Mendeleev's clapeyron equation is a derivation of the formula. Equation of state of ideal gas (Mendeleev-Clapeyron equation)
Gas is one of the four states of aggregation in which matter can be.
The particles that make up a gas are very mobile. They move almost freely and randomly, 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 size, these particles interact very weakly with each other, and their potential energy interaction is very small compared to the kinetic.
The bonds 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 consider its mathematical model - ideal gas .
It is assumed that in the ideal gas model there are no forces of attraction and repulsion 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 with 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 rebound elastically. At the same time, they change the direction of their movement, but do not change their speed. This is what the movement 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 in comparison with the kinetic energy.
- 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 container containing the gas. And this volume is also neglected.
- The average time between collisions of molecules is much longer than the time of their interaction during a collision. Therefore, the interaction time is also neglected.
A gas always takes the shape of the container it is in. The moving particles collide with each other and with the walls of the vessel. During the impact, each molecule acts on the wall with some force for a very short period of time. This is how 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 is the universal gas constant,
T - absolute temperature (degrees Kelvin).
Because V M = V / n , where V - volume, n is the amount of substance, and n= m/M , then
where m - mass of gas, M - molar mass. This equation is called the Mendeleev-Claiperon equation .
At constant mass, the equation takes the form:
This equation is called unified gas law .
Using the Mendeleev-Klaiperon law, one of the gas parameters can be determined if the other two are known.
isoprocesses
With the help of the unified gas law equation, it is possible to study processes in which the mass of the gas and one of the most important parameters - pressure, temperature or volume - remain constant. In physics, such processes are called isoprocesses .
From of the unified gas law, other important gas laws: boyle-mariotte law, Gay-Lussac's law, Charles' law, or Gay-Lussac's second law.
Isothermal process
A process in which pressure or volume changes but the temperature remains constant is called isothermal process .
In an isothermal process T = const, m = const .
The behavior of a gas in an isothermal process describes boyle-mariotte law . This law was discovered experimentally English physicist Robert Boyle in 1662 and French physicist Edme Mariotte in 1679. And they did it independently of each other. Boyle-Mariotte's law is formulated as follows: In an ideal gas at constant temperature, the product of the pressure of the gas and its volume is also constant.
The Boyle-Mariotte 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 . It can be seen from the formula 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 decrease as the volume of a gas increases?
Since the temperature of the gas does not change, the frequency of impacts of molecules on the walls of the vessel does not change either. If the volume increases, then the concentration of molecules becomes smaller. Consequently, per unit area there will be a smaller number of 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 also increases.
Graphically, the isothermal process is displayed on the plane of the curve, which is called isotherm . She has the shape hyperbole.
Each temperature value has its own isotherm. The higher the temperature, the higher is the corresponding isotherm.
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 gas on its temperature at a constant pressure was also established experimentally French chemist and physicist Joseph Louis Gay-Lussac who published it in 1802. Therefore, it is called Gay-Lussac's law : " Etc and constant pressure, the ratio of the volume of a constant mass of a gas to its absolute temperature is a constant value.
At P = const the unified gas law equation becomes Gay-Lussac equation .
An example of an isobaric process is a gas inside a cylinder in which a piston moves. As the temperature rises, the frequency of molecular collisions with the walls increases. The pressure increases and the piston rises. As a result, the volume occupied by the gas in the cylinder increases.
Graphically, the isobaric process is represented by a straight line called isobar .
The higher the pressure in the gas, the lower the corresponding isobar is located on the graph.
Isochoric process
isochoric, or isochoric, called the process of changing the pressure and temperature of an ideal gas at a constant volume.
For isochoric process m = const, V = const.
It is very easy to imagine such a process. It takes place 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 law : « For a given mass of gas at constant volume, its pressure is proportional to temperature". The French inventor and scientist Jacques Alexandre Cesar Charles established this relationship with the help of experiments in 1787. In 1802 Gay-Lussac specified it. Therefore, this law is sometimes called Gay-Lussac's second law.
At V = const from the unified gas law equation we get the equation charles law, or Gay-Lussac's second law .
At constant volume, the pressure of a gas increases when its temperature increases. .
On the graphs, the isochoric process is displayed by a line called isochore .
The larger the volume occupied by the gas, the lower is the isochore corresponding to this volume.
In reality, no gas parameter can be kept constant. 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 molecules is much greater than their size. Therefore, their properties approach those of an ideal gas.
As already mentioned, the state of a certain mass of gas is determined by three thermodynamic parameters: pressure R, volume V and temperature T.
There is a certain relationship between these parameters, called state equation, which in general view is given by
f(R,V,T)=0,
where each of the variables is a function of the other two.
