As a rule, as the temperature decreases, the density increases, although there are substances whose density behaves differently, for example, water, bronze and cast iron. Thus, the density of water has a maximum value at 4 °C and decreases both with increasing and decreasing temperature relative to this number.
When the state of aggregation changes, the density of a substance changes abruptly: the density increases during the transition from a gaseous state to a liquid and when the liquid solidifies. True, water is an exception to this rule; its density decreases as it solidifies.
The ratio of the P. of two substances under certain standard physical conditions is called relative P.: for liquid and solid substances it is usually determined in relation to the P. of distilled water at 4 °C, for gases - in relation to the P. of dry air or hydrogen under normal conditions.
The SI unit of P. is kg/m 3 , in the CGS system of units g/cm 3 . In practice, non-systemic P units are also used: g/l, t/m 3 and etc.
Density meters, pycnometers, hydrometers, and hydrostatic weighing are used to measure the density of substances (see Mora balances). . Dr. methods for determining density are based on the connection of density with the parameters of the state of a substance or with the dependence of processes occurring in a substance on its density. Thus, density ideal gas can be calculated by equation of state r = pm/RT, where p is the gas pressure, m is its molecular mass (molar mass), R - gas constant , T - absolute temperature, or determined, for example, by the speed of propagation of ultrasound (here b is adiabatic compressibility gas).
The range of P values of natural bodies and environments is exceptionally wide. Thus, the density of the interstellar medium does not exceed 10 -21 kg/m 3 , the average P. of the Sun is 1410 kg/m 3 , Earth - 5520 kg/m 3 , highest P. metals - 22,500 kg/m 3 (osmium), P. substances of atomic nuclei - 10 17 kg/m 3 , finally, the density of neutron stars can apparently reach 10 20 kg/m 3 .
Pressure gauge is a mechanical measuring device, structurally consisting of a steel or plastic dial with a spring in the form of a tube, designed to measure the pressure of liquid and gaseous substances.
In mechanical pressure gauges, the measured pressure with the help of a sensing element is converted into mechanical movement, causing mechanical deflection of the arrows or other parts of the counting mechanisms, recording the measurement results, as well as signaling devices and pressure stabilization in the systems of the controlled object. Tubular springs, harmonic (bellows) and flat membranes and other measuring mechanisms are used as sensitive elements of mechanical pressure gauges, in which elastic deformations or elasticity of special springs are caused under the influence of pressure.
According to accuracy, all mechanical pressure gauges are divided into: technical, control and standard. Technical pressure gauges have accuracy classes 1.5; 2.5; 4; control 0.5; 1.0; exemplary 0.16; 0.45.
Gauge tubular springs are hollow tubes of oval or other cross-section, bent along a circular arc, along a helical or spiral line and having one or more turns. The usual design, which is most often used in practice, uses single-turn springs. The schematic and structural diagrams of a pressure gauge with a single-turn tubular spring are presented in Fig. 2.
Fig.2. Mechanical pressure gauge and its characteristics
The end of the pressure spring 5 is soldered to fitting 1. The second soldered end K is hingedly connected by a rod 3 to the lever of the gear sector 4. The teeth of the sector are engaged with the driven gear 6, which is mounted on the axis 7 of the arrows 9. To eliminate vibrations of the arrow due to the gaps between the teeth The gear train uses a spiral spring 2, the ends of which are connected to the housing and axis 7. There is a fixed scale under the arrow.
Under the influence of the pressure difference inside and outside, the tubular spring changes the shape of its cross-section, as a result of which its sealed end K moves in proportion to the operating pressure difference.
The structural diagram of a mechanical pressure gauge (Fig. 2, b) consists of three linear links I, II, III, the static characteristics of which are presented by graphs, and, where is the movement of the free end of the tubular spring, is the initial central angle of the tubular spring. Due to the linearity of all links, the overall static characteristic of the pressure gauge is linear and the scale is uniform. The input value of link I is the measured pressure, and the output value is the movement of the free (soldered) end of the gauge spring5. Rod 3 with gear sector lever 4 forms the second link. The input value of link II is , and the output value is the angular deviation of the end of the manometric spring. The input value of link III (link III is a gear sector meshed with the driven gear 6) is the angular deviation, and the output is the angular deviation of the pointer 9 from the zero mark of scale 8.
Mechanical pressure gauges are used for measurements in the low vacuum region. In deformation pressure gauges, the elastic element associated with the indicator bends under the influence of the difference between the measured and reference pressures (atmosphere or high vacuum). In bellows industrial pressure gauges of the BC-7 series, the measured pressure causes movement of the bellows, which is transmitted to the recorder. These devices have a linear scale up to 760 torr and an accuracy of 1.6%.
PHYSICAL PROPERTIES OF GASES
1. Gas density – mass of 1 m 3 of gas at a temperature of 0 0 and a pressure of 0.1 MPa (760 mm Hg). The density of a gas depends on pressure and temperature. The density of gases varies within the range of 0.55 - 1 g/cm3.
Commonly used relative density by air (dimensionless value - the ratio of gas density to air density; under normal conditions, air density is 1.293 kg/m3).
2. Viscosity of gases – internal friction of gases that occurs during its movement. The viscosity of gases is very low 1 . 10 -5 Pa.s. Such a low viscosity of gases ensures their high mobility through cracks and pores.
3. Solubility of gases – one of the most important properties. The solubility of gases in oil or water at a pressure of no more than 5 MPa is subject to Henry's law, i.e. the amount of dissolved gas is directly proportional to pressure and solubility coefficient.
At higher pressures, gas solubility is determined by a number of indicators: temperature, chemical composition, groundwater mineralization, etc. The solubility of hydrocarbon gases in oils is 10 times greater than in water. Wet gas is more soluble in oil than dry gas. Lighter oil dissolves more gas than heavier oil.
4. Critical gas temperature. For each gas there is a temperature above which it does not transform into a liquid state, no matter how high the pressure is, i.e. critical t(for CH 4 t cr = –82.1 0 C). Homologues of methane can be in a liquid state (for C 2 H 6 t cr = 32.2 0 C, C 3 H 8 t cr = 97.0 0 C).
5. Diffusion is the spontaneous movement of gases at the molecular level in the direction of decreasing concentrations.
6. Volumetric coefficient of reservoir gas is the ratio of the volume of gas under reservoir conditions to the volume of the same gas under standard conditions
(T = 0 0 and P = 0.1 MPa).
V g = V g pl / V g st
The volume of gas in the reservoir is 100 times less than under standard conditions, because gas is supercompressible.
GAS CONDENSATES
Not only can gas dissolve in oil, but oil can also dissolve in gas. This happens under certain conditions, namely:
1) the volume of gas is greater than the volume of oil;
2) pressure 20-25 MPa;
3) temperature 90-95 0 C.
Under these conditions, liquid hydrocarbons begin to dissolve in the gas. Gradually the mixture completely turns into gas. This phenomenon is called retrograde evaporation. When one of the conditions changes, for example, when the reservoir pressure decreases during development, condensate in the form of liquid hydrocarbons begins to release from this mixture. Its composition: C 5, H 12 (pentane) and higher. This phenomenon is called retrograde condensation.
Gas condensate is the liquid part of gas condensate accumulations. Gas condensates are called light oils, since they do not contain asphalt-resinous substances. The density of gas condensate is 0.65-0.71 g/cm3. The density of gas condensates increases with depth, and it also changes (usually increases) during development.
