General gas equation. Mendeleev-Clapeyron equation - O'Five in physics! Basic equation of molecular kinetic theory
homeEquation of state ideal gas (Sometimesthe equation Clapeyron (Sometimesor - the equation Mendeleev
) - a formula establishing the relationship between pressure, molar volume and absolute temperature of an ideal gas. The equation looks like:
Since , where is the amount of substance, and , where is the mass, is the molar mass, the equation of state can be written:
This form of recording is called the Mendeleev-Clapeyron equation (law).
In the case of constant gas mass, the equation can be written as: The last equation is called united gas law
- . From it the laws of Boyle - Mariotte, Charles and Gay-Lussac are obtained:.
- Boyle's law - Mariotta.
- Gay-Lussac's Lawlaw Charles 
(Gay-Lussac's second law, 1808). And in the form of proportion
This law is convenient for calculating the transfer of gas from one state to another. From the point of view of a chemist, this law may sound slightly different: The volumes of reacting gases under the same conditions (temperature, pressure) relate to each other and to the volumes of the resulting gaseous compounds as simple integers. For example, 1 volume of hydrogen combines with 1 volume of chlorine, resulting in 2 volumes of hydrogen chloride:
- . From it the laws of Boyle - Mariotte, Charles and Gay-Lussac are obtained: 1 A volume of nitrogen combines with 3 volumes of hydrogen to form 2 volumes of ammonia:
. The Boyle-Mariotte law is named after the Irish physicist, chemist and philosopher Robert Boyle (1627-1691), who discovered it in 1662, and also after the French physicist Edme Mariotte (1620-1684), who discovered this law independently of Boyle in 1677. In some cases (in gas dynamics), it is convenient to write the equation of state of an ideal gas in the form
On the one hand, in highly compressed gases the sizes of the molecules themselves are comparable to the distances between the molecules. Thus, the free space in which the molecules move is less than the total volume of the gas. This circumstance increases the number of impacts of molecules on the wall, since it reduces the distance that a molecule must fly to reach the wall. On the other hand, in a highly compressed and therefore denser gas, molecules are noticeably attracted to other molecules much more most time than molecules in a rarefied gas. This, on the contrary, reduces the number of impacts of molecules into the wall, since in the presence of attraction to other molecules, gas molecules move towards the wall at a lower speed than in the absence of attraction. At not too high pressures, the second circumstance is more significant and the product decreases slightly. At very high pressures, the first circumstance plays a major role and the product increases.
5. Basic equation of the molecular kinetic theory of ideal gases
To derive the basic equation of molecular kinetic theory, consider a monatomic ideal gas. Let us assume that gas molecules move chaotically, the number of mutual collisions between gas molecules is negligible compared to the number of impacts on the walls of the vessel, and the collisions of molecules with the walls of the vessel are absolutely elastic. Let us select some elementary area DS on the wall of the vessel and calculate the pressure exerted on this area. With each collision, a molecule moving perpendicular to the platform transfers momentum to it m 0 v-(-m 0 v)=2m 0 v, Where T 0 - the mass of the molecule, v - its speed.
During the time Dt of the site DS, only those molecules that are enclosed in the volume of a cylinder with a base DS and height v D t .The number of these molecules is equal n D Sv D t (n- concentration of molecules).
It is necessary, however, to take into account that in reality the molecules move towards the site
DS at different angles and have different speeds, and the speed of the molecules changes with each collision. To simplify calculations, the chaotic movement of molecules is replaced by movement along three mutually perpendicular directions, so that at any moment of time 1/3 of the molecules move along each of them, with half of the molecules (1/6) moving along a given direction in one direction, half in the opposite direction . Then the number of impacts of molecules moving in given direction, about the DS pad will be 1/6 nDSvDt. When colliding with the platform, these molecules will transfer momentum to it
D R = 2m 0 v 1 / 6 n D Sv D t= 1 / 3 n m 0 v 2D S D t.
Then the pressure of the gas exerted on the wall of the vessel is
p=DP/(DtDS)= 1 / 3 nm 0 v 2 . (3.1)
If the gas volume V contains N molecules,
moving at speeds v 1 , v 2 , ..., v N, That
it is advisable to consider root mean square speed
characterizing the entire set of gas molecules.
Equation (3.1), taking into account (3.2), will take the form
p =
1
/
3
Fri 0
Expression (3.3) is called the basic equation of the molecular kinetic theory of ideal gases. Accurate calculation taking into account the movement of molecules throughout
possible directions is given by the same formula.
Considering that n = N/V we get
Where E - the total kinetic energy of the translational motion of all gas molecules.
