To describe the state of a gas, it is sufficient to set three macroscopic parameters - the volume V, pressure p and temperature T. Changing one of these parameters causes a change in the others. If the volume, pressure and temperature change simultaneously, then it is difficult to establish any regularities experimentally. It is easier to first consider a gas of constant mass ( m= const), fix the value of one of the macro parameters ( V, p or T) and consider changing the other two.

Processes in which one of the parameters p, V or Τ remains constant for a given mass of gas is called isoprocesses.

• isos means "equal" in Greek.

The laws describing isoprocesses in an ideal gas were discovered experimentally.

Isothermal process

Isothermal process is an isoprocess that occurs at constant temperature: Τ = const.

• therme - warmth.

The law was experimentally discovered independently by the English chemist and physicist Robert Boyle (1662) and the French physicist Edm Mariotte (1676).

Law of the isothermal process(Boyle-Mariotte): for a given mass of gas at a constant temperature, the product of pressure and volume is a constant:

\(~p \cdot V = \operatorname(const)\) or for two states \(~p_1 \cdot V_1 = p_2 \cdot V_2 .\)

To carry out an isothermal process, it is necessary to bring a vessel filled with gas into contact with a thermostat.

• A thermostat is a device for maintaining a constant temperature. See wikipedia for more details.

• An isothermal process can be approximately considered the process slow compression or expansion of a gas in a vessel with a piston. The thermostat in this case is the environment.

isobaric process

isobaric process is an isoprocess that occurs at constant pressure: p= const.

• baros - heaviness, weight.

• The work of J. Charles was published after the discovery of J. Gay-Lussac. But the isobaric process in Russian textbooks is called Gay-Lussac's law, in Belarusian - Charles law.

Law of the isobaric process: for a given mass of gas at constant pressure, the ratio of volume to absolute temperature is a constant:

\(~\dfrac(V)(T) = \operatorname(const),\) or \(~\dfrac(V_1)(T_1) = \dfrac(V_2)(T_2) .\)

This law can be written in terms of temperature t, measured in Celsius\[~V = V_0 \cdot (1 + \alpha \cdot t),\] where V 0 - volume of gas at 0 °С, α = 1/273 K -1 - temperature coefficient of volumetric expansion.

• Experience shows that at low densities the temperature coefficient of volume expansion does not depend on the type of gas, i.e. the same for all gases).

An isobaric process can be obtained using a cylinder with a weightless piston.

Isochoric process

Isochoric process is an isoprocess that occurs at constant volume: V= const.

• chora - occupied space, volume.

The law was experimentally investigated independently by the French physicists Jacques Charles (1787) and Joseph Gay-Lussac (1802).

• The isochoric process in Russian textbooks is called Charles's law, in Belarusian - Gay-Lussac's law.

Law of the isochoric process: for a given mass of gas at a constant volume, the ratio of pressure to absolute temperature is a constant:

\(~\dfrac(p)(T) = \operatorname(const)\), or \(~\dfrac(p_1)(T_1) = \dfrac(p_2)(T_2) .\)

If the temperature is measured on the Celsius scale, then the Gay-Lussac law will be written as \[~p = p_0 \cdot (1 + \alpha \cdot t),\] where p 0 - gas pressure at 0 °С, α - temperature coefficient of pressure, which turned out to be the same for all gases: α \u003d 1/273 K -1.

An isochoric process can be obtained in a cylinder that does not change its volume with a given change in temperature.

Careful experimental verification by modern methods has shown that the equation of state of an ideal gas and the laws of Boyle-Mariotte, Gay-Lussac and Charles following from it accurately describe the behavior of real gases at low pressures and not too low temperatures.

A bit of math

Function Graph y(x), where a, b and With- constant values:

• y = a⋅x- a straight line passing through the origin of coordinates (Fig. 1, a);
• y=c- straight line perpendicular to the axis y and passing through a point with coordinate y=c(Fig. 1b);
• \(~y = \dfrac(b)(x) \) is a hyperbola (Fig. 1c).

Rice. one

Isoprocess Plots

Since we are considering three macro parameters p, T and V, then three coordinate systems are possible: ( p, V), (V, Τ ), (p, T).