There are raw condensate and stable condensate.
Crude is a liquid phase extracted to the surface in which gaseous components are dissolved. Crude condensate is obtained directly in field separators at separation pressures and temperatures.
Stable gas condensate is obtained from raw gas by degassing it; it consists of liquid hydrocarbons (pentane) and higher ones.
GAS HYDRATES
Most gases form crystalline hydrates with water - solids. These substances are called gas hydrates and are formed at low temperatures, high pressures and at shallow depths. In appearance they resemble loose ice or snow. Deposits of this type were found in permafrost areas of Western and Eastern Siberia and in the waters of the northern seas.
The problem of using gas hydrates has not yet been sufficiently developed. All issues of gas hydrate production come down to creating conditions in the formation under which gas hydrates would decompose into gas and water.
To do this you need:
1) decrease in pressure in the reservoir;
2) increase in temperature;
3) addition of special reagents.
Patterns and changes in the properties of oil and gas in reservoirs and fields
So as a result of physical and chemical changes in oils and gases that occur under the influence of water penetrating into deposits and changes in reservoir pressure and temperature. Therefore, for reasonable forecasts of changes in the properties of oil and gas during the development process, it is necessary to have clear ideas: a) about the patterns of changes in the properties of oil and gas by volume of the deposit before the start of development; b) about the processes of physical and chemical interaction of oils and gases with waters entering the productive formation (especially with injected waters of a different composition than formation water); c) about the directions of fluid movement in the productive formation as a result of well operation; d) changes in reservoir pressure and temperature during the period of reservoir development. Patterns of changes in the properties of oil and gas according to the volume of the deposit. Complete uniformity of the properties of oil and gas dissolved in it within one deposit is a rather rare phenomenon. For oil deposits, changes in properties are usually quite natural and manifest themselves primarily in an increase in density, including optical density, viscosity, content of asphalt-resinous substances, paraffin and sulfur as the depth of the formation increases, i.e. from the roof to the wings and from the top to the bottom in thick layers. The actual change in density within most deposits usually does not exceed 0.05-0.07 g/cm3. However, very often the density gradient and its absolute values increase sharply in the immediate vicinity of the oil-water contact. Often the oil density above the insulating layer is almost constant. In “open” type deposits, confined to layers exposed to the day surface, and sealed from the top with asphalt -kirk rocks, the density of oil decreases with increasing depth, reaches a minimum, and then increases as it approaches the OWC. The described patterns are most typical for high deposits of deposits in folded regions. The main reason for their formation is the gravitational differentiation (stratification) of oils by density within the deposit, similar to the stratification of gas, oil and water within the reservoir. A significant change in the properties of oils in the OWC zone and in the upper parts of open-type oil deposits is associated with oxidative processes.
For deposits in platform areas with a low oil-bearing level and an extensive OWC zone, gravitational stratification is much weaker and the main influence on changes in the properties of oils is exerted by oxidative processes in the zone underlain by bottom water.
Simultaneously with the increase in oil density, its viscosity, as a rule, the content of asphalt-resinous substances and paraffin increases, and the gas content and saturation pressure of dissolved gases decrease.
Despite the high diffusion activity of gases, variability in their composition within a single deposit is far from a rare phenomenon. It manifests itself most sharply in the content of acidic components - carbon dioxide CO 2 and especially hydrogen sulfide H 2 S. Zoning is usually observed in the distribution of hydrogen sulfide, expressed in a regular change in the concentrations of hydrogen sulfide over the area. There are usually no obvious regular changes in concentration along the height of the deposit.
Gas-condensate deposits without an oil rim with a low level of gas content and a low condensate-gas factor, as a rule, have a fairly stable gas composition, composition and yield of condensate. However, when the height of the gas-condensate deposit is more than 300 m, the processes of gravitational stratification begin to noticeably manifest themselves, leading to an increase in the condensate content down the dip of the formation, especially sharply for deposits with a high level of gas content and an oil rim. In this case, the condensate content in the lower areas of the deposit can be several times higher than in the roof of the deposit. In particular, examples are known when the condensate-gas factor in the wells of the near-water part of the deposit was 180 cm 3 /m 3, and near the gas-oil contact - 780 cm 3 / m 3, i.e., within one deposit, the condensate content varied by 4 times. Fluctuations of 1.5--2 times are common for many fields with high levels of gas content when the condensate yield is more than 100 cm 3 /m 3.
Copyrightã L.Kourenkov
Properties of gases
Gas pressure
Gas always fills a volume limited by walls that are impenetrable to it. For example, a gas cylinder or inner tube of a car tire is almost uniformly filled with gas.
Trying to expand, the gas puts pressure on the walls of the cylinder, tire tubes or any other body, solid or liquid, with which it comes into contact. If we do not take into account the action of the Earth's gravitational field, which with the usual sizes of the vessels only changes the pressure insignificantly, then when the gas pressure in the vessel is in equilibrium, it seems to us to be completely uniform. This remark applies to the macrocosm. If we imagine what happens in the microcosm of the molecules that make up the gas in the vessel, then there can be no talk of any uniform distribution of pressure. In some places on the surface of the wall, gas molecules strike the walls, while in other places there are no impacts. This picture changes all the time in a chaotic manner. Gas molecules strike the walls of the vessels and then fly away at a speed almost equal to the speed of the molecule before the impact. Upon impact, the molecule transfers to the wall an amount of motion equal to mv, where m is the mass of the molecule and v is its speed. Reflecting from the wall, the molecule imparts to it the same amount of motion mv. Thus, with each impact (perpendicular to the wall), the molecule transfers to it an amount of motion equal to 2mv. If in 1 second there are N impacts per 1 cm 2 of the wall, then the total amount of motion transferred to this section of the wall is equal to 2Nmv. By virtue of Newton's second law, this amount of motion is equal to the product of the force F acting on this section of the wall and the time t during which it acts. In our case t = 1 sec. So F=2Nmv, there is a force acting on 1 cm 2 walls, i.e. pressure, which is usually denoted by p (and p is numerically equal to F). So we have
р=2Nmv
It’s a no brainer that the number of blows in 1 second depends on the speed of the molecules, and the number of molecules n per unit volume. For a not very compressed gas, we can assume that N is proportional to n and v, i.e. p is proportional to nmv 2.
So, in order to calculate gas pressure using molecular theory, we must know the following characteristics of the microcosm of molecules: mass m, speed v and the number of molecules n per unit volume. In order to find these micro characteristics of molecules, we must establish on what characteristics of the macro world the gas pressure depends, i.e. establish experimentally the laws of gas pressure. By comparing these experimental laws with the laws calculated using molecular theory, we will be able to determine the characteristics of the microcosm, for example, the speed of gas molecules.
So, let's establish what gas pressure depends on?
Firstly, on the degree of gas compression, i.e. depends on how many gas molecules are in a certain volume. For example, by inflating a tire or squeezing it, we force the gas to press harder on the inner tube walls.
Secondly, it depends on what the temperature of the gas is.
Typically, a change in pressure is caused by both reasons at once: a change in volume and a change in temperature. But it is possible to carry out the phenomenon in such a way that when the volume changes, the temperature will change negligibly, or when the temperature changes, the volume remains practically unchanged. We will deal with these cases first, having first made the following remark.