Since the mass of gas m =Nm 0 , then equation (3.4) can be rewritten as
pV= 1 / 3 m
For one mole of gas t = M (M - molar mass), so
pV m = 1 / 3 M
Where V m - molar volume. On the other hand, according to the Clapeyron-Mendeleev equation, pV m =RT. Thus,
RT= 1 / 3 M
Since M = m 0 N A, where m 0 is the mass of one molecule, and N A is Avogadro’s constant, it follows from equation (3.6) that
Where k = R/N A- Boltzmann constant. From here we find that at room temperature, oxygen molecules have a root mean square speed of 480 m/s, hydrogen molecules - 1900 m/s. At the temperature of liquid helium, the same speeds will be 40 and 160 m/s, respectively.
Average kinetic energy of translational motion of one ideal gas molecule
(we used formulas (3.5) and (3.7)) is proportional to the thermodynamic temperature and depends only on it. From this equation it follows that at T=0
Ideal gas
is a gas in which there are no forces of mutual attraction and repulsion between molecules and the sizes of the molecules are neglected. All real gases at high temperatures and low pressures can practically be considered ideal gases.
The equation of state for both ideal and real gases is described by three parameters according to equation (1.7).
The equation of state of an ideal gas can be derived from molecular kinetic theory or from a joint consideration of the Boyle-Mariotte and Gay-Lussac laws.
This equation was derived in 1834 by the French physicist Clapeyron and for 1 kg of gas mass has the form:
Р·υ = R·Т, (2.10)
where: R is the gas constant and represents the work done by 1 kg of gas in a process at constant pressure and with a temperature change of 1 degree.
Equation (2.7) is called t thermal equation of state
or characteristic equation
.
For an arbitrary amount of gas of mass m, the equation of state will be:
Р·V = m·R·Т. (2.11)
In 1874, D.I. Mendeleev, based on Dalton’s law ( “Equal volumes of different ideal gases at the same temperatures and pressures contain the same number of molecules.”) proposed a universal equation of state for 1 kg of gas, which is called Clapeyron-Mendeleev equation:
Р·υ = R μ ·Т/μ , (2.12)
where: μ - molar (molecular) mass of gas, (kg/kmol);
R μ = 8314.20 J/kmol (8.3142 kJ/kmol) - universal gas constant
and represents the work done by 1 kmol of an ideal gas in a process at constant pressure and with a temperature change of 1 degree.
Knowing R μ, you can find the gas constant R = R μ / μ.
For an arbitrary mass of gas, the Clapeyron-Mendeleev equation will have the form:
Р·V = m·R μ ·Т/μ . (2.13)
A mixture of ideal gases.
Gas mixture refers to a mixture of individual gases that enter into any chemical reactions with each other. Each gas (component) in the mixture, regardless of other gases, completely retains all its properties and behaves as if it alone occupied the entire volume of the mixture.
Partial pressure- this is the pressure that each gas included in the mixture would have if this gas were alone in the same quantity, in the same volume and at the same temperature as in the mixture.
The gas mixture obeys Dalton's law:
║The total pressure of the gas mixture is equal to the sum of the partial pressures║individual gases that make up the mixture.
P = P 1 + P 2 + P 3 + . . . Р n = ∑ Р i , (2.14)
where P 1, P 2, P 3. . . Р n – partial pressures.
The composition of the mixture is specified by volume, mass and mole fractions, which are determined respectively using the following formulas:
r 1 = V 1 / V cm; r 2 = V 2 / V cm; … r n = V n / V cm, (2.15)
g 1 = m 1 / m cm; g 2 = m 2 / m cm; … g n = m n / m cm, (2.16)
r 1 ′ = ν 1 / ν cm; r 2 ′ = ν 2 / ν cm; … r n ′ = ν n / ν cm, (2.17)
where V 1; V 2 ; … V n ; V cm – volumes of components and mixture;
m 1; m2; … m n ; m cm – masses of components and mixture;
ν 1; ν 2 ; … ν n ; ν cm – amount of substance (kilomoles)
components and mixtures.
For an ideal gas, according to Dalton's law:
r 1 = r 1 ′; r 2 = r 2 ′; … r n = r n ′ . (2.18)
Since V 1 +V 2 + … + V n = V cm and m 1 + m 2 + … + m n = m cm,
then r 1 + r 2 + … + r n = 1 , (2.19)
g 1 + g 2 + … + g n = 1. (2.20)
The relationship between volume and mass fractions is as follows:
g 1 = r 1 ∙μ 1 /μ cm; g 2 = r 2 ∙μ 2 /μ cm; … g n = r n ∙μ n /μ cm, (2.21)
where: μ 1, μ 2, ... μ n, μ cm – molecular weights of the components and mixture.
