The amount of air in the cylinders depends on the volume of the cylinder, the air pressure and its temperature. The ratio between air pressure and its volume at a constant temperature is determined by the relationship
where р1 and р2 - initial and final absolute pressure, kgf/cm²;
V1 and V2 - initial and final volume of air, l. The ratio between air pressure and its temperature at a constant volume is determined by the relationship
where t1 and t2 are the initial and final air temperatures.
Using these dependencies, it is possible to solve various problems that one has to face in the process of charging and operating air-breathing apparatuses.
Example 4.1. The total capacity of the cylinders of the device is 14 liters, the excess air pressure in them (by pressure gauge) is 200 kgf / cm². Determine the volume of free air, i.e., the volume reduced to normal (atmospheric) conditions.
Decision. Initial absolute pressure of atmospheric air p1 = 1 kgf/cm². Final absolute pressure of compressed air р2 = 200 + 1= 201 kgf/cm². The final volume of compressed air V 2=14 l. Volume of free air in cylinders according to (4.1)
Example 4.2. From a transport cylinder with a capacity of 40 l with a pressure of 200 kgf / cm² (absolute pressure 201 kgf / cm²), air was passed into the cylinders of the apparatus with a total capacity of 14 l and with a residual pressure of 30 kgf / cm² (absolute pressure 31 kgf / cm²). Determine the air pressure in the cylinders after air bypass.
Decision. The total volume of free air in the system of transport and equipment cylinders according to (4.1)
The total volume of compressed air in the cylinder system
Absolute pressure in the cylinder system after air bypass
excess pressure = 156 kgf / cm².
This example can also be solved in one step by calculating the absolute pressure using the formula
Example 4.3. When measuring the air pressure in the cylinders of the device in a room with a temperature of +17 ° C, the pressure gauge showed 200 kgf / cm². The device was taken outside, where a few hours later, during a working check, a pressure drop on the pressure gauge to 179 kgf / cm² was found. The outside air temperature is -13°C. There was a suspicion of air leakage from the cylinders. Check the validity of this suspicion by calculation.
Decision. Initial absolute air pressure in cylinders p1 = 200 + 1 = 201 kgf/cm², final absolute pressure p2 = 179 + 1 = 180 kgf/cm². Initial air temperature in cylinders t1 = + 17° C, final temperature t2 = - 13° C. Estimated final absolute air pressure in cylinders according to (4.2)
Suspicions are unfounded, since the actual and calculated pressure are equal.
Example 4.4. A diver under water consumes 30 l / min of air compressed to a pressure of a diving depth of 40 m. Determine the flow rate of free air, i.e., convert to atmospheric pressure.
Decision. Initial (atmospheric) absolute air pressure p1 = l kgf/cm². The final absolute pressure of compressed air according to (1.2) p2 \u003d 1 + 0.1 * 40 \u003d 5 kgf / cm². Final compressed air consumption V2 = 30 l/min. Free air flow according to (4.1)
Ideal gas law.
Experimental:
The main parameters of a gas are temperature, pressure and volume. The volume of a gas essentially depends on the pressure and temperature of the gas. Therefore, it is necessary to find the relationship between the volume, pressure and temperature of the gas. This ratio is called equation of state.
It was experimentally found that for a given amount of gas, in a good approximation, the relation is fulfilled: at a constant temperature, the volume of a gas is inversely proportional to the pressure applied to it (Fig. 1):
V~1/P , at T=const.
For example, if the pressure acting on a gas is doubled, then the volume will decrease to half of the original. This ratio is known as Boyle's law (1627-1691)-Mariotte(1620-1684), it can also be written like this:
This means that when one of the quantities changes, the other will also change, and in such a way that their product remains constant.
The dependence of volume on temperature (Fig. 2) was discovered by J. Gay-Lussac. He discovered that At constant pressure, the volume of a given amount of gas is directly proportional to the temperature:
V~T, when P = const.
The graph of this dependence passes through the origin of coordinates and, accordingly, at 0K its volume will become equal to zero, which obviously has no physical meaning. This has led to the assumption that -273 0 C is the lowest temperature that can be reached.
The third gas law, known as charles law, named after Jacques Charles (1746-1823). This law says: at a constant volume, the gas pressure is directly proportional to the absolute temperature (Fig. 3):
Р ~T, at V=const.
A well-known example of this law is the aerosol can that explodes in a fire. This is due to a sharp increase in temperature at a constant volume.
These three laws are experimental and hold well in real gases only as long as the pressure and density are not very high, and the temperature is not too close to the condensation temperature of the gas, so the word "law" is not very suitable for these properties of gases, but it has become generally accepted.