We will consider gas in a state of balance. This means; that both mechanical and thermal equilibrium have been established in the gas.
Mechanical equilibrium means that there is no movement of individual parts of the gas. To do this, it is necessary that the gas pressure be the same in all its parts, if we neglect the slight difference in pressure in the upper and lower layers of the gas that occurs under the influence of gravity.
Thermal equilibrium means that there is no transfer of heat from one part of the gas to another. To do this, it is necessary that the temperature throughout the entire volume of gas be the same.
Dependence of gas pressure on temperature
Let's start by finding out the dependence of gas pressure on temperature, provided that the volume of a certain mass of gas remains constant. These studies were first carried out in 1787 by Charles. These experiments can be reproduced in a simplified form by heating the gas in a large flask connected to a mercury manometer in the form of a narrow curved tube.
Let us neglect the insignificant increase in the volume of the flask when heated and the insignificant change in volume when the mercury is displaced in a narrow manometric tube. Thus, the volume of gas can be considered constant. By heating the water in the vessel surrounding the flask, we will note the temperature of the gas using a thermometer , and the corresponding pressure - according to the pressure gauge . Having filled the vessel with melting ice, measure the pressure corresponding to the temperature 0°C .
Experiments of this kind showed the following:
1. The increase in pressure of a certain mass of gas when heated by 1° is a certain part a of the pressure that this mass of gas had at a temperature of 0°C. If the pressure at 0°C is denoted by P, then the increase in gas pressure when heated by 1°C is aP.
When heated by t degrees, the pressure increment will be t times greater, i.e., the pressure increment proportional to the temperature increase.
2. The value a, showing by what part of the pressure at 0°C the gas pressure increases when heated by 1°, has the same value (more precisely, almost the same) for all gases, namely . The quantity a is called thermal, pressure coefficient. Thus, the thermal pressure coefficient for all gases has the same value, equal to .
The pressure of a certain mass of gas when heated to 1° V constant volume increases by part of the pressure at 0°C (Charles law).
It should be borne in mind, however, that the temperature coefficient of gas pressure obtained by measuring the temperature with a mercury thermometer is not exactly the same for different temperatures: Charles’s law is satisfied only approximately, although with a very high degree of accuracy.
Formula expressing Charles's law.
Charles's law allows you to calculate the pressure of a gas at any temperature if its pressure at 0°C is known. Let the pressure at 0°C of a given mass of gas in a given volume be , and the pressure of the same gas at temperature t There is p. There is a temperature increase t, therefore, the pressure increment is a t and the desired pressure is
P = +a t=(1+a t )= (1+ ) (1)
This formula can also be used if the gas is cooled below 0°C; wherein t will have negative values. At very low temperatures, when the gas approaches the state of liquefaction, as well as in the case of highly compressed gases, Charles’ law is not applicable and formula (1) ceases to be valid.
Charles's law from the point of view of molecular theory
What happens in the microcosm of molecules when the temperature of a gas changes, for example when the temperature of the gas rises and its pressure increases? From the point of view of molecular theory, there are two possible reasons for the increase in pressure of a given gas: firstly, the number of impacts of molecules could increase by 1 cm 2 for 1 sec; secondly, the amount of motion transmitted when one molecule hits a wall could increase. Both reasons require an increase in the speed of molecules. From here it becomes clear that an increase in gas temperature (in the macrocosm) is an increase in the average speed of the random movement of molecules (in the microcosm). Experiments to determine the velocities of gas molecules, which I will talk about a little further, confirm this conclusion.
When we are dealing not with a gas, but with a solid or liquid body, we do not have such direct methods at our disposal for determining the speed of the molecules of the body. However, even in these cases there is no doubt that with increasing temperature the speed of movement of molecules increases.
Change in gas temperature when its volume changes. Adiabatic and isothermal processes.
We have established how gas pressure depends on temperature if the volume remains unchanged. Now let's see how the pressure of a certain mass of gas changes depending on the volume it occupies if the temperature remains unchanged. However, before moving on to this issue, we need to figure out how to maintain the temperature of the gas constant. To do this, it is necessary to study what happens to the temperature of the gas if its volume changes so quickly that there is practically no heat exchange between the gas and the surrounding bodies.
Let's do this experiment. In a thick-walled tube made of transparent material, closed at one end, we place cotton wool, slightly moistened with ether, and this will create a mixture of ether vapor and air inside the tube, which explodes when heated. Then quickly push the tightly fitting piston into the tube. We will see a small explosion occur inside the tube. This means that when the mixture of ether vapor and air was compressed, the temperature of the mixture increased sharply. This phenomenon is quite understandable. By compressing a gas with an external force, we produce work, as a result of which the internal energy of the gas should increase; This is what happened - the gas heated up.
Now let's allow the gas to expand and do work against external pressure forces. This can be done. Let a large bottle contain compressed air at room temperature. By connecting the bottle with outside air, we will give the air in the bottle the opportunity to expand, leaving the small one. holes outward, and place a thermometer or flask with a tube in the stream of expanding air. The thermometer will show a temperature noticeably lower than room temperature, and a drop in the tube attached to the flask will run towards the flask, which will also indicate a decrease in the temperature of the air in the stream. This means that when a gas expands and at the same time does work, it cools and its internal energy decreases. It is clear that heating of a gas during compression and cooling during expansion are an expression of the law of conservation of energy.
If we turn to the microcosm, the phenomena of gas heating during compression and cooling during expansion will become quite clear. When a molecule hits a stationary wall and bounces off it, the speed, and therefore the kinetic energy of the molecule, is on average the same as before hitting the wall. But if a molecule hits and rebounds from a piston approaching it, its speed and kinetic energy are greater than before hitting the piston (just as the speed of a tennis ball increases if it is hit in the opposite direction with a racket). The approaching piston transfers additional energy to the molecule reflected from it. Therefore, the internal energy of a gas increases during compression. When rebounding from the retreating piston, the speed of the molecule decreases, because the molecule does work by pushing the retreating piston. Therefore, the expansion of the gas, associated with the retraction of the piston or layers of surrounding gas, is accompanied by work and leads to a decrease in the internal energy of the gas.
So, compression of a gas by an external force causes it to heat up, and expansion of the gas is accompanied by its cooling. This phenomenon always occurs to some extent, but I notice it especially sharply when the exchange of heat with surrounding bodies is minimized, because such exchange can compensate for the change in temperature to a greater or lesser extent.
Processes in which the transfer of heat is so insignificant that it can be neglected are called adiabatic.
Let's return to the question posed at the beginning of the chapter. How to ensure a constant gas temperature, despite changes in its volume? Obviously, to do this, it is necessary to continuously transfer heat to the gas from the outside if it is expanding, and to continuously remove heat from it, transferring it to surrounding bodies if the gas is compressed. In particular, the temperature of the gas remains fairly constant if the expansion or compression of the gas is very slow, and the transfer of heat from outside or outside can occur with sufficient speed. With slow expansion, heat from the surrounding bodies is transferred to the gas and its temperature decreases so little that this decrease can be neglected. With slow compression, heat, on the contrary, is transferred from the gas to the surrounding bodies, and as a result its temperature increases only negligibly.
Processes in which the temperature is maintained constant are called isothermal.