Dependence graphs between the parameters of a given mass at a constant temperature are called isotherms.

Consider two isothermal processes with temperatures T 1 and T 2 (T 2 > T one). In coordinates where there is an axis of temperature (( V, Τ) and ( p, T T, and passing through the points T 1 and T 2 (Fig. 2, a, b).

p, V). For an isothermal process \(~p \cdot V = \operatorname(const)\). Let's denote this constant by the letter z one . Then

\(~p \cdot V = z_1\) or \(~p = \dfrac(z_1)(V)\).

The graph of this function is a hyperbola (Fig. 2, c).

Rice. 2

Graphs of dependence between gas parameters at constant gas mass and pressure are called isobars.

Consider two isobaric processes with pressures p 1 and p 2 (p 2 > p one). In coordinates where there is an axis of pressure (( p, Τ) and ( p, V)), the graphs will be straight lines perpendicular to the axis p, and passing through the points p 1 and p 2 (Fig. 3, a, b).

Define the type of graph in the axes ( V, T). For an isobaric process \(~\dfrac(V)(T) = \operatorname(const)\). Let's denote this constant by the letter z 2. Then

\(~\dfrac(V)(T) = z_2\) or \(~V = z_2 \cdot T\).

The graph of this function is a straight line passing through the origin (Fig. 3, c).

Rice. 3

Graphs of the relationship between gas parameters at constant gas mass and constant volume are called isochores.

Consider two isochoric processes with volumes V 1 and V 2 (V 2 > V one). In coordinates where there is a volume axis (( V, Τ) and ( p, V)), the graphs will be straight lines perpendicular to the axis V, and passing through the points V 1 and V 2 (Fig. 4, a, b).

Define the type of graph in the axes ( p, T). For an isochoric process \(~\dfrac(p)(T) = \operatorname(const)\). Let's denote this constant by the letter z 3 . Then

\(~\dfrac(p)(T) = z_3\) or \(~p = z_3 \cdot T\).

The graph of this function is a straight line passing through the origin (Fig. 4, c).

Rice. four

• All graphs of isoprocesses are straight lines (exception, hyperbola in the axes p(V)). These lines pass either through zero or perpendicular to one of the axes.
• Since the gas pressure, its volume and temperature cannot be equal to zero, when approaching zero values, the lines of the graph are shown as dashed lines.

Ideal gas equation of state

In isoprocesses, two parameters were changed at a constant value of the third. But there are cases when three parameters change at once. For example, when air heated near the surface of the Earth rises, it expands, its pressure decreases and its temperature decreases.

The equation relating temperature T, pressure p and volume V for a given mass of an ideal gas, is called gas equation.

This equation was obtained experimentally, but it can be derived from the basic MKT equation:

\(~p = n \cdot k \cdot T.\)

By definition, the gas concentration

\(~n = \dfrac NV,\)

where N is the number of molecules. Then

\(~p = \dfrac NV \cdot k \cdot T \Rightarrow \dfrac(p \cdot V)(T) = k \cdot N . \qquad (1)\)

With a constant mass of gas, the number of molecules in it is constant and the product \(~k \cdot N = \operatorname(const).\) Therefore,

\(~\dfrac(p \cdot V)(T) = \operatorname(const)\) or for two states \(~\dfrac(p_1 \cdot V_1)(T_1) = \dfrac(p_2 \cdot V_2)( T_2).\qquad(2)\)

Relation (2) is the equation of state for an ideal gas. He is called Clapeyron's equation. It is used in cases where the mass of the gas and its chemical composition do not change and it is necessary to compare two states of the gas.

Clapeyron-Mendeleev equation

In equation (1), the number of molecules N can be expressed in terms of the Avogadro constant \(~N = \dfrac mM \cdot N_A\), where m- mass of gas, Μ is its molar mass. Then we get \(~\dfrac(p \cdot V)(T) = \dfrac mM \cdot k \cdot N_A \Rightarrow\)

\(~p \cdot V = \dfrac mM \cdot R \cdot T . \qquad (3)\)

Here \(~R = k \cdot N_A\) is the universal gas constant equal to

R\u003d 1.38 10 -23 J / K 6.02 10 23 mol -1 \u003d 8.31 J / (mol K).