The gas laws of Boyle-Mariotte, Charles and Gay-Lussac can be combined into one more general relationship between volume, pressure and temperature, which is valid for a certain amount of gas:
This shows that when one of the values P, V or T changes, the other two values will also change. This expression goes into these three laws, when one value is taken constant.
Now we should take into account one more quantity, which until now we have considered constant - the amount of this gas. It has been experimentally confirmed that: at constant temperature and pressure, the closed volume of a gas increases in direct proportion to the mass of this gas:
This dependence connects all the main quantities of the gas. If we introduce the coefficient of proportionality into this proportionality, then we get equality. However, experiments show that this coefficient is different in different gases, therefore, instead of mass m, the amount of substance n (the number of moles) is introduced.
As a result, we get:
Where n is the number of moles and R is the proportionality factor. The value R is called universal gas constant. To date, the most accurate value of this value is:
R=8.31441 ± 0.00026 J/mol
Equality (1) is called ideal gas equation of state or ideal gas law.
Avogadro's number; ideal gas law at the molecular level:
That the constant R has the same value for all gases is a magnificent reflection of the simplicity of nature. This was first realized, although in a slightly different form, by the Italian Amedeo Avogadro (1776-1856). He experimentally established that equal volumes of gas at the same pressure and temperature contain the same number of molecules. First: from equation (1) it can be seen that if different gases contain an equal number of moles, have the same pressures and temperatures, then, under the condition of constant R, they occupy equal volumes. Secondly: the number of molecules in one mole is the same for all gases, which directly follows from the definition of the mole. Therefore, we can state that the value of R is constant for all gases.
The number of molecules in one mole is called Avogadro's numberN A. It is now established that Avogadro's number is:
N A \u003d (6.022045 ± 0.000031) 10 -23 mol -1
Since the total number of molecules N of a gas is equal to the number of molecules in one mole times the number of moles (N = nN A), the ideal gas law can be rewritten as follows:
Where k is called Boltzmann constant and has a value equal to:
k \u003d R / N A \u003d (1.380662 ± 0.000044) 10 -23 J / K
Directory of compressor technology
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? |
| Decision | 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%? |
| Decision | 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. |
| Decision | The isothermal process is described by the Boyle-Mariotte law: |
Relationship between pressure, temperature, volume and number of moles of gas (the "mass" of gas). Universal (molar) gas constant R. Klaiperon-Mendeleev equation = ideal gas equation of state.
|
Limitations of practical applicability:
Within the range, the accuracy of the equation is superior to that of conventional modern engineering instruments. It is important for the engineer to understand that all gases can undergo significant dissociation or decomposition as the temperature rises. |
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Let's solve a couple of gas volume and mass flow problems assuming that the composition of the gas does not change (gas does not dissociate) - which is true for most of the gases in the above.
This problem is relevant mainly, but not only, for applications and devices in which the volume of gas is directly measured.
V 1 and V 2, at temperatures, respectively, T1 and T2 let it go T1< T2. Then we know that:
Naturally, V 1< V 2
- the indicators of a volumetric gas meter are the more "weighty" the lower the temperature
- profitable supply of "warm" gas
- profitable to buy "cold" gas
How to deal with it? At least a simple temperature compensation is required, i.e. information from an additional temperature sensor must be fed into the counting device.
This problem is relevant mainly, but not only, for applications and devices in which the gas velocity is directly measured.
Let the counter () at the delivery point give the volume accumulated costs V 1 and V 2, at pressures, respectively, P1 and P2 let it go P1< P2. Then we know that:
Naturally, V 1>V 2 for equal amounts of gas under given conditions. Let's try to formulate some practical conclusions for this case:
- the indicators of the volumetric gas meter are the more "weighty" the higher the pressure
- profitable supply of low pressure gas
- profitable to buy high pressure gas
How to deal with it? At least a simple pressure compensation is required, i.e. information from an additional pressure sensor must be supplied to the counting device.
In conclusion, I would like to note that, theoretically, each gas meter should have both temperature compensation and pressure compensation. Practically....
Annotation: traditional presentation of the topic, supplemented by a demonstration on a computer model.
Of the three aggregate states of matter, the simplest is the gaseous state. In gases, the forces acting between molecules are small and under certain conditions they can be neglected.
The gas is called perfect , if:
The size of molecules can be neglected, i.e. molecules can be considered material points;
We can neglect the forces of interaction between molecules (the potential energy of interaction of molecules is much less than their kinetic energy);
The collisions of molecules with each other and with the walls of the vessel can be considered absolutely elastic.
Real gases are close in properties to the ideal at:
Conditions close to normal conditions (t = 0 0 C, p = 1.03 10 5 Pa);
At high temperatures.
The laws that govern the behavior of ideal gases were discovered experimentally quite a long time ago. So, Boyle's law - Mariotte was established in the 17th century. We give the formulations of these laws.
