, thermodynamic process is a change in the state of a system, as a result of which at least one of its parameters (temperature, volume or pressure) changes its value. However, if we take into account that all parameters of a thermodynamic system are inextricably interconnected, then a change in any of them inevitably entails a change in at least one (ideally) or several (in reality) parameters. In the general case, we can say that the thermodynamic process is associated with an imbalance of the system, and if the system is in an equilibrium state, then no thermodynamic processes can occur in it.

The equilibrium state of a system is an abstract concept, since it is impossible to isolate anything material from the surrounding world, therefore, in any real system various thermodynamic processes inevitably occur. Moreover, in some systems such slow, almost imperceptible changes may take place that the processes associated with them can be conditionally considered to consist of a sequence of equilibrium states of the system. Such processes are called equilibrium or quasi-static.
Another possible scenario of successive changes in the system, after which it returns to its original state, is called circular process or a cycle. The concepts of equilibrium and circular processes underlie many theoretical conclusions and applied techniques of thermodynamics.

The study of a thermodynamic process consists of determining the work done in a given process, changes in internal energy, the amount of heat, as well as establishing a connection between individual quantities characterizing the state of the gas.

Of all the possible thermodynamic processes, the most interesting are isochoric, isobaric, isothermal, adiabatic and polytropic processes.

Isochoric process

Isochoric is a thermodynamic process that occurs at constant volume. This process can be accomplished by heating a gas placed in a closed vessel. As a result of the supply of heat, the gas heats up and its pressure increases.
The change in gas parameters in an isochoric process is described by Charles’s law: p 1 /T 1 = p 2 /T 2, or in the general case:

p/T = const.

The gas pressure on the walls of the vessel is directly proportional to the absolute temperature of the gas.

Since in an isochoric process the change in volume dV is zero, we can conclude that all the heat supplied to the gas is spent on changing the internal energy of the gas (no work is done).

Isobaric process

Isobaric is a thermodynamic process that occurs at constant pressure. Such a process can be carried out by placing the gas in a dense cylinder with a movable piston, which is subject to a constant external force during the removal and supply of heat.
When the gas temperature changes, the piston moves in one direction or another; in this case, the volume of gas changes in accordance with the Gay-Lussac law:

V/T = const.

This means that in an isobaric process, the volume occupied by the gas is directly proportional to the temperature.
We can conclude that a change in temperature in this process will inevitably lead to a change in the internal energy of the gas, and the change in volume is associated with the performance of work, i.e., in an isobaric process, part of the thermal energy is spent on changing the internal energy of the gas, and the other part on the performance of the gas work to overcome the action of external forces. In this case, the ratio between the heat consumption to increase internal energy and to perform work depends on the heat capacity of the gas.

Isothermal process

Isothermal is a thermodynamic process that occurs at a constant temperature.
In practice, it is very difficult to carry out an isothermal process with gas. After all, it is necessary to meet the condition that during the process of compression or expansion the gas has time to exchange temperature with the environment, maintaining its own temperature constant.
The isothermal process is described by the Boyle-Mariotte law: pV = const, i.e., at a constant temperature, the gas pressure is inversely proportional to its volume.

Obviously, during an isothermal process, the internal energy of the gas does not change, since its temperature is constant.
In order for the condition of constant gas temperature to be met, it is necessary to remove heat equivalent to the work expended on compression:

dq = dA = pdv.

Using the equation of state of a gas and having made a number of transformations and substitutions, we can conclude that the work of a gas during an isothermal process is determined by the expression:

A = RT ln(p 1 /p 2).

Adiabatic process

Adiabatic is a thermodynamic process that occurs without heat exchange between the working fluid and the environment. Like the isothermal process, the adiabatic process is very difficult to implement in practice. Such a process can occur with a working fluid placed in a vessel, for example, a cylinder with a piston, surrounded by high-quality heat-insulating material.
But no matter what high-quality heat insulator we use in this case, some, even negligible, amounts of heat will inevitably be exchanged between the working fluid and the environment.
Therefore, in practice, it is possible to create only an approximate model of the adiabatic process. However, many thermodynamic processes carried out in thermal engineering proceed so quickly that the working fluid and the medium do not have time to exchange heat, therefore, with some degree of error, such processes can be considered adiabatic.

To derive an equation relating pressure and volume 1 kg gas in an adiabatic process, we write the equation of the first law of thermodynamics:

dq = du + pdv.

Since for an adiabatic process the heat transfer dq is zero, and the change in internal energy is a function of thermal conductivity on temperature: du = c v dT, then we can write:

c v dT + pdv = 0 (3) .

Differentiating the Clapeyron equation pv = RT, we obtain:

pdv + vdp = RdT .

Let us express dT from here and substitute it into equation (3). After regrouping and transformations we get:

pdvc v /(R + 1) + c v vdp/R = 0.

Taking into account Mayer's equation R = c p – c v, the last expression can be rewritten as:

pdv(c v + c p - c v)/(c p – c v) + c v vdp/(c p – c v) = 0,

c p pdv + c v vdp = 0 (4) .

Dividing the resulting expression by c v and denoting the ratio c p /c v by the letter k, after integrating the equation (4) we get (with k = const):

ln vk + ln p = const or ln pvk = const or pvk = const.

The resulting equation is the equation of an adiabatic process, in which k is the adiabatic exponent.
If we assume that the volumetric heat capacity c v is a constant value, i.e. c v = const, then the work of the adiabatic process can be represented as the formula (given without output):

l = c v (T 1 – T 2) or l = (p 1 v 1 – p 2 v 2)/(k-1).