Boyle's Law - Mariotte
Let us now move on to a more detailed study of the question of how the pressure of a certain mass of gas changes if its temperature remains unchanged and only the volume of the gas changes. We have already found out that this isothermal the process is carried out under the condition that the temperature of the bodies surrounding the gas is constant and the volume of the gas changes so slowly that the temperature of the gas at any moment of the process does not differ from the temperature of the surrounding bodies.
We thus pose the question: how are volume and pressure related to each other during an isothermal change in the state of a gas? Daily experience teaches us that when the volume of a certain mass of gas decreases, its pressure increases. An example is the increase in elasticity when inflating a soccer ball, bicycle or car tire. The question arises: how Does the pressure of a gas increase as the volume decreases if the temperature of the gas remains unchanged?
The answer to this question was given by research carried out in the 17th century by the English physicist and chemist Robert Boyle (1627-1691) and the French physicist Eden Marriott (1620-1684).
Experiments establishing the relationship between gas volume and pressure can be reproduced: on a vertical stand , equipped with divisions, there are glass tubes A And IN, connected by a rubber tube C. Mercury is poured into the tubes. Tube B is open at the top, and tube A has a tap. Let's close this tap, thus locking a certain mass of air in the tube A. As long as we do not move the tubes, the mercury level in both tubes is the same. This means that the pressure of the air trapped in the tube A, the same as the ambient air pressure.
Let's now slowly pick up the phone IN. We will see that the mercury in both tubes will rise, but not equally: in the tube IN the mercury level will always be higher than in A. If you lower tube B, then the mercury level in both elbows decreases, but in tube IN the decrease is greater than in A.
Volume of air trapped in the tube A, can be counted by tube divisions A. The pressure of this air will differ from atmospheric pressure by the amount of pressure of the mercury column, the height of which is equal to the difference in the levels of mercury in tubes A and B. At. picking up the phone IN the pressure of the mercury column is added to atmospheric pressure. The volume of air in A decreases. When the handset goes down IN the level of mercury in it turns out to be lower than in A, and the pressure of the mercury column is subtracted from the atmospheric pressure; the volume of air in A increases accordingly.
Comparing the values of pressure and volume of air locked in tube A obtained in this way, we will be convinced that when the volume of a certain mass of air increases by a certain number of times, its pressure decreases by the same amount, and vice versa. In our experiments, the air temperature in the tube can be considered constant.
Similar experiments can be carried out with other gases. The results are the same.
So, the pressure of a certain mass of gas at a constant temperature is inversely proportional to the volume of the gas (Boyle-Mariotte law).
For rarefied gases, the Boyle-Mariotte law is fulfilled with a high degree of accuracy. For highly compressed or cooled gases, noticeable deviations from this law are found.
Formula expressing the Boyle-Mariotte law.
(2)
Graph expressing the Boyle-Mariotte law.
In physics and technology, graphs are often used that show the dependence of gas pressure on its volume. Let's draw such a graph for an isothermal process. We will plot the gas volume along the abscissa axis, and its pressure along the ordinate axis.
Let's take an example. Let the pressure of a given mass of gas with a volume of 1 m 3 be equal to 3.6 kg/cm 2 . Based on the Boyle-Mariotte law, we calculate that with a volume equal to 2 m 3 , pressure is 3.6*0.5 kg/cm 2 = 1,8kg/cm 2 . Continuing these calculations, we get the following table:
|
V (in m 3 ) |
||||||
|
P(V kg1cm 2 ) |
Plotting this data on the drawing in the form of points, the abscissas of which are the V values, and the ordinates are the corresponding values R, we obtain a curved line graph of the isothermal process in the gas (figure above).
Relationship between gas density and its pressure
Recall that the density of a substance is the mass contained in a unit volume. If we somehow change the volume of a given mass of gas, then the density of the gas will change. If, for example, we reduce the volume of a gas by a factor of five, the density of the gas will increase by a factor of five. At the same time, the gas pressure will increase; if the temperature does not change, then, as the Boyle-Mariotte law shows, the pressure will also increase five times. From this example it is clear that during an isothermal process, the gas pressure changes in direct proportion to its density.
Having designated gas densities at pressures and using the letters and , we can write:
This important result can be considered another and more significant expression of the Boyle-Mariotte law. The fact is that instead of the volume of gas, which depends on a random circumstance - on what mass of gas is chosen - formula (3) includes the density of the gas, which, like pressure, characterizes the state of the gas and does not depend at all on the random choosing its mass.
Molecular interpretation of Boyle's law - Mariotte.
In the previous chapter, we found out on the basis of the Boyle-Mariotte law that at a constant temperature, the pressure of a gas is proportional to its density. If the density of the gas changes, then the number of molecules per 1 cm 3 changes by the same amount. If the gas is not too compressed and the movement of gas molecules can be considered completely independent of each other, then the number of blows per 1 sec per 1 cm 2 of the vessel wall is proportional to the number of molecules in 1 cm 3 . Consequently, if the average speed of molecules does not change over time (we have already seen that in the macrocosm this means constant temperature), then the gas pressure should be proportional to the number of molecules in 1 cm 3 , i.e. gas density. Thus, the Boyle-Mariotte law is an excellent confirmation of our ideas about the structure of gas.
However, the Boyle-Marriott law ceases to be justified if we move to high pressures. And this circumstance can be clarified, as M.V. Lomonosov believed, on the basis of molecular concepts.
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 of the time than molecules in a rarefied gas. This, on the contrary, reduces the number of impacts of molecules on 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. Not too much pressure. the second circumstance is more significant and the product PV decreases slightly. At very high pressures, the first circumstance plays a major role and the PV product increases.
So, the Boyle-Mariotte law itself and deviations from it confirm the molecular theory.
Change in gas volume with temperature change
We studied how the pressure of a certain mass of gas depends on temperature, if the volume remains unchanged, and on the volume , occupied by the gas if the temperature remains constant. Now let's establish how a gas behaves if its temperature and volume change, but the pressure remains constant.
If we knew how the temperature of the air in the vessel changed in our experiment, and accurately measured how the volume of the gas changes, we could study this phenomenon from a quantitative perspective. Obviously, to do this, it is necessary to enclose the vessel in a shell, taking care that all parts of the device have the same temperature, accurately measure the volume of the trapped mass of gas, then change this temperature and measure the increment in the volume of gas.
Gay-Lussac's law.
A quantitative study of the dependence of gas volume on temperature at constant pressure was carried out by the French physicist and chemist Gay-Lussac (1778-1850) in 1802.
Experiments have shown that the increase in gas volume is proportional to the temperature increase. Therefore, the thermal expansion of a gas can, as for other bodies, be characterized using the volumetric expansion coefficient b. It turned out that for gases this law is observed much better than for solids and liquids, so that the coefficient of volumetric expansion of gases is a value that is practically constant even with very significant increases in Temperature, whereas for liquids and solids it is; Constancy is only approximately observed.
From here we find:
(4)
The experiments of Gay-Lussac and others revealed a remarkable result. It turned out that the coefficient of volume expansion for all gases is the same (more precisely, almost the same) and equals = 0.00366 . Thus, at heating at constant pressure by 1° the volume of a certain mass of gas increases by the volume that this mass of gas occupied at 0°C (Gay's law - Lussac ).
As can be seen, the expansion coefficient of gases coincides with their thermal pressure coefficient.