Molecular weight of the mixture:
μ cm = μ 1 r 1 + r 2 μ 2 + … + r n μ n. (2.22)
Gas constant of mixture:
R cm = g 1 R 1 + g 2 R 2 + … + g n R n =
= R μ (g 1 /μ 1 + g 2 /μ 2 + … + g n /μ n) =
= 1 / (r 1 /R 1 + r 2 /R 2 + ... + r n /R n) . (2.23)
Specific mass heat capacities of the mixture:
with р cm. = g 1 with р 1 + g 2 with р 2 + … + g n with р n. (2.24)
with v see = g 1 with p 1 + g 2 with v 2 + ... + g n with v n. (2.25)
Specific molar (molecular) heat capacities of the mixture:
with rμ cm. = r 1 with rμ 1 + r 2 with rμ 2 + … + r n with rμ n. (2.26)
with vμ cm. = r 1 with vμ 1 + r 2 with vμ 2 + … + r n with vμ n. (2.27)
Topic 3. Second law of thermodynamics.
Basic provisions of the second law of thermodynamics.
The first law of thermodynamics states that heat can be converted into work, and work into heat, and does not establish the conditions under which these transformations are possible.
The transformation of work into heat always occurs completely and unconditionally. The reverse process of converting heat into work during its continuous transition is possible only under certain conditions and not completely. Heat can naturally move from hotter bodies to colder ones. The transfer of heat from cold bodies to heated ones does not occur by itself. This requires additional energy.
Thus, for a complete analysis of phenomena and processes, it is necessary to have, in addition to the first law of thermodynamics, an additional law. This law is second law of thermodynamics
. It establishes whether a particular process is possible or impossible, in which direction the process proceeds, when thermodynamic equilibrium is achieved, and under what conditions maximum work can be obtained.
Formulations of the second law of thermodynamics.
For the existence of a heat engine, 2 sources are needed - hot spring and cold spring
(environment). If a heat engine operates from only one source, it is called perpetual motion machine of the 2nd kind.
1 formulation (Ostwald):
| "A perpetual motion machine of the 2nd kind is impossible."
A perpetual motion machine of the 1st kind is a heat engine in which L>Q 1, where Q 1 is the supplied heat. The first law of thermodynamics “allows” the possibility of creating a heat engine that completely converts the supplied heat Q 1 into work L, i.e. L = Q 1. The second law imposes more stringent restrictions and states that the work must be less than the heat supplied (L A perpetual motion machine of the 2nd kind can be realized if heat Q 2 is transferred from a cold source to a hot one. But for this, heat must spontaneously transfer from a cold body to a hot one, which is impossible. This leads to the 2nd formulation (by Clausius):
|| "Heat cannot spontaneously transfer from more
|| cold body to a warmer one."
To operate a heat engine, two sources are needed - hot and cold. 3rd formulation (Carnot):
|| "Where there is a temperature difference, it is possible to commit
|| work."
All these formulations are interconnected; from one formulation you can get another.
Entropy.
One of the functions of the state of a thermodynamic system is entropy. Entropy is a quantity defined by the expression:
dS = dQ / T. [J/K] (3.1)
or for specific entropy:
ds = dq / T. [J/(kg K)] (3.2)
Entropy is an unambiguous function of the state of a body, taking on a very specific value for each state. It is an extensive (depending on the mass of the substance) state parameter and in any thermodynamic process is completely determined by the initial and final state of the body and does not depend on the path of the process.
Entropy can be defined as a function of the basic state parameters:
S = f 1 (P,V) ; S = f 2 (P,T) ; S = f 3 (V,T) ; (3.3)
or for specific entropy:
s = f 1 (P,υ) ; s = f 2 (P,T) ; S = f 3 (υ,T) ; (3.4)
Since entropy does not depend on the type of process and is determined by the initial and final states of the working fluid, only its change in a given process is found, which can be found using the following equations:
Ds = c v ln(T 2 /T 1) + R ln(υ 2 /υ 1); (3.5)
Ds = c p ln(T 2 /T 1) - R ln(P 2 /P 1) ; (3.6)
Ds = c v ln(P 2 /P 1) + c p ln(υ 2 /υ 1) . (3.7)
If the entropy of the system increases (Ds > 0), then heat is supplied to the system.
If the entropy of the system decreases (Ds< 0), то системе отводится тепло.
If the entropy of the system does not change (Ds = 0, s = Const), then heat is not supplied or removed to the system (adiabatic process).
Carnot cycle and theorems.
The Carnot cycle is a circular cycle consisting of 2 isothermal and 2 adiabatic processes. Reversible Carnot cycle in p,υ- and T,s-diagrams shown in Fig. 3.1.
1-2 – reversible adiabatic expansion at s 1 = Const. The temperature decreases from T 1 to T 2.
2-3 – isothermal compression, heat removal q 2 to a cold source from the working fluid.