The French physicist and engineer B. Clapeyron (1799-1864) derived the equation of state for an ideal gas by combining the laws of Boyle - Mariotte and Gay-Lussac. Let some mass of gas occupy a volume V 1 , has pressure R 1 and is at a temperature T 1 . The same mass of gas in another arbitrary state is characterized by the parameters R 2 , V 2 , T 2 (fig.63). State Transition 1 into a state 2 is carried out in the form of two processes: 1) isothermal (isotherm 1 -1 "), 2) isochoric (isochore 1 "-2).
In accordance with the laws of Boyle - Mariotte (41.1) and Gay-Lussac (41.5), we write:
p 1 V 1 =p" 1 V 2 , (42.1)
p" 1 /p" 2 \u003d T 1 /T 2. (42.2)
Eliminating from equations (42.1) and (42.2) R" 1 , we get
p 1 V 1 /T 1 =p 2 V 2 / T 2 .
Since the states 1 and 2 were chosen arbitrarily, then for a given mass of gas
magnitude pV/T remains constant
pV/T=B=const.(42.3)
Expression (42.3) is Clapeyron's equation, wherein AT is the gas constant, different for different gases.
The Russian scientist D. I. Mendeleev (1834-1907) combined Clapeyron's equation with Avogadro's law, referring equation (42.3) to one mole, using the molar volume V t . According to Avogadro's law, for the same R and T moles of all gases occupy the same molar volume V m , so constant AT will be the same for all gases. This common constant for all gases is denoted R and called molar gas constant. Equation
pV m =RT(42.4)
satisfies only an ideal gas, and it is the ideal gas equation of state, also called the Clapeyron-Mendeleev equation.
The numerical value of the molar gas constant is determined from formula (42.4), assuming that a mole of gas is under normal conditions (R 0 = 1.013 10 5 Pa, T 0 \u003d 273.15 K:, V m \u003d 22.41 10 -3 m 3 / mol): R \u003d 8.31 J / (mol K).
From equation (42.4) for a mole of gas, one can pass to the Clapeyron-Mendeleev equation for an arbitrary mass of gas. If at certain given pressures and temperatures one mole of gas occupies a molar volume l/m, then under the same conditions the mass tons of gas will take the volume V = (m/M) V m , where M- molar mass(mass of one mole of substance). The unit of molar mass is the kilogram per mole (kg/mol). Clapeyron - Mendeleev equation for mass tons of gas
where v = m/M- amount of substance.
A slightly different form of the ideal gas equation of state is often used, introducing Boltzmann constant:
k \u003d R / N A \u003d 1.38 10 -2 3 J / K.
Proceeding from this, we write the equation of state (42.4) in the form
p = RT/V m = kN A T/V m = nkT,
where N A / V m = n-concentration of molecules (number of molecules per unit volume). Thus, from the equation
p = nkT(42.6)
it follows that the pressure of an ideal gas at a given temperature is directly proportional to the concentration of its molecules (or the density of the gas). At the same temperature and pressure, all gases contain the same number of molecules per unit volume. The number of molecules contained in 1 m 3 of gas at normal conditions called numberLoshmidt :
N L = P0 /(kT 0 ) = 2.68 10 25 m -3.
We take the formula and substitute in it. We get:
p= nkT.
Recall now that A , where ν - number of moles of gas:
pV= νRT.(3)
Relation (3) is called the Mendeleev-Clapeyron equation. It gives the relationship of the three most important macroscopic parameters that describe the state of an ideal gas - pressure, volume and temperature. Therefore, the Mendeleev-Clapeyron equation is also called ideal gas equation of state.
Given that where m- mass of gas, we get another form of the Mendeleev - Clapeyron equation:
There is another useful version of this equation. Let's divide both parts into V:
But - the density of the gas. From here
In problems in physics, all three forms of writing (3) - (5) are actively used.
isoprocesses
Throughout this section, we will adhere to the following assumption: mass and chemical composition gases remain unchanged. In other words, we believe that:
m= const, that is, there is no gas leakage from the vessel or, conversely, gas inflow into the vessel;
µ = const, that is, gas particles do not experience any changes (say, there is no dissociation - the decay of molecules into atoms).
These two conditions are satisfied in very many physically interesting situations (for example, in simple models of heat engines) and therefore deserve a separate consideration.
If the mass of a gas and its molar mass are fixed, then the state of the gas is determined by three macroscopic parameters: pressure, volume and temperature. These parameters are related to each other by the equation of state (the Mendeleev-Clapeyron equation).