Equation (3) is also the equation of state for an ideal gas. In this form, it was first recorded by the Russian scientist D.I. Mendeleev, so it is called Clapeyron-Mendeleev equation. It is valid for any mass of gas and connects the parameters of one state of the gas.

Laws of Avogadro and Dalton

Two consequences follow from the equation of state:

1. From formula (1) we obtain \(~N = \dfrac(p \cdot V)(k \cdot T)\), which shows that if different gases occupy equal volumes at the same temperatures and pressures, then the number N their molecules are also the same, i.e. follows established empirically Avogadro's law: at equal pressures and temperatures, equal volumes of any gas contain the same number of molecules.
2. Let there be a mixture of gases in the vessel, each of which, in the absence of others, exerts a corresponding pressure p 1 , p 2 , ... (partial pressures gases). Let us write the equation of state for each gas:
   \(~p_1 \cdot V = N_1 \cdot k\cdot T, p_2 \cdot V = N_2 \cdot k \cdot T, \ldots\)
   and add them up:
   \(~p_1+ p_2 + \ldots = \dfrac((N_1+ N_2 + \ldots) \cdot k \cdot T)(V) = \dfrac(N \cdot k \cdot T)(V),\)
   where N 1 + N 2 + ... = N is the number of molecules in the mixture of gases. But \(~\dfrac(N \cdot k \cdot T)(V) = p\) .
   Consequently, p = p 1 + p 2 + ..., i.e. the pressure of a mixture of gases is equal to the sum of the partial pressures of each of the gases- this is dalton's law, discovered by him in 1801 experimentally.

Literature

Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsia i vykhavanne, 2004. - C. 143-146.

If in some process the mass and temperature of the gas do not change, then such a process is called isothermal.

Atm= const T = const P 1 V 1 = P 2 V 2 orPV = const.

Received PV= const the equation is called isothermal process equation.

This equation was obtained by the English physicist Robert Boyle in 1662 and the French physicist Edmond Mariotte in 1676.

P equation 1 / R 2 = V 2 / V 1 is called the Boyle-Mariotte equation.

The state of the gas is characterized by three macro parameters:

P - pressure,

V - volume,

T - temperature.

When graphical representation of the process, you can specify only two parameters that change, so the same process can be represented in three coordinate planes: ( R -V), (V – T), (P – T).

The graph of an isothermal process is called an isotherm. The isotherm, depicted in a rectangular coordinate system (P - V), along the ordinate axis of which the gas pressure is measured, and along the abscissa axis - its volume, is a hyperbola (Fig. 3).

The isotherm, depicted in a rectangular coordinate system (V - T), is a straight line parallel to the y-axis (Fig. 4).

The isotherm, depicted in a rectangular coordinate system (P - T), is a straight line parallel to the y-axis (Fig. 5).

The graphs of the isothermal process are depicted as follows:

ISOCHORIC PROCESS

Isochoric process is a process that takes place at constant volume (V = const) and under the condition m = const and M = const.

Under these conditions, from the equation of state of an ideal gas for two values ​​of temperature T 0 and T follows:

P 0 V = mRT 0

RV= MRTor R / R 0 = T / T 0

For a gas of a given mass, the ratio of pressure to temperature is constant if the volume of the gas does not change. When P 1 / P 2 = T 1 / T 2 (this equation is called Charles' law), it is applicable for an isochoric process : V = const.

This is the equation for an isochoric process.

If V is the volume of gas at absolute temperature T, V 0 is the volume of gas at a temperature of 0 0 C; coefficient a, equal to 1/273 K -1, called the temperature coefficient of volumetric expansion of gases, then the equation for the isochoric process can be written as P = P 0 × a ×T.

The curve of an isochoric process is called an isochore.

Isochore depicted P – V), along the ordinate axis of which the gas pressure is measured, and along the abscissa axis - its volume, is a straight line parallel to the ordinate axis (Fig. 6).