Polytropic process

Unlike the thermodynamic processes discussed above, when any of the gas parameters remained unchanged, the polytropic process is characterized by the possibility of changing any of the main gas parameters. All thermodynamic processes discussed above are special cases of polytropic processes.
The general equation of a polytropic process has the form pv n = const, where n is the polytropic index - a constant value for a given process that can take values ​​from - ∞ to + ∞.

It is obvious that by giving the polytropic index certain values, one can obtain one or another thermodynamic process - isochoric, isobaric, isothermal or adiabatic.
So, if we take n = 0, we get p = const - an isobaric process, if we take n = 1, we get an isothermal process described by the dependence pv = const; for n = k the process is adiabatic, and for n equal to - ∞ or + ∞. we get an isochoric process.

What is an isobaric process

Definition

An isobaric (or isobaric) process is a process occurring in a constant mass of gas at constant pressure.

Let us write the equation for two states of an ideal gas:

\ \

Dividing equation (2) by equation (1), we obtain the equation of the isobaric process:

\[\frac(V_2)(V_1)=\frac(T_2)(T_1)\ (3)\]

\[\frac(V)(T)=const\ \left(4\right).\]

Equation (4) is called Gay-Lussac's law.

Internal energy and amount of heat of an isobaric process

This process occurs with heat input if the volume increases, or heat removal to decrease the volume. Let us write down the first law of thermodynamics and consistently obtain expressions for work, internal energy and the amount of heat of an isobaric process:

\[\delta Q=dU+dA=\frac(i)(2)\nu RdT+pdV,\ \left(5\right).\] \[\triangle Q=\int\limits^(T_2)_ (T_1)(dU)+\int\limits^(V_2)_(V_1)(dA)(6)\]

where $\delta Q\ $ is the elementary heat supplied to the system, $dU$ is the change in the internal energy of the gas in the ongoing process, $dA$ is the elementary work performed by the gas in the process, i is the number of degrees of freedom of the gas molecule, R - - universal gas constant, d - number of moles of gas.

Change in internal energy of gas:

\[\triangle U=\frac(i)(2)\nu R((T)_2-T_1)\ (7)\] \

Equation (8) determines the work for an isobaric process. Subtracting equation (1) from (2), we obtain another equation for the work of a gas in an isobaric process:

\ \[\triangle Q=\frac(i)(2)nR((T)_2-T_1)+\nu R((T)_2-T_1)=c_(\mu p)\nu \triangle T\ ( 10),\]

where $c_(\mu p)$ is the molar heat capacity of the gas in an isobaric process. Equation (10) determines the amount of heat imparted to a gas of mass m in an isobaric process with an increase in temperature by $\triangle T.$

Isoprocesses are very often depicted in thermodynamic diagrams. Thus, the line depicting an isobaric process on such a diagram is called an isobar (Fig. 1).

Example 1

Task: Determine how the pressures $p_1$ and $p_2$ relate in the V(T) diagram in Fig. 1c.

Let's draw the $T_1$ isotherm

At points A and B, the temperatures are the same, therefore, the gas obeys the Boyle-Mariotte law:

\ \

Let's convert these volumes into SI: $V_1=2l=2(\cdot 10)^(-3)m^3$, $V_2=4l=4( 10)^(-3)m^3$

Let's carry out the calculations:

Answer: The work done by a gas in an isobaric process is 600 J.

Example 3

Assignment: Compare the work done on gas in the ABC process and the work done on gas in the CDA process (Figure 3).

As a basis for the solution, we will take the formula that determines the work of the gas:

From the geometric meaning of the definite integral it is known that work is the area of ​​the figure, which is limited by the function of the integrand, the abscissa axis, and isochores at points $V_1\ and\ V_2$ (p(V) axis). Let's convert the process graphs to p(V) axes.

Let's consider each segment of the process graphs shown in Figure (3).

AB: Isochoric process (p=const), $V\uparrow \left(\Volume\increases\right),\ T\uparrow $;

VS: Isochoric process (V =const), $T\uparrow $ (from the graph), p$\uparrow $, from the law for an isochoric process ($\frac(p)(T)=const$);

CD: (p=const), $V\downarrow ,\ T\downarrow ;$

DA: (V =const), $T\downarrow ,\p\downarrow .$

Let us depict the graphs of the processes in the p(V) axes (Fig. 4):

Gas work $A_(ABC)=S_(ABC)$ ($S_(ABC)$ is the area of ​​the rectangle ABFE) (Fig. 3). Work on gas $A_(CDA)=S_(CDA)$ ($S_(CDA)$)$\ -area\ rectangle\ $EFCD. Obviously, $A_(CDA)>A_(ABC).$

Isobaric process

Graphs of isoprocesses in different coordinate systems

Isobaric process(ancient Greek ισος, isos - “same” + βαρος, baros - “weight”) - the process of changing the state of a thermodynamic system at constant pressure ()

The dependence of gas volume on temperature at constant pressure was experimentally studied in 1802 by Joseph Louis Gay-Lussac. Gay-Lussac's law: At constant pressure and constant values ​​of the mass of the gas and its molar mass, the ratio of the volume of the gas to its absolute temperature remains constant: V/T = const.

Isochoric process

Isochoric process(from the Greek chorus - occupied space) - the process of changing the state of a thermodynamic system at constant volume (). For ideal gases, the isochoric process is described by Charles' law: for a given mass of gas at constant volume, pressure is directly proportional to temperature:

The line depicting an isochoric process on a diagram is called an isochore.