Thermodynamic process
Thermodynamic process(or simply process) is the change in the state of the gas over time. During the thermodynamic process, the values of macroscopic parameters change - pressure, volume and temperature.
Of particular interest are isoprocesses- thermodynamic processes in which the value of one of the macroscopic parameters remains unchanged. Fixing each of the three parameters in turn, we get three types of isoprocesses.
1. Isothermal process runs at a constant gas temperature: T= const.
2. isobaric process runs at constant gas pressure: p= const.
3. Isochoric process runs at a constant volume of gas: V= const.
Isoprocesses are described by very simple laws of Boyle - Mariotte, Gay-Lussac and Charles. Let's move on to studying them.
Isothermal process
In an isothermal process, the temperature of the gas is constant. During the process, only the pressure of the gas and its volume change.
Establish a relationship between pressure p and volume V gas in an isothermal process. Let the gas temperature be T. Let us consider two arbitrary states of the gas: in one of them, the values of the macroscopic parameters are equal to p 1 ,V 1 ,T, and in the second p 2 ,V 2 ,T. These values are related by the Mendeleev-Clapeyron equation:
As we said from the very beginning, the mass of gas m and its molar mass µ assumed to be unchanged. Therefore, the right parts of the written equations are equal. Therefore, the left-hand sides are also equal: p 1V 1 = p 2V 2.
Since the two states of the gas were chosen arbitrarily, we can conclude that during an isothermal process, the product of gas pressure and volume remains constant:
pV= const .
This statement is called Boyle's Law - Mariotte. Having written the Boyle-Mariotte law in the form
p= ,
one can also formulate it like this: In an isothermal process, the pressure of a gas is inversely proportional to its volume.. If, for example, during isothermal expansion of a gas, its volume increases three times, then the pressure of the gas decreases three times.
How to explain the inverse relationship between pressure and volume from a physical point of view? At a constant temperature, the average remains unchanged kinetic energy gas molecules, that is, simply speaking, the force of impacts of molecules on the walls of the vessel does not change. With an increase in volume, the concentration of molecules decreases, and, accordingly, the number of molecular impacts per unit time per unit area of the wall decreases - the gas pressure drops. On the contrary, with a decrease in volume, the concentration of molecules increases, their impacts are more frequent, and the pressure of the gas increases.
As already mentioned, the state of a certain mass is determined by three thermodynamic parameters: pressure p, volume V and temperature T. There is a certain relationship between these parameters, called equation of state.
The French physicist B. Clapeyron derived the equation of state for an ideal gas by combining the laws of Boyle-Mariotte and Gay-Lussac.
1) isothermal (isotherm 1-1¢),
2) isochoric (isochore 1¢-2).
In accordance with the laws of Boyle-Mariotte (1.1) and Gay-Lussac (1.4), we write:
Eliminating p 1 " from equations (1.5) and (1.6), we obtain
Since states 1 and 2 were chosen arbitrarily, for a given mass of gas, the value remains constant, i.e.
. (1.7)
Expression (1.7) is the Clapeyron equation, in which B is the gas constant, which is different for different gases.
The Russian scientist DIMendeleev combined Clapeyron's equation with Avogadro's law, referring equation (1.7) to one mole, using the molar volume V m . According to Avogadro's law, for the same p and T, moles of all gases occupy the same molar volume V m, so the constant B will be the same for all gases. This common constant for all gases is denoted R and is called molar gas constant. Equation
satisfies only an ideal gas, and it is ideal gas equation of state also called the Mendeleev-Clapeyron equation.
Numeric value molar gas constant is determined from formula (1.8), assuming that a mole of gas is at normal conditions(p 0 =1.013×10 5 Pa, T 0 =273.15 K, V m =22.41×10 -3 m 3 /mol): R=8.31 J/(mol K).
From equation (1.8) for a mole of gas, one can pass to the Clapeyron-Mendeleev equation for an arbitrary mass of gas. If, at a given pressure and temperature, one mole of gas occupies a volume V m, then under the same conditions, the mass m of gas will occupy a volume, where M - molar mass(mass of one mole of substance). Unit molar mass- kilogram per mole (kg/mol). Clapeyron-Mendeleev equation for mass m of gas
where is the amount of matter.
