (that is, we supplied additional energy to it). The thermodynamic state of the gas changed: its temperature and pressure increased. This process occurs without changing the volume. The internal energy of our gas has increased. After this, our gas gave up the supplied energy, cooling to its original state. A factor such as, for example, the speed of these processes will not have any significance. The resulting change in the internal energy of the gas at any rate of heating and cooling is zero.

An important point is that the same value of thermal energy can correspond to not one, but several thermodynamic states.

The nature of changes in thermal energy

In order to change energy, work must be done. Work can be done by the gas itself or by an external force.

In the first case, energy expenditure to perform work is due to the internal energy of the gas. For example, we had compressed gas in a tank with a piston. If you release the piston, the expanding gas will begin to lift it, doing work (for it to be useful, let the piston lift some kind of weight). The internal energy of the gas will decrease by the amount spent on work against gravity and friction forces: U 2 = U 1 - A. In this case, the work of the gas is positive, since the direction of the force applied to the piston coincides with the direction of movement of the piston.

Let's begin to lower the piston, doing work against the gas pressure force and again against the friction forces. Thus, we impart a certain amount of energy to the gas. Here the work of external forces is already considered positive.

In addition to mechanical work, there is also a way to take away energy from a gas or give it energy, as we have already encountered in the example of heating a gas. The energy transferred to the gas during heat exchange processes is called the amount of heat. There are three types of heat transfer: conduction, convection and radiative transfer. Let's look at them in a little more detail.

Thermal conductivity

The ability of a substance to exchange heat, carried out by its particles by transferring kinetic energy to each other during mutual collisions during thermal motion, is thermal conductivity. If a certain region of a substance is heated, that is, a certain amount of heat is imparted to it, the internal energy after some time, through collisions of atoms or molecules, will be distributed on average uniformly among all particles.

It is clear that thermal conductivity strongly depends on the frequency of collisions, and that, in turn, on the average distance between particles. Therefore, gas, especially ideal gas, is characterized by very low thermal conductivity, and this property is often used for thermal insulation.

Of the real gases, thermal conductivity is higher in those whose molecules are the lightest and at the same time polyatomic. This condition is met to the greatest extent by molecular hydrogen, and to the least by radon, as the heaviest monatomic gas. The more rarefied the gas, the worse a conductor of heat it is.

In general, energy transfer by thermal conduction for an ideal gas is a very inefficient process.

Convection

Much more effective for gas is convection, in which internal energy is distributed through a flow of matter circulating in a gravitational field. hot gas is formed due to the Archimedean force, since it is less dense due to the hot gas moving upward is constantly replaced by colder gas - circulation of gas flows is established. Therefore, in order to ensure effective, that is, the fastest, heating through convection, it is necessary to heat the gas tank from below - just like a kettle with water.

If it is necessary to remove a certain amount of heat from the gas, then it is more efficient to place the refrigerator at the top, since the gas that has given up energy to the refrigerator will rush down under the influence of gravity.

An example of convection in gas is heating indoor air using heating systems (they are placed in the room as low as possible) or cooling using air conditioning, and in natural conditions the phenomenon of thermal convection causes the movement of air masses and affects the weather and climate.

In the absence of gravity (with weightlessness in a spaceship), convection, that is, the circulation of air currents, is not established. So there is no point in lighting gas burners or matches on board a spacecraft: hot combustion products will not be vented upward, and oxygen will not be supplied to the fire source, and the flame will go out.

Radiant transfer

A substance can also heat up under the influence of thermal radiation, when atoms and molecules acquire energy by absorbing electromagnetic quanta - photons. At low photon frequencies this process is not very efficient. Let us remember that when we open a microwave oven, we find hot food there, but not hot air. As the frequency of radiation increases, the effect of radiation heating increases; for example, in the upper atmosphere of the Earth, highly rarefied gas is intensely heated and ionized by solar ultraviolet radiation.

Different gases absorb thermal radiation to varying degrees. So, water, methane, carbon dioxide absorb it quite strongly. The phenomenon of the greenhouse effect is based on this property.