It is also worth pointing out that the energy supplied to the gas is spent on changing the internal energy, that is, Q = 3* ν*R*T/2=3*V*ΔP, where R is the universal gas constant, ν is the number of moles in the gas, T is the temperature in Kelvin , V volume of gas, ΔP increment of pressure change. and the line depicting the isochoric process on the diagram, in the P(T) axes, should be extended and connected with a dotted line to the origin of coordinates, since misunderstandings may arise.

Isothermal process

Isothermal process(from the Greek “thermos” - warm, hot) - the process of changing the state of a thermodynamic system at a constant temperature ()(). The isothermal process is described by the Boyle-Mariotte law:

At a constant temperature and constant values ​​of the gas mass and its molar mass, the product of the gas volume and its pressure remains constant: PV = const.

Isoentropic process

Isoentropic process- the process of changing the state of a thermodynamic system at constant entropy (). For example, a reversible adiabatic process is isentropic: in such a process there is no heat exchange with the environment. An ideal gas in such a process is described by the following equation:

where is the adiabatic index, determined by the type of gas.

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See what “Isoprocesses” are in other dictionaries:

Isoprocesses are thermodynamic processes during which mass and another physical quantity of state parameters: pressure, volume or temperature remain unchanged. So, constant pressure corresponds to an isobaric process, volume is isochoric... Wikipedia

Molecular kinetic theory (abbreviated as MKT) is a theory that considers the structure of matter from the point of view of three main approximately correct provisions: all bodies consist of particles whose size can be neglected: atoms, molecules and ions; particles... ...Wikipedia

- (abbreviated MKT) a theory that considers the structure of matter from the point of view of three main approximately correct provisions: all bodies consist of particles whose size can be neglected: atoms, molecules and ions; particles are in continuous... ... Wikipedia

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• Statistical forecasting of the deformation-strength characteristics of structural materials, G. Pluvinazh, V. T. Sapunov, This book presents a new method that offers a general methodology for predicting the characteristics of kinetic processes, unified for metal and polymer materials. Method… Category: Textbooks for universities Publisher:

Basic thermodynamic properties of ideal gases

When studying thermodynamic processes, the equation of state is used

and the mathematical expression of the first law of thermodynamics

When studying thermodynamic processes of ideal gases, in the general case it is necessary to determine the equation of the process curve in PV , P.T. , VT diagram, establish a connection between thermodynamic parameters and determine the following quantities:

− change in internal energy of the working fluid

(the formula is valid not only for V = const, but also for any process)

− determine the external (thermodynamic) specific work

and available specific work

−the amount of heat involved in the thermodynamic process

Where is the heat capacity of the process

– enthalpy change in a thermodynamic process

(the formula is valid not only for p = const, but also in any process)

– the proportion of heat that is spent on changing the internal energy in this process:

– the fraction of heat converted into useful work in a given process

In general, any two thermodynamic parameters out of three ( P , V , T) can be changed arbitrarily. For practice, the following processes are of greatest interest:

Processes at constant volume ( V = const) – isochoric.

At constant pressure ( P = const) – isobaric.

At constant temperature ( T = const) – isothermal.

Process dq =0 (proceeding without heat exchange of the working fluid with the environment) – an adiabatic process.

A polytropic process, which, under certain conditions, can be considered as generalizing in relation to all basic processes.

In the future, we will consider the 1st law of thermodynamics and the quantities included in it as related to 1 kg of mass.

Constant volume process

(isochoric process)

Such a process can be performed by a working fluid, for example, located in a vessel that does not change its volume, if heat is supplied to the working fluid from a heat source or heat is removed from the working fluid to the refrigerator.

In an isochoric process V = const And dV =0 . The equation of the isochoric process is obtained from the equation of state at V = const .

– Charles’s law (*)

That is, when V = const gas pressure is proportional to absolute temperature. When heat is supplied, the pressure increases, and when heat is removed, it decreases.

Let us depict the process at V = const V pV , pT And VT diagrams.

IN p V – isochore diagram 1-2 – vertical straight line parallel to the axis p . In process 1-2, heat is supplied to the gas, the pressure increases, and therefore, from equation (*), the temperature increases. In the reverse process 2-1, heat is removed from the gas, as a result of which the internal energy of the gas decreases and its temperature decreases, i.e. process 1-2 – heating, 2-1 – gas cooling.

IN p T-diagram isochores - straight lines coming out from the origin with an angular coefficient (proportionality coefficient)

Moreover, the higher the volume level, the lower the isochore lies.

In the VT diagram, isochores are straight lines parallel to the T axis.

External gas work in an isochoric process:

because the

Available specific work

The change in internal energy of a gas in an isochoric process, if

Specific heat supplied to the working fluid, at

Since when V = const gas does no work ( dl =0 ), then the equation of the first law of thermodynamics will take the form:

That is, in the process V = const all the heat supplied to the working fluid is spent on increasing the internal energy, that is, on increasing the temperature of the gas. When a gas is cooled, its internal energy decreases by the amount of heat removed.

The share of heat spent on changing internal energy

The share of heat spent on doing work

Constant pressure process

(isobaric process)

An isobaric process, for example, can take place in a cylinder under a piston that moves without friction so that the pressure in the cylinder remains constant.

In an isobaric process p = const , dp =0

The equation of the isobaric process is obtained when p = const from the equation of state:

– Gay-Lussac’s law (*)

In process at p = const The volume of a gas is proportional to the temperature, that is, when the gas expands, the temperature, and therefore the internal energy, increases, and when it contracts, it decreases.

Let's depict the process in pV , pT , VT – diagrams.