Isochore depicted in a rectangular coordinate system (V – T), is a straight line parallel to the abscissa axis (Fig. 7).

Isochore depicted in a rectangular coordinate system (P – T), along the ordinate axis of which the gas pressure is measured, and along the abscissa axis - its absolute temperature, is a straight line passing through the origin of coordinates (Fig. 8).

Experimentally, the dependence of gas pressure on temperature was studied by the French physicist Jacques Charles in 1787

The isochoric process can be carried out, for example, by heating air at a constant volume.

The graphs of the isochoric process are depicted as follows:

DEFINITION

Processes in which one of the parameters of the state of the gas remains constant are called isoprocesses.

DEFINITION

Gas laws are the laws describing isoprocesses in an ideal gas.

The gas laws were discovered experimentally, but they can all be derived from the Mendeleev-Clapeyron equation.

Let's consider each of them.

Boyle-Mariotte's law (isothermal process)

Isothermal process A change in the state of a gas so that its temperature remains constant is called.

For a constant mass of gas at a constant temperature, the product of gas pressure and volume is a constant value:

The same law can be rewritten in another form (for two states of an ideal gas):

This law follows from the Mendeleev-Clapeyron equation:

Obviously, at a constant mass of gas and at a constant temperature, the right side of the equation remains constant.

Graphs of dependence of gas parameters at constant temperature are called isotherms.

Denoting the constant by the letter , we write down the functional dependence of pressure on volume in an isothermal process:

It can be seen that the pressure of a gas is inversely proportional to its volume. Inversely proportional graph, and, consequently, the graph of the isotherm in coordinates is a hyperbola(Fig. 1, a). Figure 1 b) and c) shows isotherms in coordinates and respectively.

Fig.1. Graphs of isothermal processes in various coordinates

Gay-Lussac's law (isobaric process)

isobaric process A change in the state of a gas so that its pressure remains constant is called.

For a constant mass of gas at constant pressure, the ratio of gas volume to temperature is a constant value:

This law also follows from the Mendeleev-Clapeyron equation:

isobars.

Consider two isobaric processes with pressures and title="(!LANG:Rendered by QuickLaTeX.com" height="18" width="95" style="vertical-align: -4px;">. В координатах и изобары будут иметь вид прямых линий, перпендикулярных оси (рис.2 а,б).!}

Let's determine the type of graph in coordinates. Denoting the constant with the letter, we write down the functional dependence of the volume on temperature during the isobaric process:

It can be seen that at constant pressure, the volume of a gas is directly proportional to its temperature. Direct proportionality graph, and, consequently, the graph of the isobar in coordinates is a straight line passing through the origin(Fig. 2, c). In reality, at sufficiently low temperatures, all gases turn into liquids, to which gas laws are no longer applicable. Therefore, near the origin, the isobars in Fig. 2, c) are shown by dotted lines.

Fig.2. Graphs of isobaric processes in various coordinates

Charles' law (isochoric process)

Isochoric process A change in the state of a gas so that its volume remains constant is called.

For a constant mass of gas at a constant volume, the ratio of gas pressure to its temperature is a constant value:

For two states of a gas, this law can be written as:

This law can also be obtained from the Mendeleev-Clapeyron equation:

Graphs of dependence of gas parameters at constant pressure are called isochores.

Consider two isochoric processes with volumes and title="(!LANG:Rendered by QuickLaTeX.com" height="18" width="98" style="vertical-align: -4px;">. В координатах и графиками изохор будут прямые, перпендикулярные оси (рис.3 а, б).!}

To determine the type of graph of the isochoric process in coordinates, we denote the constant in Charles's law by the letter , we get:

Thus, the functional dependence of pressure on temperature at constant volume is a direct proportionality, the graph of such a dependence is a straight line passing through the origin (Fig. 3, c).