3-4 – reversible adiabatic compression at s 2 =Const. The temperature rises from T 3 to T 4.
4-1 – isothermal expansion, supply of heat q 1 to the hot source to the working fluid.
The main characteristic of any cycle is thermal efficiency(t.k.p.d.).
h t = L c / Q c, (3.8)
h t = (Q 1 – Q 2) / Q 1.
For a reversible Carnot cycle t.k.p.d. determined by the formula:
h tk = (T 1 – T 2) / T 1. (3.9)
this implies Carnot's 1st theorem
:
|| "The thermal efficiency of a reversible Carnot cycle does not depend on
|| properties of the working fluid and is determined only by temperatures
|| sources."
From a comparison of an arbitrary reversible cycle and a Carnot cycle it follows Carnot's 2nd theorem:
|| "The reversible Carnot cycle is the best cycle in || a given temperature range"
Those. t.k.p.d. The Carnot cycle is always greater than the coefficient of efficiency. arbitrary loop:
h tк > h t . (3.10)
Topic 4. Thermodynamic processes.
Details Category: Molecular kinetic theory Published 05.11.2014 07:28 Views: 14155Gas is one of four states of aggregation in which a substance can exist.
The particles that make up the gas are very mobile. They move almost freely and chaotically, periodically colliding with each other like billiard balls. Such a collision is called elastic collision . During a collision, they dramatically change the nature of their movement.
Since in gaseous substances the distance between molecules, atoms and ions is much greater than their sizes, these particles interact very weakly with each other, and their potential interaction energy is very small compared to the kinetic energy.
The connections between molecules in a real gas are complex. Therefore, it is also quite difficult to describe the dependence of its temperature, pressure, volume on the properties of the molecules themselves, their quantity, and the speed of their movement. But the task is greatly simplified if, instead of real gas, we consider it mathematical model - ideal gas .
It is assumed that in the ideal gas model there are no attractive or repulsive forces between molecules. They all move independently of each other. And the laws of classical Newtonian mechanics can be applied to each of them. And they interact with each other only during elastic collisions. The time of the collision itself is very short compared to the time between collisions.
Classical ideal gas
Let's try to imagine the molecules of an ideal gas as small balls located in a huge cube at a great distance from each other. Because of this distance, they cannot interact with each other. Therefore, their potential energy is zero. But these balls move at great speed. This means they have kinetic energy. When they collide with each other and with the walls of the cube, they behave like balls, that is, they bounce elastically. At the same time, they change the direction of their movement, but do not change their speed. This is roughly what the motion of molecules in an ideal gas looks like.
- The potential energy of interaction between molecules of an ideal gas is so small that it is neglected compared to kinetic energy.
- The molecules in an ideal gas are also so small that they can be considered material points. And this means that they total volume is also negligible compared to the volume of the vessel in which the gas is located. And this volume is also neglected.
- The average time between collisions of molecules is much greater than the time of their interaction during a collision. Therefore, the interaction time is also neglected.
Gas always takes the shape of the container in which it is located. Moving particles collide with each other and with the walls of the container. During an impact, each molecule exerts some force on the wall for a very short period of time. This is how it arises pressure . The total gas pressure is the sum of the pressures of all molecules.
Ideal gas equation of state
The state of an ideal gas is characterized by three parameters: pressure, volume And temperature. The relationship between them is described by the equation:
Where R - pressure,
V M - molar volume,
R - universal gas constant,
T - absolute temperature (degrees Kelvin).
Because V M = V / n , Where V - volume, n - the amount of substance, and n= m/M , That
Where m - mass of gas, M - molar mass. This equation is called Mendeleev-Clayperon equation .
At constant mass the equation becomes:
This equation is called united gas law .
Using the Mendeleev-Cliperon law, one of the gas parameters can be determined if the other two are known.
Isoprocesses
Using the equation of the unified gas law, it is possible to study processes in which the mass of gas and one of the most important parameters- pressure, temperature or volume - remain constant. In physics such processes are called isoprocesses .
From The unified gas law leads to other important gas laws: Boyle-Mariotte law, Gay-Lussac's law, Charles's law, or Gay-Lussac's second law.
Isothermal process
A process in which pressure or volume changes but temperature remains constant is called isothermal process .
In an isothermal process T = const, m = const .
The behavior of a gas in an isothermal process is described by Boyle-Mariotte law . This law was discovered experimentally English physicist Robert Boyle in 1662 and French physicist Edme Mariotte in 1679. Moreover, they did this independently of each other. The Boyle-Marriott law is formulated as follows: In an ideal gas at a constant temperature, the product of the gas pressure and its volume is also constant.