It should be noted that the thermal expansion of gases is very significant, so the volume of gas at 0°C is noticeably different from the volume at another, for example, room temperature. Therefore, as already mentioned, in the case of gases it is impossible to replace the volume in formula (4) without a noticeable error volume V. In accordance with this, it is convenient to give the expansion formula for gases the following form. For the initial volume we take the volume at a temperature of 0°C. In this case, the increment in gas temperature t is equal to the temperature measured on the Celsius scale t . Consequently, the coefficient of volumetric expansion
Where 
(5)
Formula (6) can be used to calculate volume both at temperatures above O o C and at temperatures below 0 ° C. In this last case I negative. It should, however, be borne in mind that Gay-Lussac's law does not hold true when the gas is highly compressed or so cooled that it approaches a state of liquefaction. In this case, formula (6) cannot be used.
Graphs expressing the laws of Charles and Gay-Lussac
We will plot the temperature of the gas located in a constant volume along the abscissa axis, and its pressure along the ordinate axis. Let the gas pressure at 0°C be 1 kg|cm 2 . Using Charles' law, we can calculate its pressure at 100 0 C, at 200 ° C, at 300 ° C, etc.
Let's plot this data on a graph. We will get a slanted straight line. We can continue this graph towards negative temperatures. However, as already indicated, Charles’s law is applicable only to temperatures that are not very low. Therefore, the continuation of the graph until the intersection with the abscissa axis, i.e., to the point where the pressure is zero, will not correspond to the behavior of a real gas.
Absolute temperature
It is easy to see that the pressure of a gas enclosed in a constant volume is not directly proportional to the temperature measured on the Celsius scale. This is clear, for example, from the table given in the previous chapter. If at 100°C the gas pressure is 1.37 kg1cm 2 , then at 200° C it is equal to 1.73 kg/cm 2 . The temperature measured by the Celsius thermometer doubled, but the gas pressure increased only 1.26 times. There is nothing surprising, of course, in this, since the Celsius thermometer scale is set arbitrarily, without any connection with the laws of gas expansion. It is possible, however, using gas laws, to establish a temperature scale such that gas pressure will directly proportional to temperature, measured on this new scale. Zero in this new scale is called absolute zero. This name was adopted because, as was proven by the English physicist Kelvin (William Thomson) (1824-1907), no body can be cooled below this temperature. In accordance with this, this new scale is called absolute temperature scale. Thus, absolute zero indicates a temperature equal to -273° Celsius, and represents the temperature below which no body can be cooled under any circumstances. A temperature expressed as 273°+ represents the absolute temperature of a body that has a temperature on the Celsius scale equal to. Absolute temperatures are usually denoted by the letter T. Thus, 273 o + = . The absolute temperature scale is often called the Kelvin scale and is written T° K. Based on what has been said
The result obtained can be expressed in words: the pressure of a given mass of gas enclosed in a constant volume is directly proportional to the absolute temperature. This is a new expression of Charles's law.
Formula (6) is also convenient to use in the case when the pressure at 0°C is unknown.
Gas volume and absolute temperature
From formula (6), you can get the following formula:
- the volume of a certain mass of gas at constant pressure is directly proportional to the absolute temperature. This is a new expression of Gay-Lussac's law.
Dependence of gas density on temperature
What happens to the density of a certain mass of gas if the temperature rises and the pressure remains unchanged?
Recall that density is equal to the mass of a body divided by volume. Since the mass of the gas is constant, when heated, the density of the gas decreases as many times as the volume increases.
As we know, the volume of a gas is directly proportional to the absolute temperature if the pressure remains constant. Hence, The density of a gas at constant pressure is inversely proportional to the absolute temperature. If and - gas densities at temperatures and , That there is a relation
Unified gas law
We considered cases when one of the three quantities characterizing the state of a gas (pressure, temperature and volume) does not change. We have seen that if the temperature is constant, then pressure and volume are related to each other by the Boyle-Mariotte law; if the volume is constant, then pressure and temperature are related by Charles' law; If the pressure is constant, then the volume and temperature are related by the Gay-Lussac law. Let us establish a connection between the pressure, volume and temperature of a certain mass of gas if all three of these quantities change.
Let the initial volume, pressure and absolute temperature of a certain mass of gas be equal to V 1, P 1 and T 1 final - V 2, P 2 and T 2 - One can imagine that the transition from the initial to the final state occurred in two stages. Let, for example, first change the volume of gas from V 1 to V 2 , and the temperature T 1 remained unchanged. The resulting gas pressure will be denoted by P avg. . Then the temperature changed from T 1 to T 2 at a constant volume, and the pressure changed from P avg to P 2 . Let's make a table:
Boyle's Law - Mariotte
P 1 V 1 t 1
P cp V 2 T 1
Charles's Law
P cp V 2 T 1
Applying the Boyle-Mariotte law to the first transition, we write
Applying Charles' law to the second transition, we can write
Multiplying these equalities term by term and reducing by P cp we get:
(10)
So, the product of the volume of a certain mass of gas and its pressure is proportional to the absolute temperature of the gas. This is the unified law of the gas state or the equation of state of the gas.
Law Dalton
Until now we have talked about the pressure of any one gas - oxygen, hydrogen, etc. But in nature and in technology we very often deal with a mixture of several gases. The most important example of this is air, which is a mixture of nitrogen, oxygen, argon, carbon dioxide and other gases. What does pressure depend on? mix-si gases?
Let's place a piece of a substance in the flask that chemically binds oxygen from the air (for example, phosphorus), and quickly close the flask with a stopper with a tube. connected to a mercury pressure gauge. After some time, all the oxygen in the air will combine with phosphorus. We will see that the pressure gauge will show less pressure than before the oxygen was removed. This means that the presence of oxygen in the air increases its pressure.
An accurate study of the pressure of a mixture of gases was first carried out by the English chemist John Dalton (1766-1844) in 1809. The pressure that each of the gases composing the mixture would have if the other gases were removed from the volume occupied by the mixture is called partial pressure this gas. Dalton found that the pressure of a mixture of gases is equal to the sum of their partial pressures(Dalton's law). Note that Dalton's law is not applicable to highly compressed gases, just like the Boyle-Mariotte law.
I’ll tell you a little further how to interpret Dalton’s law from the point of view of molecular theory.
Gas densities
The density of a gas is one of the most important characteristics of its properties. When talking about the density of a gas, we usually mean its density under normal conditions(i.e. at a temperature of 0 ° C and a pressure of 760 mm rt. Art.). In addition, they often use relative density gas, which means the ratio of the density of a given gas to the density of air under the same conditions. It is easy to see that the relative density of a gas does not depend on the conditions in which it is located, since, according to the laws of the gas state, the volumes of all gases change equally with changes in pressure and temperature.
Densities of some gases
|
Density under normal conditions in g/l or in kg/m 3 |
Relation to air density |
Relation to hydrogen density |
Molecular or atomic weight |
|
|
0,0899 1,25 1,43 1,977 0,179 |
0,0695 0,967 1.11 1,53 0,139 |
29 (medium) |
||
|
Hydrogen (H2) |
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|
Nitrogen (N2) |
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Oxygen (O 2) |
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Carbon dioxide (CO 2 ) |
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|
Helium(He) |
The gas density can be determined as follows. Let's weigh the flask with the tap twice: once by pumping out as much air as possible from it, and another time by filling the flask with the test gas to a pressure that should be known. Dividing the difference in weights by the volume of the flask, which must be determined in advance, we find the density of the gas under these conditions. Then, using the equation of state of gases, we can easily find the gas density under normal conditions d n. Indeed, let us put in formula (10) P 2 == P n, V 2 = V n, T 2 = T n and, multiplying the numerator and denominator
formula for gas mass m, we get:
Hence, taking into account what we find:
The results of measurements of the density of some gases are given in the table above.