IN pV–diagram of processes at p = const are depicted as straight lines parallel to the axis V . The area of ​​the rectangle 12 gives the gas work on the appropriate scale l. In process 1-2, heat is supplied to the gas, since the specific volume increases, and therefore, according to equation (*), the temperature increases. In the reverse process 2-1, heat is removed from the gas, as a result the internal energy and temperature of the gas decrease, i.e. process 1-2 is heating, and 2-1 is cooling the gas.

IN VT– in a diagram, isobars are straight lines extending from the origin, with an angular coefficient of .

IN pT– in the diagram, isobars are straight lines parallel to the axis T .

Gas work in an isobaric process ( p = const )

Since then

That is, if the gas temperature increases, then the work is positive.

Available work

because the ,.

Change in internal energy of gas if

The amount of heat imparted to a gas when heated (or given off by it when cooled), if

That is, the heat supplied to the working fluid in an isobaric process goes to increase its enthalpy, i.e. in an isobaric process is a total differential.

The equation of the first law of thermodynamics is

The fraction of heat spent on changing internal energy in an isobaric process is

Where k – adiabatic index.

The fraction of heat consumed to perform work at p = const ,

In MKT, n – number of degrees of freedom.

For monatomic gas n =3 and then φ=0.6, ψ=0.4, that is, 40% of the heat imparted to the gas is used to perform external work, and 60% is used to change the internal energy of the body.

For diatomic gas n =5 and then φ=0.715, ψ=0.285, that is, ≈28.5% of the heat imparted to the gas is used to perform external work and 71.5% is used to change the internal energy.

For triatomic gas n =6 and then φ=0.75, ψ=0.25, that is, 25% of the heat is used to perform external work (steam engine).

Constant temperature process

(isothermal process)

Such a thermodynamic process can occur in the cylinder of a piston machine if, as heat is supplied to the working fluid, the piston of the machine moves, increasing the volume so much that the temperature of the working fluid remains constant.

In an isothermal process T = const , dT =0.

From the equation of state

−Boyle-Mariotte law.

Consequently, in a process at constant temperature, gas pressure is inversely proportional to volume, i.e. During isothermal expansion the pressure drops and during compression it increases.

Let us depict the isothermal process in pV , pT , VT − diagrams.

IN pV- diagram - an isothermal process is depicted by an equilateral hyperbola, and the higher the temperature, the higher the isotherm is located.

IN pT − diagram - isotherms - straight lines parallel to the axis p .

IN VT − diagram - straight lines parallel to the axis V .

dT =0, That

That is U = const , i = const – internal energy and enthalpy are unchanged.

The equation of the first law of thermodynamics takes the form ( T = const)

That is, all the heat imparted to the gas in an isothermal process is spent on expansion work. In the reverse process - during the compression process, heat is removed from the gas, equal to the external work of compression.

Specific work in an isothermal process

Specific available work

From the last two equations it follows that in an isothermal process for an ideal gas, the available work is equal to the work of the process.

The heat imparted to the gas in process 1-2 is

1st law of thermodynamics

It follows that when T = const l = l 0= q , those. the work, the work available and the amount of heat received by the system are equal.

Since in an isothermal process dT =0, q = l = some finite value, then from

we find that in an isothermal process C =∞. Therefore, it is impossible to determine the amount of heat imparted to the gas in an isothermal process using specific heat capacity.

The fraction of heat spent on changing internal energy at T = const

and the fraction of heat expended to perform work is

Process without heat exchange with the external environment

(adiabatic process)

In an adiabatic process, energy exchange between the working fluid and the environment occurs only in the form of work. The working fluid is assumed to be thermally insulated from the environment, i.e. There is no heat transfer between it and the environment, i.e.

q =0, and consequently dq =0

Then, the equation of the first law of thermodynamics will take the form

Thus, the change in internal energy and work in the adiabatic process are equivalent in magnitude and opposite in sign.

Consequently, the work of the adiabatic expansion process occurs due to a decrease in the internal energy of the gas and, consequently, the temperature of the gas will decrease. The work of adiabatic compression goes entirely to increasing the internal energy, i.e. to increase its temperature.

We obtain the adiabatic equation for an ideal gas. From the first law of thermodynamics

at dq =0 we get ( du = CV dT )

Heat capacity , where

Differentiating the equation of state pV = RT we get

Substituting RdT from (**) to (*)

or, dividing by pV ,

Integrating at k = const , we get

The last equation is called Poisson's equation and is the adiabatic equation for .

From the Poisson equation it follows that

that is, during adiabatic expansion the pressure drops, and during compression it increases.

Let us depict the isochoric process in pV , pT And VT – diagrams

Square V 1 12 V 2 under adiabatic 1-2 on pV – the diagram gives work l equal to the change in internal energy of the gas

Comparing the adiabatic equation with the Boyle-Mariotte law ( T = const ) we can conclude that, since k >1, then when expanding along an adiabat, the pressure drops more than along an isotherm, i.e. V pV – the adiabatic diagram is larger than the isotherm, i.e. an adiabatic is a non-equilateral hyperbola that does not intersect the coordinate axes.

We obtain the adiabatic equation in pT And VT − diagrams. In an adiabatic process, all three parameters change ( p , V , T ).

We get the dependence between T And V . Equations of state for points 1 and 2

whence, dividing the second equation by the first

Substituting the pressure ratio from the Poisson adiabatic equation

or TVk -1= const – adiabatic equation in VT - diagram.