A slightly different form of the ideal gas equation of state is often used, introducing Boltzmann's constant:
Proceeding from this, we write the equation of state (1.8) in the form
where is the concentration of molecules (the number of molecules per unit volume). Thus, from the equation
p=nkT (1.10)
it follows that the pressure of an ideal gas at a given temperature is directly proportional to the concentration of its molecules (or gas density). At the same temperature and pressure, all gases contain the same number of molecules per unit volume. The number of molecules contained in 1 m 3 of gas under normal conditions is called Loschmidt number:
Basic equation of molecular kinetic
Theories of ideal gases
To derive the basic equation of the molecular kinetic theory, we consider a monatomic ideal gas. Let us assume that the gas molecules move randomly, the number of mutual collisions between them is negligibly small compared to the number of impacts on the walls of the vessel, and the collisions of the molecules with the walls of the vessel are absolutely elastic. Let's select some elementary area DS on the wall of the vessel (Fig. 50) and calculate the pressure exerted on this area.
During the time Dt, only those molecules reach the platform DS that are contained in the volume of the cylinder with the base DS and the height Dt (Fig. 50).
The number of these molecules is equal to nDSDt (n-concentration of molecules). However, it must be taken into account that the molecules actually move towards the area DS under different angles and have different speeds, and the speed of the molecules changes with each collision. To simplify the calculations, the chaotic motion of molecules is replaced by motion along three mutually perpendicular directions, so that at any time 1/3 of the molecules move along each of them, with half (1/6) moving along this direction in one direction, half in the opposite direction. Then the number of impacts of molecules moving in a given direction on the platform DS will be 1/6nDS Dt. When colliding with the platform, these molecules will transfer momentum to it.
As already mentioned, the state of a certain mass of gas is determined by three thermodynamic parameters: pressure R, volume V and temperature T. There is a certain relationship between these parameters, called the equation of state, which is generally given by the expression: Fig.7.4.
F(p,V, T)=0,
where each of the variables is a function of the other two.
The French physicist and engineer B. Clapeyron derived the equation of state for an ideal gas by combining the laws of Boyle - Mariotte and Gay-Lussac. Let some mass of gas occupy a volume V 1 , has pressure R 1 and is at a temperature T one . The same mass of gas in another arbitrary state is characterized by the parameters R 2 ,V 2 ,T 2 (fig.7.4).
The transition from state 1 to state 2 is carried out in the form of two processes: 1) isothermal (isotherm 1 - 1 /), 2) isochoric (isochore 1 / – 2).
In accordance with the laws of Boyle-Mariotte (7.1) and Gay-Lussac (7.5), we write:
R 1 V 1 =p / 1 V 2 , (7.6)
. (7.7)
Eliminating from equations (7.6) and (7.7) p/ 1 we get:
Since states 1 and 2 were chosen arbitrarily, for a given mass of gas, the value pV/T remains constant, i.e.
pV/T= AT= const. (7.8)
Expression (7.8) is Clapeyron's equation, wherein AT- gas constant, different for different gases.
D. I. Mendeleev combined Clapeyron's equation with Avogadro's law, referring equation (7.8) to one mole, using the molar volume V m . According to Avogadro's law, for the same p and Τ moles of all gases occupy the same molar volume Vm, so the constant AT will be the same for all gases . This common constant for all gases is denoted R and called molar gas constant. Equation
pV m = RT(7.9)
satisfies only an ideal gas, and it is ideal gas equation of state also called Clapeyron - Mendeleev equation.
The numerical value of the molar gas constant is determined from formula (7.9), assuming that a mole of gas is under normal conditions ( R 0 = 1.013×10 5 Pa, T 0 \u003d 273.15 K, Vm\u003d 22.41 × 10 -3 m 3 / mol): R\u003d 8.31 J / (mol K).
From equation (7.9) for a mole of gas, one can pass to the Clapeyron-Mendeleev equation for an arbitrary mass of gas. If for some given p and T one mole of gas occupies a molar volume V m , then mass t gas will take the volume V=(m/M)Vm,where Μ – molar mass(mass of one mole of substance). The unit of molar mass is the kilogram per mole (kg/mol). Clapeyron - Mendeleev equation for mass t gas
pV= RT= vRT,(7.10)
where: v=m/M- amount of substance.
A slightly different form of the ideal gas equation of state is often used, introducing Boltzmann's constant
k=R/N A= 1.38∙10 -23 J/K.
Proceeding from this, we write the equation of state (2.4) in the form
p=RT/Vm= kN A T/V m= nkT,
where N A / V m \u003d n- concentration of molecules(number of molecules per unit volume). Thus, from the equation
p=nkT(7.11)
it follows that the pressure of an ideal gas at a given temperature is directly proportional to the concentration of its molecules (or gas density). At the same temperature and pressure, all gases contain the same number of molecules per unit volume. The number of molecules contained in 1m 3 of gas under normal conditions , called Loschmidt number:
N l \u003d p 0 / (kT 0)= 2.68∙10 25 m -3.