The last two columns indicate the proportionality between the density of a gas and its molecular weight (in the case of helium, atomic weight).
Avogadro's law
Comparing the numbers in the penultimate column of the table with the molecular weights of the gases under consideration, it is easy to notice that the densities of gases under the same conditions are proportional to their molecular weights. A very significant conclusion follows from this fact. Since molecular weights are related to the masses of molecules, then
, where d is the density of gases, and m is the mass of their molecules.
the masses of their molecules. On the other hand, the masses of gases M 1 and M 2 , enclosed in equal volumes V, relate as their densities:
designating the number of molecules of the first and second gases contained in the volume V, letters N 1 and N 2, we can write that the total mass of a gas is equal to the mass of one of its molecules multiplied by the number of molecules: M 1 =t 1 N 1 And M 2 =t 2 N 2 That's why
Comparing this result with the formula , we'll find
that N 1 = N 2. So , at the same pressure and temperature, equal volumes of different gases contain the same number of molecules.
This law was discovered by the Italian chemist Amedeo Avogadro (1776-1856) based on chemical research. It refers to gases that are not very highly compressed (for example, gases under atmospheric pressure). In the case of highly compressed gases, it cannot be considered valid.
Avogadro's law means that the pressure of a gas at a certain temperature depends only on the number of molecules per unit volume of gas, but does not depend on whether the molecules are heavy or light. Having understood this, it is easy to understand the essence of Dalton's law. According to the Boyle-Mariotte law, if we increase the density of a gas, that is, we add a certain number of molecules of this gas to a certain volume, we increase the pressure of the gas. But according to Avogadro's law, the same increase in pressure should be obtained if, instead of adding molecules of the first gas, we add the same number of molecules of another gas. This is exactly what Dalton's law is, which states that you can increase the pressure of a gas by adding molecules of another gas to the same volume, and if the number of added molecules is the same as in the first case, then the same increase in pressure will be obtained. It is clear that Dalton's law is a direct consequence of Avogadro's law.
Gram molecule. Avogadro's number.
The number that gives the ratio of the masses of two molecules also indicates the ratio of the masses of two portions of a substance containing the same number of molecules. Therefore, 2 g of hydrogen (molecular weight of Ha is 2), 32 G oxygen (molecular weight Od is 32) and 55.8 G iron (its molecular weight coincides with the atomic weight, equal to 55.8), etc. contain the same number of molecules.
An amount of a substance containing a number of grams equal to its molecular weight is called gram molecule or we pray.
From the above it follows that moles of different substances contain the same number of molecules. Therefore, it is often convenient to use the mole as a special unit containing a different number of grams for different substances, but the same number of molecules.
The number of molecules in one mole of a substance, called Avogadro's number is an important physical quantity. Numerous and varied studies have been done to determine Avogadro's number. They relate to Brownian movement, to the phenomena of electrolysis and a number of others. These studies have produced fairly consistent results. Currently, it is accepted that Avogadro's number is equal to
N= 6,02*10 23 mol -1 .
So, 2 g of hydrogen, 32 g of oxygen, etc. each contain 6.02 * 10 23 molecules. To imagine the enormity of this number, imagine a desert with an area of 1 million square kilometers, covered with a layer of sand 600 thick. m. Then, if for each grain of sand there is a volume of 1 mm 3 , then the total number of grains of sand in the desert will be equal to Avogadro's number.
From Avogadro's law it follows that moles of different gases have the same volumes under the same conditions. The volume of one mole under normal conditions can be calculated by dividing the molecular weight of a gas by its density under normal conditions.
Thus, The volume of a mole of any gas under normal conditions is equal to 22400 cm 3.
Speeds gas molecules
What are the speeds at which molecules, in particular gas molecules, move? This question naturally arose as soon as ideas about molecules were developed. For a long time, the velocities of molecules could only be estimated by indirect calculations, and only relatively recently were methods developed for directly determining the velocities of gas molecules.
First of all, let us clarify what is meant by the speed of molecules. Let us recall that due to incessant collisions, the speed of each individual molecule changes all the time: the molecule moves sometimes quickly, sometimes slowly, and for some time the speed of the molecule takes on many different values. On the other hand, at any given moment, in the enormous number of molecules that make up the volume of gas under consideration, there are molecules with very different velocities. Obviously, to characterize the state of a gas we must talk about some average speed. We can assume that this is the average velocity of one of the molecules over a sufficiently long period of time or that this is the average velocity of all gas molecules in a given volume at some point in time.
Let us dwell on the reasoning that makes it possible to calculate the average speed of gas molecules.
Gas pressure is proportional Friv 2 , Where T - molecular mass, v- average speed and P - number of molecules per unit volume. A more accurate calculation leads to the formula
From formula (12) a number of important consequences can be deduced. Let us rewrite formula (12) in this form:
where e is the average kinetic energy of one molecule. Let us denote the gas pressure at temperatures T 1 and T 2 by the letters p 1 and p 2 and the average kinetic energies of molecules at these temperatures e 1 and e 2 . In this case
Comparing this relationship with Charles's law
So, The absolute temperature of a gas is proportional to the average kinetic energy of gas molecules. Since the average kinetic energy of molecules is proportional to the square of the average speed of molecules, our comparison leads to the conclusion that the absolute temperature of a gas is proportional to the square of the average speed of gas molecules and that the speed of molecules increases in proportion to the square root of the absolute temperature.
Average velocities of molecules of some gases
As can be seen, the average velocities of molecules are very significant. At room temperature they usually reach hundreds of meters per second. In a gas, the average speed of molecules is approximately one and a half times greater than the speed of sound in the same gas.
At first glance, this result seems very strange. It seems that molecules cannot move at such high speeds: after all, diffusion even in gases, and even more so in liquids, proceeds relatively very slowly, in any case much slower than sound propagates. The point, however, is that when moving, molecules very often collide with each other and at the same time change the direction of their movement. As a result, they move first in one direction and then in the other, mostly crowding around in one place. As a result, despite the high speed of movement in the intervals between collisions, despite the fact that the molecules do not linger anywhere, they move in any particular direction rather slowly.
The table also shows that the difference in the speeds of different molecules is associated with the difference in their masses. This circumstance is confirmed by a number of observations. For example, hydrogen penetrates through narrow openings (pores) at a higher rate than oxygen or nitrogen. You can discover this from such an experience.
The glass funnel is closed with a porous vessel or sealed with paper and the end is lowered into water. If we cover the funnel with a glass, under which hydrogen (or illuminating gas) is introduced, we will see that the water level at the end of the funnel will drop and bubbles will begin to come out of it. How to explain this?