First law of thermodynamics

Generally speaking, a change in internal energy through heating a gas (heat exchange) also comes down to doing work either on gas molecules or on them through an external force (which is denoted in the same way, but with the opposite sign). What kind of work is done with this method of transition from one state to another? The law of conservation of energy will help us answer this question, or more precisely, its concretization in relation to the behavior of thermodynamic systems - the first law of thermodynamics.

The law, or the universal principle of conservation of energy, in its most general form states that energy is not born from nothing and does not disappear without a trace, but only passes from one form to another. In relation to a thermodynamic system, this must be understood in such a way that the work done by the system is expressed through the difference between the amount of heat imparted to the system (ideal gas) and the change in its internal energy. In other words, the amount of heat imparted to the gas is spent on this change and on the operation of the system.

This is written much simpler in the form of formulas: dA = dQ - dU, and, accordingly, dQ = dU + dA.

We already know that these quantities do not depend on the way in which the transition between states occurs. The speed of this transition and, as a consequence, efficiency depend on the method.

As for the second law of thermodynamics, it specifies the direction of change: heat cannot be transferred from a colder (and therefore less energetic) gas to a hotter one without additional energy expenditure from the outside. The second law also indicates that part of the energy spent by the system to perform work inevitably dissipates, is lost (does not disappear, but goes into an unusable form).

Thermodynamic processes

Transitions between energy states of an ideal gas can have different character of changes in certain of its parameters. Internal energy in transition processes of different types will also behave differently. Let us briefly consider several types of such processes.

• An isochoric process occurs without a change in volume, therefore, the gas does not do any work. The internal energy of a gas changes as a function of the difference between the final and initial temperatures.
• An isobaric process occurs at constant pressure. The gas does work, and its thermal energy is calculated in the same way as in the previous case.
• An isothermal process is characterized by a constant temperature, which means that the thermal energy does not change. The amount of heat received by the gas is entirely spent on doing work.
• An adiabatic or adiabatic process occurs in a gas without heat transfer, in a thermally insulated reservoir. Work is done only due to the expenditure of thermal energy: dA = - dU. During adiabatic compression, thermal energy increases, and during expansion, it decreases accordingly.

Various isoprocesses underlie the functioning of heat engines. Thus, an isochoric process takes place in a gasoline engine at the extreme positions of the piston in the cylinder, and the second and third strokes of the engine are examples of an adiabatic process. When producing liquefied gases, adiabatic expansion plays an important role - thanks to it, gas condensation becomes possible. Isoprocesses in gases, in the study of which one cannot do without the concept of the internal energy of an ideal gas, are characteristic of many natural phenomena and find application in a wide variety of branches of technology.

Internal energy body (denoted as E or U) is the sum of the energies of molecular interactions and thermal motions of the molecule. Internal energy is a unique function of the state of the system. This means that whenever a system finds itself in a given state, its internal energy takes on the value inherent in this state, regardless of the previous history of the system. Consequently, the change in internal energy during the transition from one state to another will always be equal to the difference between its values ​​in the final and initial states, regardless of the path along which the transition took place.

The internal energy of a body cannot be measured directly. You can only determine the change in internal energy:

This formula is a mathematical expression of the first law of thermodynamics

For quasi-static processes the following relation holds:

Ideal gases

According to Joule's law, derived empirically, the internal energy of an ideal gas does not depend on pressure or volume. Based on this fact, we can obtain an expression for the change in internal energy of an ideal gas. By definition of molar heat capacity at constant volume, . Since the internal energy of an ideal gas is a function only of temperature, then

The same formula is also true for calculating changes in the internal energy of any body, but only in processes with constant volume (isochoric processes); in general, it is a function of both temperature and volume.

If we neglect the change in molar heat capacity with a change in temperature, we obtain:

where is the amount of substance, is the change in temperature.

Literature

• Sivukhin D.V. General physics course. - 5th edition, revised. - M.: Fizmatlit, 2006. - T. II. Thermodynamics and molecular physics. - 544 p. - ISBN 5-9221-0601-5

Notes

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See what “Internal energy” is in other dictionaries: internal energy - A function of the state of a closed thermodynamic system, determined by the fact that its increment in any process occurring in this system is equal to the sum of the heat imparted to the system and the work done on it. Note Internal energy... ...