Substituting into (*) (3) the volume ratio from the adiabatic equation (Poisson)

or − adiabatic equation in pT - diagram. These equations are obtained under the assumption that k = const .

Work in an adiabatic process at CV = const

Considering the relationship between temperature T And V

Considering the relationship between T And p

Change in internal energy u=- l.

Available work, taking into account that

,

Those. available work in k times more work of the adiabatic process l .

φ And ψ we don't find it.

Polytropic process

A polytropic process is any arbitrary process that occurs at a constant heat capacity, i.e.

Then, the equation of the 1st law of thermodynamics will take the form

(*) (1)

Thus, if C = const And CV = const , then the quantitative distribution of heat between internal energy and work in a polytropic process remains constant (for example, 1:2).

The share of heat spent on changing the internal energy of the working fluid

The fraction of heat spent on external work is

We obtain the equation of the polytropic process. To do this, we use the equation of the 1st law of thermodynamics (*)

From here, from (*) and (**)

Dividing the second equation (4) by the first (3)

Let us introduce a quantity called the polytropic index. Then,

Integrating this expression, we get

This equation is the polytropic equation in pV − diagram. Potlitrope indicator n is constant for a particular process, and can vary from -∞ to +∞.

Using the equation of state, we can obtain the polytropic equation in VT And pT– diagrams.

From - polytropic equation in VT - diagram.

From

− polytropic equation in pT - diagram.

The polytropic process is a general one, and the main processes (isochoric, isothermal, adiabatic) are special cases of the polytropic process, each of which has its own meaning n . Thus, for each isochoric process n =±∞, isobaric n =0, isothermal n =1, adiabatic n = k .

Since the polytropic and adiabatic equations are identical in form and differ only in magnitude n(polytropic index instead of k − adiabatic index), then we can write

work of a polytropic process

available work of a polytropic process

Heat capacity of gas from , from where

Moreover, depending on n The heat capacity of the process can be positive, negative, equal to zero and varies from -∞ to +∞.

In C processes<0 всегда l> q those. To perform the expansion work, in addition to the supplied heat, part of the internal energy of the gas is consumed.

Change in internal energy of a polytropic process

Heat imparted to gas in a polytropic process

Changing the enthalpy of the working fluid

Second law of thermodynamics

The first law of thermodynamics characterizes the processes of energy transformation from the quantitative side, i.e. he claims that heat can be converted into work, and work into heat, without establishing the conditions under which these transformations are possible. Thus, it only establishes the equivalence of different forms of energy.

The second law of thermodynamics establishes the direction and conditions for the process

As the first law of thermodynamics, the second law was derived from experimental data.

Experience shows that the transformation of heat into useful work can only occur when heat passes from a heated body to a cold one, i.e. when there is a temperature difference between the heat transferr and the heat receiver. It is possible to change the natural direction of heat transfer to the opposite direction only at the expense of work (for example, in refrigeration machines).

According to the 2nd law of thermodynamics

A process in which heat would spontaneously transfer from cold bodies to heated bodies is impossible.

Not all of the heat received from the heat transfer can go into work, but only part of it. Part of the heat must go to the heat sink.

Thus, the creation of a device that, without compensation, would completely convert the heat of any source into work, and called perpetual motion machine of the second kind, impossible!

Reversible and irreversible processes

For any thermodynamic system, one can imagine two states, between which two processes will occur (Fig. 1): one from the first state to the second and the other, vice versa, from the second state to the first.

The first process is called direct process, and the second - reverse

If a direct process is followed by a reverse one and the thermodynamic system returns to its original state, then such processes are generally considered reversible.

In reversible processes, the system in the reverse process goes through the same equilibrium states as in the forward process. In this case, no residual phenomena occur either in the environment or in the system itself (no changes in parameters, work performed, etc.). As a result of a direct process AB , and then the reverse B.A. the final state of the system will be identical to the initial state.

The figure shows the setup of a mechanically reversible process. The installation consists of cylinder 1, piston 2 with table 3 and sand on it. Under the piston, the cylinder contains gas, which is under pressure from the sand on the table.

To create a reversible process, one grain of sand must be removed infinitely slowly. Then the process will be isothermal, and the pressure will be equal to the external pressure and the system will be constantly in an equilibrium state. If the process is carried out in the opposite direction, i.e. Infinitely slowly throw grains of sand onto table 3, then the system will successively pass through the same equilibrium states and return to the original state (if there is no friction).

When expanding, the working fluid in a reversible process produces maximum work.

An isobaric process (also called an isobaric process) is one of the thermodynamic processes that occurs at a constant pressure. The gas mass of the system also remains constant. A visual representation of a graph demonstrating an isobaric process is given by a thermodynamic diagram in the corresponding coordinate system.

Examples

The simplest example of an isobaric process is the heating of a certain volume of water in an open vessel. Another example is the expansion of an ideal gas in a cylindrical volume where the piston has a free stroke. In each of these cases the pressure will be constant. It is equal to ordinary atmospheric pressure, which is quite obvious.

Reversibility

An isobaric process can be considered reversible if the pressure in the system coincides with the external pressure and is equal at all times of the process (that is, it is constant in value), and the temperature changes very slowly. Thus, thermodynamic equilibrium in the system is maintained at every moment of time. It is the combination of the above factors that gives us the opportunity to consider the isobaric process reversible.

To carry out an isobaric process in a system, heat must be either supplied or removed. In this case, heat must be spent on the work of expansion of an ideal gas and on changing its internal energy. The formula demonstrating the dependence of quantities on each other during an isobaric process is called Gay-Lussac's law. It shows that volume is proportional to temperature. Let's derive this formula based on superficial knowledge.