Boyle's Law - Mariotte. Let the gas be under conditions where its temperature is kept constant (such conditions are called isothermal ). Then for a given mass of gas, the product of pressure and volume is a constant value:
This formula is called isotherm equation. Graphically, the dependence of p on V for various temperatures is shown in the figure.
The property of a body to change pressure with a change in volume is called compressibility. If the change in volume occurs at T=const, then the compressibility is characterized by isothermal compressibility factor which is defined as the relative change in volume that causes a change in pressure per unit.
For an ideal gas, it is easy to calculate its value. From the isotherm equation we get:
The minus sign indicates that as the volume increases, the pressure decreases. Thus, the isothermal compressibility of an ideal gas is equal to the reciprocal of its pressure. With increasing pressure, it decreases, because. the greater the pressure, the less the gas has the ability to further compress.
Gay-Lussac law. Let the gas be under conditions where its pressure is maintained constant (such conditions are called isobaric ). They can be carried out by placing gas in a cylinder closed by a movable piston. Then a change in the temperature of the gas will move the piston and change the volume. The pressure of the gas will remain constant. In this case, for a given mass of gas, its volume will be proportional to the temperature:
where V 0 - volume at temperature t = 0 0 C, - volume expansion coefficient gases. It can be represented in a form similar to the compressibility factor:
Graphically, the dependence of V on T for various pressures is shown in the figure.
Moving from temperature in the Celsius scale to absolute temperature, Gay-Lussac's law can be written as:
Charles' Law. If the gas is under conditions where its volume remains constant ( isochoric conditions), then for a given mass of gas, the pressure will be proportional to the temperature:
where p 0 - pressure at temperature t \u003d 0 0 C, - pressure coefficient. It shows the relative increase in gas pressure when it is heated by 10:
Charles' law can also be written as:
Avogadro's law: One mole of any ideal gas at the same temperature and pressure occupies the same volume. Under normal conditions (t = 0 0 C, p = 1.03 10 5 Pa), this volume is equal to m -3 / mol.
The number of particles contained in 1 mole of various substances, called. Avogadro's constant :
It is easy to calculate the number n 0 particles in 1 m 3 under normal conditions:
This number is called Loschmidt number.
Dalton's law: the pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the gases included in it, i.e.
where - partial pressures- the pressure that the components of the mixture would exert if each of them occupied a volume equal to the volume of the mixture at the same temperature.
Equation of Clapeyron - Mendeleev. From the laws of an ideal gas, one can obtain equation of state , linking T, p and V of an ideal gas in a state of equilibrium. This equation was first obtained by the French physicist and engineer B. Clapeyron and Russian scientists D.I. Mendeleev, therefore bears their name.
Let some mass of gas occupies a volume V 1 , has a pressure p 1 and is at a temperature T 1 . The same mass of gas in a different state is characterized by the parameters V 2 , p 2 , T 2 (see figure). The transition from state 1 to state 2 is carried out in the form of two processes: isothermal (1 - 1") and isochoric (1" - 2).
For these processes, one can write down the laws of Boyle - Mariotte and Gay - Lussac:
Eliminating p 1 " from the equations, we get
Since states 1 and 2 were chosen arbitrarily, the last equation can be written as:
This equation is called Clapeyron's equation , in which B is a constant, different for different masses of gases.
Mendeleev combined Clapeyron's equation with Avogadro's law. According to Avogadro's law, 1 mole of any ideal gas at the same p and T occupies the same 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 universal gas constant. Then
This equation is ideal gas equation of state , which is also called Clapeyron - Mendeleev equation .
The numerical value of the universal gas constant can be determined by substituting the values of p, T and V m into the Clapeyron - Mendeleev equation under normal conditions:
The Clapeyron - Mendeleev equation can be written for any mass of gas. To do this, recall that the volume of a gas of mass m is related to the volume of one mole by the formula V \u003d (m / M) V m, where M is molar mass of gas. Then the Clapeyron - Mendeleev equation for a gas of mass m will look like:
where is the number of moles.
The equation of state for an ideal gas is often written in terms of Boltzmann's constant :
Based on this, the equation of state can be represented as
where is the concentration of molecules. From the last equation it can be seen that the pressure of an ideal gas is directly proportional to its temperature and concentration of molecules.
Small demo ideal gas laws. After pressing the button "Let's start" You will see the host's comments on what is happening on the screen (black color) and a description of the computer's actions after you press the button "Further"(Brown color). When the computer is "busy" (i.e., experience is in progress), this button is not active. Move on to the next frame only after understanding the result obtained in the current experiment. (If your perception does not match the host's comments, write!)
You can verify the validity of the ideal gas laws on the existing