Technical Translator's Guide Physical energy system, depending on its internal. condition. V. e. includes the energy of the chaotic (thermal) movement of all microparticles of the system (molecules, atoms, ions, etc.) and the energy of the action of these particles. Kinetic. energy of motion of the system as a whole and...

Physical encyclopedia INTERNAL ENERGY - the energy of a body or system, depending on its internal state; consists of the kinetic energy of body molecules and their structural units (atoms, electrons, nuclei), the interaction energy of atoms in molecules, the interaction energy of electronic... ...

Big Polytechnic Encyclopedia A body consists of the kinetic energy of the molecules of the body and their structural units (atoms, electrons, nuclei), the energy of interaction of atoms in molecules, etc. The internal energy does not include the energy of motion of the body as a whole and potential energy...

See what “Internal energy” is in other dictionaries: Big Encyclopedic Dictionary - ▲ energy material body, in accordance with, state, internal temperature internal en...

See what “Internal energy” is in other dictionaries: Ideographic Dictionary of the Russian Language Chemical terms

Modern encyclopedia

Internal energy- body, includes the kinetic energy of the molecules, atoms, electrons, nuclei that make up the body, as well as the energy of interaction of these particles with each other. The change in internal energy is numerically equal to the work done on the body (for example, when... ... Illustrated Encyclopedic Dictionary

See what “Internal energy” is in other dictionaries:- a thermodynamic quantity that characterizes the number of all types of internal movements performed in the system. It is impossible to measure the absolute internal energy of a body. In practice, only the change in internal energy is measured... ... Encyclopedic Dictionary of Metallurgy

A body consists of the kinetic energy of the molecules of the body and their structural units (atoms, electrons, nuclei), the energy of interaction of atoms in molecules, etc. The internal energy does not include the energy of motion of the body as a whole and potential energy... encyclopedic Dictionary

Books

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If you pump into a thick-walled jar closed with a stopper, the bottom of which is covered with water, then after some time the stopper will fly out of the jar and fog will form in the jar. The cork flew out of the jar because the air there acted on it with a certain force. The air did work when the plug came out. It is known that a body can do work if it has energy. Therefore, the air in the jar has energy.

As the air performed work, its temperature decreased and its condition changed. At the same time, the mechanical energy of the air did not change: neither its speed nor its position relative to the Earth changed. Consequently, the work was done not due to mechanical, but due to other energy. This energy is the internal energy of the air in the jar.

Internal energy a body is the sum of the kinetic energy of movement of its molecules and the potential energy of their interaction. Kinetic energy ( Ek) molecules have, since they are in motion, potential energy ( Ep) as they interact. Internal energy is denoted by the letter U. The unit of internal energy is 1 joule (1 J). U = Ek + En.

Ways to change internal energy

The greater the speed of movement of molecules, the higher the temperature of the body, therefore, the internal energy depends on body temperature . To transform a substance from a solid to a liquid state, for example, to turn ice into water, you need to supply energy to it. Therefore, water will have more internal energy than ice of the same mass, and therefore internal energy depends on the physical state of the body .

Internal energy can be changed when performing work . If you hit a piece of lead several times with a hammer, you can even tell by touch that the piece of lead will heat up. Consequently, his internal energy, as well as the internal energy of the hammer, increased. This happened because work was done on a piece of lead.

If the body itself does work, then its internal energy decreases, and if work is done on it, then its internal energy increases.

If you pour hot water into a glass of cold water, the temperature of the hot water will decrease, and the temperature of the cold water will increase. In the example considered, mechanical work is not performed; the internal energy of the bodies changes by heat transfer, as evidenced by the decrease in its temperature.

Hot water molecules have more kinetic energy than cold water molecules. Hot water molecules transfer this energy to cold water molecules during collisions, and the kinetic energy of cold water molecules increases. The kinetic energy of hot water molecules decreases.

Heat transfer is a way of changing the internal energy of a body when transferring energy from one part of the body to another or from one body to another without doing work.