The Boyle-Marriott equation can be derived from the unified gas law. Substituting into the formula T = const , we get
p · V = const
That's what it is Boyle-Mariotte law . From the formula it is clear that the pressure of a gas at constant temperature is inversely proportional to its volume. The higher the pressure, the lower the volume, and vice versa.
How to explain this phenomenon? Why does the pressure of a gas decrease as the volume of a gas increases?
Since the temperature of the gas does not change, the frequency of collisions of molecules with the walls of the vessel does not change. If the volume increases, the concentration of molecules becomes less. Consequently, per unit area there will be fewer molecules that collide with the walls per unit time. The pressure drops. As the volume decreases, the number of collisions, on the contrary, increases. Accordingly, the pressure increases.
Graphically, an isothermal process is displayed on a curve plane, which is called isotherm . She has a shape hyperboles.
Each temperature value has its own isotherm. The higher the temperature, the higher the corresponding isotherm is located.
Isobaric process
The processes of changing the temperature and volume of a gas at constant pressure are called isobaric . For this process m = const, P = const.
The dependence of the volume of a gas on its temperature at constant pressure was also established experimentally French chemist and physicist Joseph Louis Gay-Lussac, who published it in 1802. That is why it is called Gay-Lussac's law : " Etc and constant pressure, the ratio of the volume of a constant mass of gas to its absolute temperature is a constant value."
At P = const the equation of the unified gas law turns into Gay-Lussac equation .
An example of an isobaric process is a gas located inside a cylinder in which a piston moves. As the temperature rises, the frequency of molecules hitting the walls increases. The pressure increases and the piston rises. As a result, the volume occupied by the gas in the cylinder increases.
Graphically, an isobaric process is represented by a straight line, which is called isobar .
The higher the pressure in the gas, the lower the corresponding isobar is located on the graph.
Isochoric process
Isochoric, or isochoric, is the process of changing the pressure and temperature of an ideal gas at constant volume.
For an isochoric process m = const, V = const.
It is very simple to imagine such a process. It occurs in a vessel of a fixed volume. For example, in a cylinder, the piston in which does not move, but is rigidly fixed.
The isochoric process is described Charles's law : « For a given mass of gas at constant volume, its pressure is proportional to temperature" The French inventor and scientist Jacques Alexandre César Charles established this relationship through experiments in 1787. In 1802, it was clarified by Gay-Lussac. Therefore this law is sometimes called Gay-Lussac's second law.
At V = const from the equation of the unified gas law we get the equation Charles's law or Gay-Lussac's second law .
At constant volume, the pressure of a gas increases if its temperature increases. .
On graphs, an isochoric process is represented by a line called isochore .
The larger the volume occupied by the gas, the lower the isochore corresponding to this volume is located.
In reality, no gas parameter can be maintained unchanged. This can only be done in laboratory conditions.
Of course, an ideal gas does not exist in nature. But in real rarefied gases at very low temperatures and pressures not exceeding 200 atmospheres, the distance between the molecules is much greater than their sizes. Therefore, their properties approach those of an ideal gas.
Let's take a certain amount of gas of a certain chemical composition, for example nitrogen, oxygen or air, and enclose it in a vessel, the volume of which can be changed at our discretion. Let's assume that we have a pressure gauge, i.e. a device for measuring gas pressure, and a thermometer for measuring its temperature. Experience shows that the listed macroscopic parameters fully characterize a gas as a thermodynamic system in the case when this gas consists of neutral molecules that do not have their own dipole moment.
In a state of thermodynamic equilibrium, not all of these parameters are independent; they are interconnected by an equation of state. To obtain this equation, you need to use
experimentally established patterns of gas behavior when any external parameters change.
A gas in a vessel is a simple thermodynamic system. Let us first assume that neither the amount of gas nor its chemical composition do not change during the experiment, so we will only talk about three macroscopic parameters - pressure, volume V and temperature. To establish the patterns connecting these parameters, it is convenient to fix the value of one of the parameters and monitor changes in the other two. We will assume that the changes we cause in the gas occur so slowly that at any moment of time the macroscopic parameters characterizing the entire gas in a state of thermodynamic equilibrium have very definite values.
Isoprocesses. As already noted, from any nonequilibrium state, a thermodynamic system reaches a state of equilibrium after some time - the relaxation time. In order for macroscopic parameters to have well-defined values during changes occurring in the system, the characteristic time of these changes must be much longer than the relaxation time. This condition imposes restrictions on the permissible rate of the process in a gas, at which its macroscopic parameters retain their meaning.
Processes that occur with a constant value of one of the parameters are usually called isoprocesses. Thus, a process occurring at a constant temperature is called isothermal, at a constant volume - isochoric (isochoric), at a constant pressure - isobaric (isobaric).