Fig.3. Graphs of isochoric processes in various coordinates

Examples of problem solving

EXAMPLE 1

Exercise	To what temperature must a certain mass of gas with an initial temperature be cooled isobarically so that the volume of the gas decreases by one quarter?
Solution	The isobaric process is described by the Gay-Lussac law: According to the condition of the problem, the volume of gas due to isobaric cooling decreases by one quarter, therefore: whence the final temperature of the gas: Let's convert the units to the SI system: initial gas temperature. Let's calculate:
Answer	The gas must be cooled to a temperature

EXAMPLE 2

Exercise	A closed vessel contains a gas at a pressure of 200 kPa. What will be the pressure of the gas if the temperature is increased by 30%?
Solution	Since the gas container is closed, the volume of the gas does not change. The isochoric process is described by Charles' law: According to the condition of the problem, the gas temperature increased by 30%, so we can write: Substituting the last relation into Charles's law, we get: Let's convert the units to the SI system: the initial gas pressure kPa \u003d Pa. Let's calculate:
Answer	The gas pressure will become equal to 260 kPa.

EXAMPLE 3

Exercise	The oxygen system that the aircraft is equipped with has oxygen at a pressure of Pa. At the maximum lifting height, the pilot connects this system with an empty cylinder with a crane using a crane. What pressure will be established in it? The process of gas expansion occurs at a constant temperature.
Solution	The isothermal process is described by the Boyle-Mariotte law:

In this lesson, we will continue to study the relationship between the three macroscopic gas parameters, and more specifically, their relationship in gas processes occurring at a constant value of one of these three parameters, or isoprocesses: isothermal, isochoric and isobaric.

Consider the following isoprocess - the isobaric process.

Definition. isobaric(or isobaric) process- the process of transition of an ideal gas from one state to another at a constant pressure value. For the first time, such a process was considered by the French scientist Joseph-Louis Gay-Lussac (Fig. 4), so the law bears his name. Let's write this law

And now considering: and

Gay-Lussac's law

From this law, a directly proportional relationship between temperature and volume obviously follows: with an increase in temperature, an increase in volume is observed, and vice versa. The plot of the changing quantities in the equation, that is, T and V, has the following form and is called an isobar (Fig. 3):

Rice. 3. Graphs of isobaric processes in V-T coordinates ()

It should be noted that since we are working in the SI system, that is, with an absolute temperature scale, there is an area on the graph that is close to absolute zero temperatures, in which this law is not fulfilled. Therefore, a straight line in a region close to zero should be represented by a dotted line.

Rice. 4. Joseph Louis Gay-Lussac ()

Finally, consider the third isoprocess.

Definition. isochoric(or isochoric) process- the process of transition of an ideal gas from one state to another at a constant value of volume. The process was considered for the first time by the Frenchman Jacques Charles (Fig. 6), so the law bears his name. Let's write Charles's law:

We write the usual equation of state again:

And now considering: and

We get: for any different states of the gas, or simply:

Charles' law

From this law, a directly proportional relationship between temperature and pressure obviously follows: with an increase in temperature, an increase in pressure is observed, and vice versa. The plot of the changing quantities in the equation, that is, T and P, has the following form and is called the isochore (Fig. 5):

Rice. 5. Graphs of isochoric processes in coordinates V-T

In the region of absolute zero for the graphs of the isochoric process, there is also only a conditional dependence, so the straight line should also be brought to the origin by a dotted line.

Rice. 6. Jacques Charles ()

It is worth noting that it is precisely this dependence of temperature on pressure and volume in isochoric and isobaric processes, respectively, that determines the efficiency and accuracy of temperature measurement using gas thermometers.

It is also interesting that the isoprocesses we are considering were historically the first to be discovered, which, as we have shown, are special cases of the equation of state, and only then the Clapeyron and Mendeleev-Clapeyron equations. Chronologically, processes occurring at a constant temperature were first studied, then at a constant volume, and lastly, isobaric processes.

Now, to compare all isoprocesses, we have collected them in one table (see Fig. 7). Please note that the graphs of isoprocesses in coordinates containing a constant parameter, in fact, look like a dependence of a constant on some variable.

Rice. 7.

In the next lesson, we will consider the properties of such a specific gas as saturated steam, and consider the boiling process in detail.