Internal body energy cannot be a constant value. It can change in any body. If you increase the body temperature, then its internal energy will increase, because the average speed of molecular movement will increase. Thus, the kinetic energy of the molecules of the body increases. And, conversely, as the temperature decreases, the internal energy of the body decreases.

We can conclude: The internal energy of a body changes if the speed of movement of the molecules changes. Let's try to determine what method can be used to increase or decrease the speed of movement of molecules. Consider the following experiment. Let's attach a brass tube with thin walls to the stand. Fill the tube with ether and close it with a stopper. Then we tie a rope around it and begin to move the rope intensively in different directions. After a certain time, the ether will boil, and the force of the steam will push out the plug. Experience demonstrates that the internal energy of the substance (ether) has increased: after all, it has changed its temperature, at the same time boiling.

The increase in internal energy occurred due to the work done when the tube was rubbed with a rope.

As we know, heating of bodies can also occur during impacts, flexion or extension, or, more simply, during deformation. In all the examples given, the internal energy of the body increases.

Thus, the internal energy of the body can be increased by doing work on the body.

If the work is performed by the body itself, its internal energy decreases.

Let's consider another experiment.

We pump air into a glass vessel that has thick walls and is closed with a stopper through a specially made hole in it.

After some time, the cork will fly out of the vessel. At the moment when the stopper flies out of the vessel, we will be able to see the formation of fog. Consequently, its formation means that the air in the vessel has become cold. The compressed air that is in the vessel does a certain amount of work when pushing the plug out. He performs this work due to his internal energy, which is reduced. Conclusions about the decrease in internal energy can be drawn based on the cooling of the air in the vessel. Thus, The internal energy of a body can be changed by performing certain work.

However, internal energy can be changed in another way, without doing work. Let's consider an example: water in a kettle that is standing on the stove is boiling. The air, as well as other objects in the room, are heated by a central radiator. In such cases, the internal energy increases, because body temperature increases. But the work is not done. So, we conclude a change in internal energy may not occur due to the performance of a certain amount of work.

Let's look at another example.

Place a metal knitting needle in a glass of water. The kinetic energy of hot water molecules is greater than the kinetic energy of cold metal particles. The hot water molecules will transfer some of their kinetic energy to the cold metal particles. Thus, the energy of the water molecules will decrease in a certain way, while the energy of the metal particles will increase. The water temperature will drop, and the temperature of the knitting needle will slowly will increase. In the future, the difference between the temperature of the knitting needle and the water will disappear. Due to this experience, we saw a change in the internal energy of various bodies. We conclude: The internal energy of various bodies changes due to heat transfer.

The process of converting internal energy without performing specific work on the body or the body itself is called heat transfer.

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Energy is a general measure of various forms of motion of matter. According to the forms of motion of matter, types of energy are also distinguished - mechanical, electrical, chemical, etc. Any thermodynamic system in any state has a certain amount of energy, the existence of which was proven by R. Clausius (1850) and was called internal energy.

Internal energy (U) is the energy of all types of movement of microparticles that make up the system, and the energy of their interaction with each other.

Internal energy consists of the energy of translational, rotational and vibrational motion of particles, the energy of intermolecular and intramolecular, intraatomic and intranuclear interactions, etc.

The energy of intramolecular interaction, i.e. energy of interaction of atoms in a molecule, often called chemical energy . A change in this energy occurs during chemical transformations.

For thermodynamic analysis, there is no need to know from which forms of motion of matter the internal energy is composed.

The amount of internal energy depends only on the state of the system. Consequently, internal energy can be considered as one of the characteristics of this state, along with such quantities as pressure, temperature.

Each state of the system corresponds to a strictly defined value of each of its properties.

If a homogeneous system in the initial state has volume V 1, pressure P 1, temperature T 1, internal energy U 1, electrical conductivity æ 1, etc., and in the final state these properties are respectively equal to V 2, P 2, T 2, U 2, æ 2, etc., then the change in each property during the transition of the system from the initial state to the final state will be the same, regardless of which way the system passes from one state to another: first, second or third (Fig. .1.4).