Boyle-Marriott law. Historically, the isothermal process in gas was the first to be experimentally studied. The English physicist R. Boyle and, independently of him, the French physicist E. Mariotte established the law of volume change with pressure changes: for a given amount of any gas at a constant temperature, the volume is inversely proportional to pressure. Usually the Boyle-Mariotte law is written in the form
To maintain a constant temperature, the gas under study must be in good thermal contact with the environment, which has a constant temperature. In this case, the gas is said to be in contact with a thermostat - a large thermal reservoir, the state of which is not affected by any changes occurring in the gas under study.
The Boyle-Marriott law holds true for all gases and their mixtures over a wide range of temperatures and pressures. Deviations from
of this law become significant only at pressures several hundred times higher than atmospheric pressure and at sufficiently low temperatures.
You can check the validity of the Boyle-Mariotte law using very simple means. To do this, it is enough to have a glass tube sealed at one end, in which a column of mercury closes a certain amount of air (Melde tube). The volume of air can be measured with a ruler along the length of the air column in the tube (Fig. 45), and the pressure can be judged by the height of the mercury column at different orientations of the tube in the gravitational field.
To visually depict changes in the state of a gas and the processes occurring with it, it is convenient to use so-called -diagrams, where volume values are plotted along the abscissa axis, and pressure values are plotted along the ordinate axis. The curve on the -diagram corresponding to an isothermal process is called an isotherm.
Rice. 45. The simplest device for testing the Boyle-Mariotte law (Melde tube)
Rice. 46. Gas isotherms on the -diagram
As follows from the Boyle-Mariotte law, gas isotherms are hyperbolas (Fig. 46). The higher the temperature, the further the corresponding isotherm is located from the coordinate axes.
Charles's law. The dependence of gas pressure on temperature at a constant volume was experimentally established by the French physicist J. Charles. According to Charles's law, gas pressure at constant volume depends linearly on temperature:
where is the gas pressure at O °C. It turns out that the temperature coefficient of pressure a is the same for all gases and is equal to
Gay-Lussac's law. The dependence of gas volume on temperature at constant pressure has a similar form. This was established experimentally by the French physicist Gay-Lussac, who found that the temperature coefficient of expansion is the same for all gases. The value of this coefficient turned out to be the same as the coefficient a in Charles’s law. Thus, Gay-Lussac's law can be written as
where is the volume of gas at O °C.
The coincidence of temperature coefficients in the laws of Charles and Gay-Lussac is not accidental and indicates that these gas laws established experimentally are not independent. Below we will go into more detail on this.
Gas thermometer. The fact that the dependence of pressure or volume on temperature, expressed by the laws of Charles and Gay-Lussac, is the same for all gases, makes it especially convenient choice gas as a thermometric body. Although in practice it is inconvenient to use gas thermometers due to their bulkiness and thermal inertia, they are used to calibrate other thermometers that are more convenient for practical applications.
Kelvin scale. The dependence of pressure or volume on temperature in the laws of Charles and Gay-Lussac becomes even simpler if we move to a new temperature scale, requiring that the linear dependence become direct proportionality.
By depicting the dependence of gas volume on temperature expressed by formula (3) (Fig. 47) and continuing the graph to the left until it intersects with the temperature axis, it is easy to verify that the continuation of the graph intersects the Γ axis at a temperature value equal to since It is at this point that the beginning of the new temperature scale should be placed , so that equations (2) and (3) can be written as direct proportionality. This point is called absolute zero temperature. The scale of the new scale, i.e. the temperature unit, is selected in the same way as in the Celsius scale. On the new temperature scale, zero degrees Celsius corresponds to a temperature of a degree (more precisely 273.15), and any other temperature T is related to the corresponding temperature on the Celsius scale by the relation
The temperature scale introduced here is called the Kelvin scale, and the unit of measurement that is the same as the degree Celsius scale is called the kelvin and is symbolized by the letter K. This scale is sometimes called the International Practical Temperature Scale.
When using the Kelvin temperature scale, the graph of Gay-Lussac's law takes the form shown in Fig. 48, and formulas (2) and (3) can be written in the form
Rice. 47. Dependence of gas volume on temperature at constant pressure expressed by Gay-Lussac’s law
Rice. 48. Graph of Gay-Lussac's law on the Kelvin temperature scale
The proportionality coefficient in (6) characterizes the slope of the graph in Fig. 48.