Bibliography

1. Myakishev G.Ya., Sinyakov A.Z. Molecular physics. Thermodynamics. - M.: Bustard, 2010.
2. Gendenstein L.E., Dick Yu.I. Physics grade 10. - M.: Ileksa, 2005.
3. Kasyanov V.A. Physics grade 10. - M.: Bustard, 2010.

1. slideshare.net().
2. e-science.ru ().
3. mathus.ru ().

Homework

1. Page 70: No. 514-518. Physics. Task book. 10-11 grades. Rymkevich A.P. - M.: Bustard, 2013. ()
2. What is the relationship between temperature and density of an ideal gas in an isobaric process?
3. When the cheeks are puffed up, both volume and pressure in the mouth increase at a constant temperature. Does this contradict the Boyle-Mariotte law? Why?
4. *What will the graph of this process look like in P-V coordinates?

, a 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 linked, 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 a violation of the equilibrium 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 various thermodynamic processes inevitably occur in any real system. At the same time, in some systems such slow, almost imperceptible changes can 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 methods of thermodynamics.

The study of a thermodynamic process consists in determining the work done in this process, the change in internal energy, the amount of heat, and also in establishing a relationship between individual quantities characterizing the state of a gas.

Of all the possible thermodynamic processes, the isochoric, isobaric, isothermal, adiabatic, and polytropic processes are of greatest interest.

Isochoric process

An isochoric process is a thermodynamic process that occurs at constant volume. Such a process can be performed by heating a gas placed in a closed vessel. The gas heats up as a result of the supply of heat, and its pressure increases.
The change in gas parameters in an isochoric process describes Charles's law: p 1 /T 1 \u003d p 2 /T 2, or in the general case:

p/T = const .

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

Since in an isochoric process the change in volume dV is equal to 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

An isobaric process 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 acted upon by a constant external force during the removal and supply of heat.
When the temperature of the gas changes, the piston moves in one direction or another; while the volume of gas changes in accordance with Gay-Lussac's law:

V/T = const .

This means that in an isobaric process the volume occupied by the gas is directly proportional to the temperature.
It can be concluded that a change in temperature in this process will inevitably lead to a change in the internal energy of the gas, and a 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 is spent on the performance of the gas work to overcome the action of external forces. In this case, the ratio between the heat costs for increasing internal energy and for performing work depends on the heat capacity of the gas.

Isothermal process

An isothermal process is a thermodynamic process that occurs at a constant temperature.
It is very difficult to carry out an isothermal process with gas in practice. After all, it is necessary to comply with the condition that in 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 \u003d const, i.e. at a constant temperature, the gas pressure is inversely proportional to its volume.

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

dq = dA = pdv .

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

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



adiabatic process

An adiabatic process is a thermodynamic process that proceeds without heat exchange between the working fluid and the environment. Like an isothermal process, it is very difficult to implement an adiabatic process in practice. Such a process can proceed with the working medium placed in a vessel, for example, a cylinder with a piston, surrounded by a high-quality heat-insulating material.
But no matter what high-quality heat insulator we use in this case, some, even if negligible, amount 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. Nevertheless, many thermodynamic processes carried out in heat engineering proceed so quickly that the working fluid and the medium do not have time to exchange heat, therefore, with a certain degree of error, such processes can be considered as 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 equal to zero, and the change in internal energy is a function of thermal conductivity of temperature: du = c v dT , then we can write:

c v dT + pdv = 0 (3) .

Differentiating the Clapeyron equation pv = RT , we get:

pdv + vdp = RdT .

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

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

Taking into account the Mayer 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 (at 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 \u003d const, then the work of the adiabatic process can be represented as the formula (given without output):

l \u003d c v (T 1 - T 2) or l \u003d (p 1 v 1 - p 2 v 2) / (k-1).

Polytropic process

Unlike the thermodynamic processes considered 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 the above thermodynamic processes are special cases of polytropic processes.
The general equation of the polytropic process has the form pv n = const , where n is the polytropic index - a constant value for this process, which can take values ​​from - ∞ to + ∞ .

It is obvious that by giving certain values ​​to the polytropic index, one or another thermodynamic process can be obtained - 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.