Both air molecules (from inside the funnel under the glass) and hydrogen molecules (from under the glass into the funnel) can pass through narrow pores in a vessel or paper. But the speed of these processes varies. The difference in the sizes of molecules does not play a significant role in this case, because the difference is small, especially compared to the size of the pores: a hydrogen molecule has a “length” of about 2.3 * 10 -8 cm, and a molecule of oxygen or nitrogen is about 3*10 -8 cm, the diameter of the holes, which are pores, is thousands of times larger. The high speed of hydrogen penetration through the porous wall is explained by the higher speed of movement of its molecules. Therefore, hydrogen molecules penetrate faster from the glass into the funnel. As a result, molecules accumulate in the funnel, the pressure increases and the mixture of gases comes out in the form of bubbles.
Such devices are used to detect the presence of mine gases in the air, which can cause an explosion in mines.
Heat capacity of gases
Suppose we have 1 G gas How much heat must be supplied to it in order for its temperature to increase by 1°C, in other words, how much specific heat capacity of gas? As experience shows, this question cannot be given an unambiguous answer. The answer depends on the conditions under which the gas is heated. If its volume does not change, then a certain amount of heat is needed to heat the gas; At the same time, the gas pressure also increases. If heating is carried out in such a way that its pressure remains unchanged, then a different, larger amount of heat will be required than in the first case; this will increase the volume of gas. Finally, other cases are possible when both volume and pressure change during heating; in this case, an amount of heat will be required, depending on the extent to which these changes occur. According to what has been said, gas can have a wide variety of specific heat capacities, depending on the heating conditions. Two of all these specific heat capacities are usually distinguished: specific heat capacity at constant volume (C v ) and specific heat capacity at constant pressure (C p ).
To determine Cv, it is necessary to heat the gas placed in a closed vessel. The expansion of the vessel itself during heating can be neglected. When determining C p, it is necessary to heat the gas placed in a cylinder closed by a piston, the load on which remains unchanged.
The heat capacity at constant pressure C p is greater than the heat capacity at constant volume C v . Indeed, when heated 1 G of gas by 1° at a constant volume, the heat supplied only goes to increase the internal energy of the gas. To heat the same mass of gas by 1° at a constant pressure, it is necessary to impart heat to it, due to which not only the internal energy of the gas will increase, but also work will be done associated with the expansion of the gas. To obtain C p, to the value of C v it is necessary to add another amount of heat equivalent to the work done during the expansion of the gas.
Page 5
Absolute temperature
It is easy to see that the pressure of a gas enclosed in a constant volume is not directly proportional to the temperature measured on the Celsius scale. This is clear, for example, from the table given in the previous chapter. If at 100°C the gas pressure is 1.37 kg/cm2, then at 200°C it is 1.73 kg/cm2. The temperature measured by the Celsius thermometer doubled, but the gas pressure increased only 1.26 times. There is nothing surprising, of course, in this, since the Celsius thermometer scale is set arbitrarily, without any connection with the laws of gas expansion. It is possible, however, using gas laws, to establish a temperature scale such that the gas pressure will be directly proportional to the temperature measured on this new scale. Zero on this new scale is called absolute zero. This name was adopted because, as was proven by the English physicist Kelvin (William Thomson) (1824-1907), no body can be cooled below this temperature.
In accordance with this, this new scale is called the absolute temperature scale. Thus, absolute zero indicates a temperature equal to -273° Celsius and represents the temperature below which no body can be cooled under any circumstances. The temperature expressed as 273°+t1 represents the absolute temperature of a body that has a temperature on the Celsius scale equal to t1. Absolute temperatures are usually denoted by the letter T. Thus, 2730+t1=T1. The absolute temperature scale is often called the Kelvin scale and is written T° K. Based on the above
The result obtained can be expressed in words: the pressure of a given mass of gas enclosed in a constant volume is directly proportional to the absolute temperature. This is a new expression of Charles's law.
Formula (6) is also convenient to use in the case when the pressure at 0°C is unknown.
Gas volume and absolute temperature
From formula (6), we can obtain the following formula:
The volume of a certain mass of gas at constant pressure is directly proportional to the absolute temperature. This is a new expression of Gay-Lussac's law.
Dependence of gas density on temperature
What happens to the density of a certain mass of gas if the temperature increases but the pressure remains unchanged?
Recall that density is equal to the mass of a body divided by volume. Since the mass of the gas is constant, when heated, the density of the gas decreases as many times as the volume increases.
As we know, the volume of a gas is directly proportional to the absolute temperature if the pressure remains constant. Consequently, the density of a gas at constant pressure is inversely proportional to the absolute temperature. If d1 and d2 are the gas densities at temperatures t1 and t2, then the relation holds
Unified gas law
We considered cases when one of the three quantities characterizing the state of a gas (pressure, temperature and volume) does not change. We have seen that if the temperature is constant, then pressure and volume are related to each other by the Boyle-Mariotte law; if the volume is constant, then pressure and temperature are related by Charles' law; If the pressure is constant, then the volume and temperature are related by the Gay-Lussac law. Let us establish a connection between the pressure, volume and temperature of a certain mass of gas if all three of these quantities change.
Let the initial volume, pressure and absolute temperature of a certain mass of gas be equal to V1, P1 and T1, and the final ones - V2, P2 and T2 - You can imagine that the transition from the initial to the final state occurred in two stages. Let, for example, first change the volume of gas from V1 to V2, and the temperature T1 remains unchanged. The resulting gas pressure will be denoted by Pav. Then the temperature changed from T1 to T2 at a constant volume, and the pressure changed from Pav. to P. Let's make a table:
Boyle's Law - Mariotte
Charles's Law
Changing, for the first transition we write the Boyle-Mariotte law
Applying Charles' law to the second transition, we can write
Multiplying these equalities term by term and reducing by Pcp we get:
So, the product of the volume of a certain mass of gas and its pressure is proportional to the absolute temperature of the gas. This is the unified law of the gas state or the equation of state of the gas.
Law Dalton
Until now, we have talked about the pressure of any one gas - oxygen, hydrogen, etc. But in nature and in technology, we very often deal with a mixture of several gases. The most important example of this is air, which is a mixture of nitrogen, oxygen, argon, carbon dioxide and other gases. What does the pressure of a gas mixture depend on?
Place in the flask a piece of a substance that chemically binds oxygen from the air (for example, phosphorus), and quickly close the flask with a stopper and tube. connected to a mercury manometer. After some time, all the oxygen in the air will combine with phosphorus. We will see that the pressure gauge will show less pressure than before the oxygen was removed. This means that the presence of oxygen in the air increases its pressure.
An accurate study of the pressure of a mixture of gases was first carried out by the English chemist John Dalton (1766-1844) in 1809. The pressure that each of the gases making up the mixture would have if the other gases were removed from the volume occupied by the mixture is called the partial pressure of this gas. Dalton found that the pressure of a mixture of gases is equal to the sum of their partial pressures (Dalton's law). Note that Dalton's law is not applicable to highly compressed gases, just like the Boyle-Mariotte law.
Abstract on the topic:
Air density
Plan:
- Introduction
- 1
Relationships within the ideal gas model
- 1.1 Temperature, pressure and density
- 1.2 Effect of air humidity
- 1.3 Effect of altitude in the troposphere
Notes
Introduction
Air density- mass of gas in the Earth’s atmosphere per unit volume or specific gravity of air under natural conditions. Magnitude air density is a function of the height of the measurements taken, its temperature and humidity. Usually the standard value is considered to be 1.225 kg ⁄ m 3 , which corresponds to the density of dry air at 15°C at sea level.