Equation of gas state. Experimental gas laws make it possible to establish the equation of state of a gas. To do this, it is enough to use any two of the given laws. Let a certain amount of gas be in a state with volume pressure and temperature. Let us transfer it to another (intermediate) state, characterized by the same temperature value and some new values of volume V and pressure. In an isothermal process, the Boyle-Mariotte law is satisfied, therefore
Now let’s transfer the gas from the intermediate state to the final state with the same volume value as in the intermediate state, and some values of pressure and temperature. In an isochoric process, Charles’ law is satisfied, therefore
since Substituting into from (7) and taking into account that we finally get
We have changed all three macroscopic parameters and T, and yet relation (9) shows that for a given amount of gas (number of moles), the combination of parameters has the same value, no matter what state this gas is in. This means that the equation (9) is the equation of state of a gas. It is called the Clapeyron equation.
In the above derivation of equation (9), the Gay-Lussac law was not used. However, it is easy to see that it contains all three gas laws. Indeed, assuming in we obtain for an isobaric process the relation that corresponds to the Gay-Lussac law.
Mendeleev-Clapeyron equation. Let's take one mole of gas under normal conditions, i.e. at normal atmospheric pressure. In accordance with Avogadro's law, established experimentally, one mole of any gas (helium, nitrogen, oxygen, etc.) occupies the same volume of a liter under normal conditions. Therefore, for one mole of any gas, the combination denoted by and called the universal gas constant (or molar gas constant) has the same value:
Taking into account (10), the equation of state of one mole of any gas can be written in the form
Equation (11) is easy to generalize for an arbitrary amount of gas. Since at the same values of temperature and pressure, moles of gas occupy a volume greater than 1 mole, then
In this form, the equation of state of a gas was first obtained by the Russian scientist D.I. Mendeleev. Therefore, it is called the Mendeleev-Clapeyron equation.
Ideal gas. The gas equation of state (11) or (12) was obtained on the basis of experimentally established gas laws. These laws are satisfied approximately: the conditions for their applicability
different for different gases. For example, for helium they are valid in a wider range of temperatures and pressures than for carbon dioxide. The equation of state obtained from approximate gas laws is also approximate.
Let us introduce a physical model - an ideal gas. By this we mean a system for which equation (11) or (12) is exact. A remarkable feature of an ideal gas is that its internal energy is proportional to the absolute temperature and does not depend on the volume occupied by the gas.
As in all other use cases physical models, the applicability of the ideal gas model to a particular real gas depends not only on the properties of the gas itself, but also on the nature of the question to which the answer needs to be found. This model does not allow us to describe the behavior of various gases, but it reveals properties common to all gases.
You can get acquainted with the application of the equation of state of an ideal gas using the example of specific problems.
Tasks
1. One volume cylinder contains nitrogen at a pressure of . Another volume cylinder contains oxygen at pressure The temperature of the gases coincides with the temperature environment. What gas pressure will be established if you open the valve of the tube connecting these cylinders to each other?
Solution. After opening the tap, gas from a cylinder with a higher pressure will flow into another cylinder. Eventually, the pressure in the cylinders will equalize and the gases will mix. Even if the temperature changes during the flow of gases, after thermal equilibrium is established it will again become equal to the temperature of the surrounding air.
To solve the problem, you can use the equation of state of an ideal gas. Denoting the amount of gases in the cylinders before opening the tap, we have
In the final state, the mixture of gases contains moles, occupies a volume and is at a pressure that needs to be determined. Applying the Mendeleev-Clapeyron equation to a mixture of gases, we have
Expressing from equations (13) and substituting into (14), we find
In the particular case when the initial gas pressures are the same, the pressure of the mixture after equilibrium is established remains the same. An interesting limiting case is the one corresponding to the replacement of the second vessel by the atmosphere. From (15) we obtain where is the atmospheric pressure. This result is obvious from general considerations.
Let us pay attention to the fact that the result expressed by formula (15) corresponds to the fact that the pressure of a mixture of gases is equal to the sum of the partial pressures of each of the gases, i.e., the pressures that each of the gases would have if it occupied the entire volume at the same temperature. Indeed, the partial pressures of each gas can be found using the Boyle-Mariotte law:
It can be seen that the total pressure equal to the sum partial pressure is expressed by formula (15). The statement that the pressure of a mixture of chemically non-interacting gases is equal to the sum of the partial pressures is called Dalton's law.
2. Having heated the stove, the air temperature in the country house was increased from 0 to How did the air density change?
Solution. It is clear that the volume of the room did not change when the furnace was heated, since the thermal expansion of the walls can be neglected. If we heated air with a constant volume V in a closed vessel, its pressure would increase, but the density would remain unchanged. But country house not sealed, so the air pressure remains unchanged, equal to the external one atmospheric pressure. It is clear that with an increase in temperature T the mass of air in the room must change: some of it must escape through the cracks to the outside. It is clear that a column of water will not be pushed out of the tube only with very small changes in temperature. To estimate the temperature change at which the column rises a given distance, we rewrite (19) as follows:
Assuming for the estimate, we obtain The above estimate shows that with the help of this very simple device it is possible to detect a temperature change of up to 0.01 K, since it is easy to replace a change in the position of the column by 1 mm.