Derivation of Gay-Lussac's law (primary understanding)

A person with at least a little understanding of molecular physics knows that many problems involve certain parameters. Their names are gas pressure, gas volume and gas temperature. In certain cases, molecular and molar mass, amount of substance, universal gas constant and other indicators are used. And there is a certain connection here. Let's talk about the universal gas constant in more detail. In case anyone doesn't know how they got it.

Obtaining the universal gas constant

This constant (a constant number with a certain dimension) is also called the Mendeleev constant. It is also present in the Mendeleev-Clapeyron equation for an ideal gas. How did our famous physicist obtain this constant?

As we know, the ideal gas equation has the following form: PV/T (which is pronounced as: “the product of pressure and volume divided by temperature”). The so-called Avogadro's law applies to the universal gas constant. It says that if we take any gas, then the same number of moles at the same temperature and the same pressure will occupy the same volume.

In fact, this is a verbal formulation of the equation of state of an ideal gas, which was written down as a formula a little earlier. If we take normal conditions (and this is when the gas temperature is 273.15 Kelvin, the pressure is 1 atmosphere, respectively, 101325 Pascals, and the volume of a mole of gas is 22.4 liters) and substitute them into the equation, multiply everything and divide, we get , that the totality of such actions gives us a numerical indicator equal to 8.31. The dimension is given in Joules divided by the product of a mole times Kelvin (J/mol*K).

Mendeleev-Clapeyron equation

Let's take the equation of state of an ideal gas and rewrite it in a new form. The initial equation, recall, has the form PV/T=R. Now let’s multiply both parts by the temperature indicator. We get the formula PV(m)=RT. That is, the product of pressure and volume is equal to the product of the universal gas constant and temperature.

Now let's multiply both sides of the equation by one or another number of moles. Let us denote their number by a letter, say, X. Thus, we obtain the following formula: PV(m)X=XRT. But we know that the product V with the subscript “m” gives us the result simply as volume V, and the number of moles X is revealed in the form of dividing the partial mass by the molar mass, that is, it has the form m/M.

Thus, the final formula will look like this: PV=MRT/m. This is the same Mendeleev-Clapeyron equation, which both physicists came to almost simultaneously. We can multiply the right-hand side of the equation (and at the same time divide) by Avogadro's number. Then we get: PV = XN(a)RT/N(a). But the product of the number of moles by Avogadro’s number, that is, XN(a), gives us nothing more than the total number of gas molecules, denoted by the letter N.

At the same time, the quotient of the universal gas constant and Avogadro’s number – R/N(a) will give the Boltzmann constant (denoted by k). As a result, we will get another formula, but in a slightly different form. Here it is: PV=NkT. You can expand this formula and get the following result: NkT/V=P.

Gas work in an isobaric process

As we found out earlier, an isobaric process is a thermodynamic process in which the pressure remains constant. And to find out how work will be determined during an isobaric process, we will have to turn to the first law of thermodynamics. The general formula is as follows: dQ = dU + dA, where dQ is the amount of heat, dU is the change in internal energy, and dA is the work done during the thermodynamic process.

Now let's look specifically at the isobaric process. Let's take into account the fact that the pressure remains constant. Now let's try to rewrite the first law of thermodynamics for an isobaric process: dQ = dU + pdV. To get a clear idea of ​​the process and work, you need to depict it in a coordinate system. Let's denote the abscissa axis p, the ordinate axis V. Let the volume increase. At two distinct points with a corresponding value of p (fixed, of course), we note the states representing V1 (initial volume) and V2 (final volume). In this case, the graph will be a straight line parallel to the x-axis.

Finding work in this case is easier than ever. This will simply be the area of ​​the figure, limited on both sides by projections onto the abscissa axis, and on the third side by a straight line connecting the points lying, respectively, at the beginning and end of the isobar line. Let's try to calculate the value of the work using the integral.

It will be calculated as follows: A = p (integral from V1 to V2) dV. Let's expand the integral. We find that the work will be equal to the product of pressure and the difference in volumes. That is, the formula will look like this: A = p (V2 – V1). If we expand on some quantities, we get another formula. It looks like this: A = xR (T2 – T2), where x is the amount of substance.

Universal gas constant and its meaning

We can say that the last expression will determine the physical meaning of R - the universal gas constant. To make it clearer, let's look at specific numbers. Let's take one mole of any substance to check. At the same time, let the temperature difference be 1 Kelvin. In this case, it is easy to notice that the work of the gas will be equal to the universal gas constant (or vice versa).

Conclusion

This fact can be presented in a slightly different light by rephrasing the wording. For example, the universal gas constant will be numerically equal to the work done in the isobaric expansion of one mole of an ideal gas if it is heated by one Kelvin. Calculating work for other isoprocesses will be somewhat more difficult, but the main thing is to apply logic. Then everything will quickly fall into place, and deriving the formula will be easier than you think.

Isochoric process

An isochoric process occurs at constant volume. The dependence of pressure on temperature is described by the equation:

- Charles's law, (2)

which reads: for a given mass of gas at constant volume, the gas pressure increases linearly with increasing temperature .

Isobaric process

Isobaric process. This is a process that occurs under constant pressure, R = const.

The dependence of volume on temperature is described by the law:

- Gay-Lusac law, (3)

which reads: for a given mass of gas at constant pressure, the volume of gas increases linearly with increasing temperature .

Adiabatic process

An adiabatic process is a process that occurs without heat exchange with the environment ( dQ= 0). It is described by the Poisson equation:

, (4)

where  is the constant of the adiabatic process. The adiabatic process constant is:

. (5)

During an adiabatic process, all gas parameters change: pressure, volume and temperature.