Rice. 1.4 Independence of system properties from its transition path

from normal state to another

Those. (U 2 - U 1) I = (U 2 - U 1) II = (U 2 - U 1) III (1.4)

Where are the numbers I, II, III, etc. indicate process paths. Consequently, if the system moves from the initial state (1) to the final state (2) along one path, and from the final state at the beginning - along another path, i.e. If a circular process (cycle) is completed, then the change in each property of the system will be equal to zero.

Thus, the change in the system state function does not depend on the process path, but depends only on the initial and final states of the system. An infinitesimal change in the properties of a system is usually denoted by the differential sign d. For example, dU is an infinitely small change in internal energy, etc.

Forms of energy exchange

In accordance with different forms of motion of matter and different types of energy, there are different forms of energy exchange (energy transfer) - forms of interaction. Thermodynamics considers two forms of energy exchange between a system and its environment. This is work and warmth.

Job. The most obvious form of energy exchange is mechanical work, corresponding to the mechanical form of motion of matter. It is produced when the body moves under the influence of mechanical force. In accordance with other forms of motion of matter, other types of work are also distinguished: electrical, chemical, etc. Work is a form of transmission of orderly, organized motion, since when work is performed, the particles of the body move in an organized manner in one direction. For example, work done during gas expansion. The molecules of the gas located in the cylinder under the piston are in chaotic, disordered motion. When the gas begins to move the piston, that is, to perform mechanical work, organized movement will be superimposed on the random movement of gas molecules: all molecules receive some displacement in the direction of movement of the piston. Electrical work is also associated with the organized movement of charged particles of matter in a certain direction.

Since work is a measure of energy transferred, its quantity is measured in the same units as energy.

Heat. The form of energy exchange corresponding to the chaotic movement of microparticles that make up the system is called heat exchange, and the amount of energy transferred during heat exchange is called warmth.

Heat transfer is not associated with a change in the position of the bodies that make up a thermodynamic system, and consists of the direct transfer of energy from the molecules of one body to the molecules of another upon their contact.

P Let us imagine an insulated vessel (system) divided into two parts by a heat-conducting partition AB (Fig. 1.5). Let us assume that there is gas in both parts of the vessel.

Rice. 1.5. To the concept of heat

In the left half of the vessel the gas temperature is T 1, and in the right half T 2. If T 1 > T 2, then the average kinetic energy ( ) gas molecules on the left side of the vessel will be greater than the average kinetic energy ( ) in the right half of the vessel.

As a result of continuous collisions of molecules with the partition in the left half of the vessel, part of their energy is transferred to the molecules of the partition. The molecules of the gas located in the right half of the vessel, colliding with the partition, will acquire some part of the energy from its molecules.

As a result of these collisions, the kinetic energy of molecules in the left half of the vessel will decrease, and in the right half it will increase; temperatures T 1 and T 2 will be equalized.

Since heat is a form of energy, its quantity is measured in the same units as energy. Thus, heat exchange and work are forms of energy exchange, and the amount of heat and the amount of work are measures of the energy transferred. The difference between them is that heat is a form of transfer of microphysical, disordered movement of particles (and, accordingly, the energy of this movement), and work is a form of transfer of energy of ordered, organized movement of matter.

Sometimes they say: heat (or work) is supplied or removed from the system, but it should be understood that it is not heat and work that is supplied or removed, but energy, therefore one should not use such expressions as “heat reserve” or “heat contained.”

Being forms of energy exchange (forms of interaction) of a system with the environment, heat and work cannot be associated with any specific state of the system, cannot be its properties, and, therefore, functions of its state. This means that if the system passes from the initial state (1) to the final state (2) in different ways, then heat and work will have different values ​​for different transition paths (Fig. 1.6)

The finite amounts of heat and work are denoted by Q and A, and infinitesimal values ​​by δQ and δA, respectively. The quantities δQ and δA, unlike dU, are not a complete differential, because Q and A are not state functions.

When the path of the process is predetermined, work and heat will acquire the properties of functions of the state of the system, i.e. their numerical values ​​will be determined only by the initial and final states of the system.