1. Relationships within the ideal gas model
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1.1. Temperature, pressure and density
The density of dry air can be calculated using Clapeyron's equation for an ideal gas at a given temperature. and pressure:
Here ρ - air density, p- absolute pressure, R- specific gas constant for dry air (287.058 J ⁄ (kg K)), T- absolute temperature in Kelvin. Thus, by substitution we get:
- at a standard atmosphere of the International Union of Pure and Applied Chemistry (temperature 0°C, pressure 100 kPa, zero humidity), the air density is 1.2754 kg ⁄ m³;
- at 20 °C, 101.325 kPa and dry air, the density of the atmosphere is 1.2041 kg ⁄ m³.
The table below shows various air parameters, calculated on the basis of the corresponding elementary formulas, depending on the temperature (pressure taken as 101.325 kPa)
1.2. Effect of air humidity
Humidity refers to the presence of gaseous water vapor in the air, the partial pressure of which does not exceed the saturated vapor pressure for given atmospheric conditions. Adding water vapor to air leads to a decrease in its density, which is explained by the lower molar mass of water (18 g ⁄ mol) compared to the molar mass of dry air (29 g ⁄ mol). Humid air can be considered as a mixture of ideal gases, the combination of densities of each of which allows to obtain the required value for their mixture. This interpretation allows the density value to be determined with an error level of less than 0.2% in the temperature range from −10 °C to 50 °C and can be expressed as follows:
where is the density of moist air (kg ⁄ m³); p d- partial pressure of dry air (Pa); R d- universal gas constant for dry air (287.058 J ⁄ (kg K)); T- temperature (K); p v- water vapor pressure (Pa) and R v- universal constant for steam (461.495 J ⁄ (kg K)). Water vapor pressure can be determined from relative humidity:
Where p v- water vapor pressure; φ - relative humidity and p sat is the partial pressure of saturated vapor, the latter can be represented as the following simplified expression:
which gives the result in millibars. Dry air pressure p d determined by a simple difference:
Where p denotes the absolute pressure of the system under consideration.
1.3. Effect of altitude in the troposphere
Dependence of pressure, temperature and air density on altitude compared to the standard atmosphere ( p 0 =101325 Pa, T0=288.15 K, ρ 0 =1.225 kg/m³).
To calculate the air density at a certain altitude in the troposphere, the following parameters can be used (atmosphere parameters indicate the value for a standard atmosphere):
- standard atmospheric pressure at sea level - p 0 = 101325 Pa;
- standard temperature at sea level - T0= 288.15 K;
- acceleration of free fall above the Earth's surface - g= 9.80665 m ⁄ sec 2 (for these calculations it is considered a height-independent value);
- rate of temperature drop (English) Russian. with height, within the troposphere - L= 0.0065 K ⁄ m;
- universal gas constant - R= 8.31447 J ⁄ (Mol K);
- molar mass of dry air - M= 0.0289644 kg ⁄ Mol.
For the troposphere (i.e. the region of linear decrease in temperature - this is the only property of the troposphere used here) temperature at altitude h above sea level can be given by the formula:
Pressure at altitude h:
Then the density can be calculated by substituting the temperature T and pressure P corresponding to a given height h into the formula:
These three formulas (dependence of temperature, pressure and density on height) are used to construct the graphs shown on the right. The graphs are normalized - they show the general behavior of the parameters. “Zero” values for correct calculations must be substituted each time in accordance with the readings of the corresponding instruments (thermometer and barometer) at the moment at sea level.
The derived differential equations (1.2, 1.4) contain parameters that characterize a liquid or gas: density r , viscosity m , as well as parameters of the porous medium - porosity coefficients m and permeability k . For further calculations, it is necessary to know the dependence of these coefficients on pressure.
Density of droplet liquid. With steady filtration of a droplet liquid, its density can be considered independent of pressure, that is, the liquid can be considered incompressible: r = const .
In unsteady processes, it is necessary to take into account the compressibility of the liquid, which is characterized volumetric compression ratio of the liquid b . This coefficient is usually considered constant:
Having integrated the last equality from the initial pressure values p 0 and density r 0 to current values, we get:
In this case, we obtain a linear dependence of density on pressure.
Density of gases. Compressible liquids (gases) with small changes in pressure and temperature can also be characterized by the coefficients of volumetric compression and thermal expansion. But with large changes in pressure and temperature, these coefficients change within wide limits, so the dependence of the density of an ideal gas on pressure and temperature is based on Clayperon–Mendeleev equations of state:
Where R' = R/M m– gas constant, depending on the composition of the gas.
The gas constant for air and methane are respectively equal, R΄ air = 287 J/kg K˚; R΄ methane = 520 J/kg K˚.
The last equation is sometimes written as:
 | (1.50) |
From the last equation it is clear that the density of a gas depends on pressure and temperature, so if the density of the gas is known, then it is necessary to indicate the pressure, temperature and composition of the gas, which is inconvenient. Therefore, the concepts of normal and standard physical conditions are introduced.
Normal conditions correspond to temperature t = 0°C and pressure p at = 0.1013°MPa. The air density under normal conditions is equal to ρ v.n.us = 1.29 kg/m 3.
Standard terms correspond to temperature t = 20°C and pressure p at = 0.1013°MPa. The air density under standard conditions is equal to ρ w.st.us = 1.22 kg/m 3.
Therefore, from the known density under given conditions, it is possible to calculate the gas density at other values of pressure and temperature:
Excluding the reservoir temperature, we obtain the ideal gas equation of state, which we will use in the future:
Where z – coefficient characterizing the degree of deviation of the state of a real gas from the law of ideal gases (supercompressibility coefficient) and depending for a given gas on pressure and temperature z = z(p, T) . Supercompressibility coefficient values z are determined according to D. Brown's graphs.
Oil viscosity. Experiments show that the viscosity coefficients of oil (at pressures above saturation pressure) and gas increase with increasing pressure. With significant changes in pressure (up to 100 MPa), the dependence of the viscosity of reservoir oils and natural gases on pressure can be assumed to be exponential:
 | (1.56) |
For small changes in pressure, this dependence is linear.
Here m 0 – viscosity at fixed pressure p 0 ; β m – coefficient determined experimentally and depending on the composition of oil or gas.
Reservoir porosity. To find out how the porosity coefficient depends on pressure, let us consider the question of stresses acting in a porous medium filled with liquid. As the pressure in the liquid decreases, the force on the skeleton of the porous medium increases, so porosity decreases.
Due to the low deformation of the solid phase, it is usually believed that the change in porosity depends linearly on the change in pressure. The law of rock compressibility is written as follows, introducing coefficient of volumetric elasticity of the formation b c:
Where m 0 – porosity coefficient at pressure p 0 .
Laboratory experiments for different granular rocks and field studies show that the coefficient of volumetric elasticity of the formation is (0.3 - 2) 10 -10 Pa -1.
With significant changes in pressure, the change in porosity is described by the equation:
and for large ones – exponential:
 | (1.61) |
In fractured formations, permeability changes depending on pressure more intensively than in porous ones, therefore, in fractured formations, taking into account the dependence k(p) more necessary than in granular ones.
The equations of state of the liquid or gas saturating the formation and the porous medium close the system of differential equations.