What is the relaxation time for a thermodynamic system?
What restrictions should be imposed on the rate of processes in a gas so that at any moment in time the macroscopic parameters describing the gas in a state of equilibrium make sense?
What determines the numerical value of the constant on the right side of the equation of the Boyle-Mariotte law (1)?
What do they mean when they say that the system under study is in contact with a thermostat?
Suggest a way to test the Boyle-Mariotte law using the device described in the text (see Fig. 45).
What are the advantages of choosing gas as a thermometric body?
How is the choice of the temperature reference point in the Kelvin scale related to the value of the temperature coefficient of gas expansion?
How is the relationship between temperatures measured on the Celsius scale and the Kelvin scale established?
Derive the equation of state of the gas using the Boyle-Mariotte and Gay-Lussac laws.
The Clapeyron equation was obtained using only two gas laws, but contains all three laws. How does this relate to the fact that gases have the same temperature coefficients of pressure and volume?
What is the universal gas constant? How is it related to Avogadro's law?
Which physical system called an ideal gas? What determines the conditions for the applicability of this model? What does the internal energy of an ideal gas depend on?
Is it possible to explain Dalton's law, established experimentally, for a mixture of gases, based on the Mendeleev-Clapeyron equation?
How will the sensitivity to temperature changes of the simple device described in problem 3 change if the upper hole of the tube is plugged?
« Physics - 10th grade"
This chapter will discuss the implications that can be drawn from the concept of temperature and other macroscopic parameters. The basic equation of the molecular kinetic theory of gases has brought us very close to establishing connections between these parameters.
We examined in detail the behavior of an ideal gas from the point of view of molecular kinetic theory. The dependence of gas pressure on the concentration of its molecules and temperature was determined (see formula (9.17)).
Based on this dependence, it is possible to obtain an equation connecting all three macroscopic parameters p, V and T, characterizing the state of an ideal gas of a given mass.
Formula (9.17) can only be used up to a pressure of the order of 10 atm.
The equation relating three macroscopic parameters p, V and T is called ideal gas equation of state.
Let us substitute the expression for the concentration of gas molecules into the equation p = nkT. Taking into account formula (8.8), the gas concentration can be written as follows:
where N A is Avogadro's constant, m is the mass of the gas, M is its molar mass. After substituting formula (10.1) into expression (9.17) we will have
The product of Boltzmann's constant k and Avogadro's constant N A is called the universal (molar) gas constant and is denoted by the letter R:
R = kN A = 1.38 10 -23 J/K 6.02 10 23 1/mol = 8.31 J/(mol K). (10.3)
Substituting the universal gas constant R into equation (10.2) instead of kN A, we obtain the equation of state of an ideal gas of arbitrary mass
The only quantity in this equation that depends on the type of gas is its molar mass.
The equation of state implies a relationship between the pressure, volume and temperature of an ideal gas, which can be in any two states.
If index 1 denotes the parameters related to the first state, and index 2 denotes the parameters related to the second state, then according to equation (10.4) for a gas of a given mass
The right-hand sides of these equations are the same, therefore, their left-hand sides must also be equal:
It is known that one mole of any gas under normal conditions (p 0 = 1 atm = 1.013 10 5 Pa, t = 0 °C or T = 273 K) occupies a volume of 22.4 liters. For one mole of gas, according to relation (10.5), we write:
We have obtained the value of the universal gas constant R.
Thus, for one mole of any gas
The equation of state in the form (10.4) was first obtained by the great Russian scientist D.I. Mendeleev. He is called Mendeleev-Clapeyron equation.
The equation of state in the form (10.5) is called Clapeyron equation and is one of the forms of writing the equation of state.
B. Clapeyron worked in Russia for 10 years as a professor at the Institute of Railways. Returning to France, he participated in the construction of many railways and drew up many projects for the construction of bridges and roads.
His name is included in the list of the greatest scientists of France, placed on the first floor of the Eiffel Tower.
The equation of state does not need to be derived every time, it must be remembered. It would be nice to remember the value of the universal gas constant:
R = 8.31 J/(mol K).
So far we have talked about the pressure of an ideal gas. But in nature and in technology, we very often deal with a mixture of several gases, which under certain conditions can be considered ideal.
The most important example of a mixture of gases is air, which is a mixture of nitrogen, oxygen, argon, carbon dioxide and other gases. What is the pressure of the gas mixture?
Dalton's law is valid for a mixture of gases.
Dalton's law
The pressure of a mixture of chemically non-interacting gases is equal to the sum of their partial pressures
p = p 1 + p 2 + ... + p i + ... .
where p i - partial pressure i-th components mixtures.