2. Heat capacity of gas

Quantity of heat dQ, communicated to the body when heated is equal to

,

Where With - specific heat capacity of a substance, equal to the amount of heat imparted to a unit mass of a substance to heat it by one degree.

In addition to specific heat capacity, the concept of molar heat capacity is introduced. Molar heat capacity WITH- equal to the amount of heat imparted to one mole of a substance to heat it by one degree.

Molar and specific heat capacities are related to each other by the relationship:

C =  With, (6)

Where WITH- molar heat capacity,  - molar mass.

Gas can be heated at constant pressure and at constant volume, therefore two heat capacities are introduced for gas: isobaric and isochoric. The molar isobaric and molar isochoric heat capacities of a gas are related to the corresponding relationships:

;

.

This shows that the ratio of the molar heat capacities of the gas is equal to the ratio of the specific heat capacities

.

The amount of heat imparted to 1 mole of gas during an isochoric process is equal to:

and in an isobaric process

3First law of thermodynamics

Quantity of heat dQ, communicated to the thermodynamic system, is spent on increasing its internal energy dU and to work dA systems against external forces.

dQ = dU + dA . (9)

Internal energy U- the total energy of all molecules in a gas for an ideal gas - the kinetic energy of rotational and translational motion. For one mole of gas is determined by the expression

. (10)

The work done by the gas is equal to

dA= pdV . (11)

Where dV- change in its volume.

Application of the first law of thermodynamics Isothermal process

During this process the temperature remains constant ( T=const) IN in this case dT=0 and internal energy does not change dU=0 dQ= dA, those. all the heat supplied is consumed by the gas to perform work against external forces.

Isochoric process

For an isochoric process, V=const, dV=0 and dA=0. Those. during this process, no work is done, because the volume does not change. Then 1 start will be recorded :

dQ= dU.

Those. the amount of heat is spent to change internal energy. But by definition

(for 1 mole). Hence,

.

From this formula it is clear that the change in the internal energy of a gas is determined only by the change in its temperature. The heat capacity at constant volume (isochoric heat capacity) is equal to:

Isobaric process

A thermodynamic process in which the pressure does not change is called isobaric.

In an isobaric process With n = c p .. For this process the polytropic index n= 0.

Isochoric process

A thermodynamic process in which the specific volume does not change is called isochoric.

In an isochoric process With n = c v . This process occurs when n= .

Isothermal process

A thermodynamic process in which the temperature does not change is called isothermal.

In an isothermal process With n = c T = . In an isothermal process, the polytropic index n= 1.

Adiabatic process

A thermodynamic process that occurs without heat exchange with the environment is called adiabatic.

In an adiabatic process With n = c q = , then the polytropic index n= k. Here via To denotes the ratio of heat capacity at constant pressure to heat capacity at constant volume, that is k =. This ratio in thermodynamics is called the adiabatic exponent.

The adiabatic equation has the form:

pvTo= const.

Wet air

A mixture of dry air and water vapor is called moist air.

Absolute humidity

Absolute humidity refers to the mass of water vapor contained in one cubic meter of moist air.

From the definition it follows that absolute humidity is the vapor density in moist air, i.e. = The unit of measurement for absolute humidity is kilogram per cubic meter (kg/m3).

Relative humidity

The ratio of the actual value of absolute humidity to its maximum possible value at the same temperature is called

relative humidity.

Relative humidity is indicated by: or =.

Moisture content

The mass of water vapor per 1 kg of dry air is called

Indicate moisture content through d, measured in g/kg. From the definition

tion follows: d=

Dryness degree

The mass fraction of dry steam in wet steam is called the degree of dryness.

Indicate the degree of dryness through x and calculate as x= m c / m,

Where m c– mass of dry steam; m– mass of wet steam.

Throttling

Throttling is the process of reducing pressure in a gas

flow while overcoming local resistance (examples of local resistances: faucet, valve, gate valve, capillary tube).

Throttle effect

The ratio of an infinitesimal change in temperature to an infinitesimal change in pressure during throttling is called throttling

effect.

This relation is denoted by , then =

The experiments of Joule and Thomson showed that for a real gas it can change sign: be less than zero, equal to zero or greater than zero.

Inversion temperature

The change in the sign of the throttling effect is called inversion, and the temperature at which= 0 , is called the inversion temperature. It is designated T inv .

Carnot heat engine

An exotic heat engine that has the highest possible value of thermal efficiency due to the fact that heat is supplied and removed during an isothermal process, and compression and expansion of the working fluid occurs in an adiabatic process.

Heat engine

A heat engine is a machine that uses heat to produce mechanical work.

Internal combustion engine

A heat engine in which the chemical energy of the fuel is converted into heat inside an expansion machine is called an internal combustion engine (ICE).

Isochoric internal combustion engine

Isochoric is an internal combustion engine in which fuel combustion occurs at a constant volume, and a piston machine is used to perform work.

Isobaric internal combustion engine

An isobaric internal combustion engine is one in which fuel combustion occurs at constant pressure, and a piston machine is used to perform work.

Gas turbine engine

A gas turbine engine refers to an internal combustion engine in which fuel combustion is carried out, in most cases, in an isobaric process, and a gas turbine is used as an expansion machine.

Turbojet engine

A turbojet engine refers to an internal combustion engine in which combustion

fuel expansion is carried out in an isobaric process, and a gas turbine and a jet nozzle are used as an expansion machine.

Compression ratio

The compression ratio in piston internal combustion engines is understood as the ratio of the total volume of the cylinder to the volume of the combustion chamber.

Indicates the degree of compression .

Pressure increase rate

The ratio of the final pressure to the initial one during the compression process separately in a stage (or in the compressor as a whole) is called the pressure increase ratio, denoted st ,that is st = p con. /R beginning .

Compressor volume flow

By volumetric flow we mean the number of cubic meters of gas leaving the compressor per unit time and reduced to the pressure and temperature at the inlet to the compressor. Denotes the compressor flow, and is expressed in m3 /With.

Refrigeration machine

A machine that performs artificial cooling using sub-

energy supplied is called a refrigeration machine.

Refrigerant

Refrigerant is the working fluid of a refrigeration machine.

Cooling effect

The refrigeration effect is the amount of heat (q 2 ), taken away from

cooled object with one kilogram of refrigerant.

Cooling capacity

The amount of heat removed from the cooled object per unit time is called cooling capacity. Indicates cooling capacity N x , expressed in watts (W).

For determining N x use the expression:

N x = q 2 ,

Where q 2 – cooling effect ;

– second mass flow of refrigerant.

Coefficient of performance

The coefficient of performance establishes the energy efficiency of refrigeration units and is numerically equal to the ratio of the amount of heat removed from the cooled body to the amount of energy spent on cooling.

Designate the cooling coefficient, from the definition =.

2. Heat transfer theory

Heat exchange

Heat exchange – this is a spontaneous irreversible process of heat transfer in space with a non-uniform temperature field.

Temperature field

The temperature field is the set of temperature values ​​at all points of the space under consideration at some fixed point in time.

Temperature gradient

The temperature gradient is a vector directed normal to the isothermal surface in the direction of increasing temperature and numerically equal to the frequency derivative of the temperature normal to the surface.

details:

Heat flow

The amount of heat passing per unit time through an isothermal

ical surface is calledheat flow.

Heat flow is denoted by the unit watt (W).

Heat flux density

The heat flux per unit surface area is called heat flux density.

Designate heat flux density , expressed in watts per square meter (W/m2). From the definition:

Thermal conductivity

Heat exchange through the thermal movement of microstructural particles of matter (molecules, atoms, electrons, ions) in a particular environment is called thermal conductivity.

Law of Heat Conduction

The heat flow passing through an isothermal surface element

ness dF, proportional gradT:

= qrad T dF .

Since the directions of heat flow and temperature gradient are opposite, a minus is placed after the equal sign in the expression. The value of the proportionality coefficient is called the thermal conductivity coefficient.

Coefficient of thermal conductivity

Thermal conductivity coefficient is a value characterizing the heat-conducting properties of a material. DesignationcThe unit of measurement is watt per meter kelvin (W/(m  TO)).

The numerical value of the thermal conductivity coefficient determines the amount

the quality of heat passing through a unit of isothermal surface per unit of time, provided that grad T = 1.

Differential heat equation

The differential equation of thermal conductivity for three-dimensional

of a nonstationary temperature field is called an equation of the form:

where T is temperature;

–time;

a – thermal diffusivity coefficient;

x, y, z– coordinates.

This equation in general establishes a connection between temporal and spatial changes in temperature at any point of the body.

Thermal diffusivity coefficient

Thermal diffusivity coefficient is a value characterizing the speed of propagation of isothermal surfaces in non-stationary thermal processes.

Designates the coefficient of thermal diffusivity a and is expressed in square meters per second (m2/s).

To calculate the value of the thermal diffusivity coefficient, the expression is used a =

Convection

Underconvection (from lat. conviction– movement, delivery) understand the heat exchange carried out by macroscopic elements of a liquid or gaseous medium when they move.

Convective heat transfer

The transfer of heat in a coolant by convection and thermal conductivity is called convective heat transfer.

Heat dissipation

Convective heat exchange between the coolant and the surface of the body flowing around it is calledheat transfer.

Basic law of heat transfer

The heat flux density is proportional to the temperature difference:

where the proportionality coefficient, called the coefficient

heat transfer;

temperature difference equal to the temperature difference between the coolant and the surface.

Heat transfer coefficient

The heat transfer coefficient characterizes the intensity of convective heat exchange at the coolant-wall interface.

Denotes the heat transfer coefficient and is expressed in watts per square meter-kelvin. ( W/(m2 K)). Numerically, the heat transfer coefficient is equal to the heat flux per unit surface per unit time at a temperature pressure equal to unity.

Differential heat transfer equation

The differential heat transfer equation is the expression

type:

Thermal similarity theory

The theory of thermal similarity is a system of concepts and rules that provide the ability to transfer the results of experiments to determine heat transfer coefficients from one object to another.

The theory of thermal similarity allows, without integrating the differential equations describing heat transfer, to obtain similarity criteria from them and, using experimental data, to establish criterion dependencies for determining heat transfer processes in all experimentally similar processes.

Thermal similarity criteria

The thermal similarity criteria are understood as dimensionless complexes made up of certain combinations of quantities that describe a particular heat transfer process.

In most tasks to determine the heat transfer coefficient, the following thermal similarity criteria are used:

Nusselt criterion, Nu= ,

where α is the heat transfer coefficient,

l

λ – thermal conductivity coefficient.

The Nusselt criterion characterizes heat exchange at the wall-coolant interface and establishes a numerical relationship between the intensity of heat transfer and thermal conductivity (λ/l) coolant.

Reynolds criterion, Re= ,

Where c– coolant speed;

l– characteristic geometric size;