Relationship between pressure, temperature, volume and number of moles of gas (the "mass" of gas). Universal (molar) gas constant R
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.
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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....
The physical properties of gases and the laws of the gaseous state are based on the molecular-kinetic theory of gases. Most of the laws of the gas state were derived for an ideal gas, the molecular forces of which are equal to zero, and the volume of the molecules themselves is infinitesimal compared to the volume of the intermolecular space.
The molecules of real gases, in addition to the energy of rectilinear motion, have the energy of rotation and vibration. They occupy a certain volume, that is, they have a finite size. The laws for real gases are somewhat different from the laws for ideal gases. This deviation is the greater, the higher the pressure of the gases and the lower their temperature, it is taken into account by introducing a correction factor for compressibility into the corresponding equations.
When transporting gases through pipelines under high pressure, the compressibility factor is of great importance.
At gas pressures in gas networks up to 1 MPa, the laws of the gas state for an ideal gas quite accurately reflect the properties of natural gas. At higher pressures or low temperatures, equations are used that take into account the volume occupied by molecules and the forces of interaction between them, or correction factors are introduced into the equations for an ideal gas - gas compressibility factors.
Boyle's Law - Mariotte.
Numerous experiments have established that if you take a certain amount of gas and subject it to various pressures, then the volume of this gas will change inversely with the pressure. This relationship between pressure and volume of a gas at constant temperature is expressed by the following formula:
p 1 / p 2 \u003d V 2 / V 1, or V 2 \u003d p 1 V 1 / p 2,
where p1 and V 1- initial absolute pressure and volume of gas; p2 and V 2 - pressure and volume of gas after the change.
From this formula, you can get the following mathematical expression:
V 2 p 2 = V 1 p 1 = const.
That is, the product of the value of the volume of gas by the value of the gas pressure corresponding to this volume will be a constant value at a constant temperature. This law has practical application in the gas industry. It allows you to determine the volume of a gas when its pressure changes and the pressure of a gas when its volume changes, provided that the temperature of the gas remains constant. The more the volume of a gas increases at a constant temperature, the lower its density becomes.
The relationship between volume and density is expressed by the formula:
V 1/V 2 = ρ 2 /ρ 1 ,
where V 1 and V 2- volumes occupied by gas; ρ 1 and ρ 2 are the gas densities corresponding to these volumes.
If the ratio of gas volumes is replaced by the ratio of their densities, then we can get:
ρ 2 /ρ 1 = p 2 /p 1 or ρ 2 = p 2 ρ 1 /p 1.
It can be concluded that at the same temperature, the densities of gases are directly proportional to the pressures under which these gases are located, that is, the density of a gas (at a constant temperature) will be the greater, the greater its pressure.
Example. The volume of gas at a pressure of 760 mm Hg. Art. and a temperature of 0 ° C is 300 m 3. What volume will this gas occupy at a pressure of 1520 mm Hg. Art. and at the same temperature?
760 mmHg Art. = 101329 Pa = 101.3 kPa;
1520 mmHg Art. = 202658 Pa = 202.6 kPa.
Substituting given values V, p 1, p 2 into the formula, we get, m 3:
V 2= 101, 3-300/202,6 = 150.
Gay-Lussac's law.
At constant pressure, with increasing temperature, the volume of gases increases, and with decreasing temperature, it decreases, that is, at constant pressure, the volumes of the same amount of gas are directly proportional to their absolute temperatures. Mathematically, this relationship between the volume and temperature of a gas at constant pressure is written as follows:
V 2 / V 1 \u003d T 2 / T 1
where V is the volume of gas; T is the absolute temperature.
It follows from the formula that if a certain volume of gas is heated at constant pressure, then it will change as many times as its absolute temperature changes.
It has been established that when a gas is heated by 1 °C at constant pressure, its volume increases by a constant value equal to 1/273.2 of the initial volume. This value is called the thermal expansion coefficient and is denoted p. With this in mind, the Gay-Lussac law can be formulated as follows: the volume of a given mass of gas at constant pressure is a linear function of temperature:
V t = V 0 (1 + βt or V t = V 0 T/273.
Charles' law.
At constant volume, the absolute pressure of a constant amount of gas is directly proportional to its absolute temperatures. Charles' law is expressed by the following formula:
p 2 / p 1 \u003d T 2 / T 1 or p 2 \u003d p 1 T 2 / T 1
where p 1 and p 2- absolute pressures; T1 and T 2 are the absolute temperatures of the gas.
From the formula, we can conclude that at a constant volume, the pressure of a gas during heating increases as many times as its absolute temperature increases.
Let's make sure that the gas molecules are really located far enough from each other, and therefore the gases are well compressible. Let's take a syringe and place its piston approximately in the middle of the cylinder. We connect the syringe hole with a tube, the second end of which is tightly closed. Thus, some air will be trapped in the syringe barrel under the plunger and in the tube. Some air will be trapped in the barrel under the plunger. Now let's put a load on the movable piston of the syringe. It is easy to notice that the piston will drop a little. This means that the volume of air has decreased. In other words, gases are easily compressed. Thus, there are sufficiently large gaps between gas molecules. Placing a weight on the piston causes the volume of the gas to decrease. On the other hand, after the weight is set, the piston, having lowered slightly, stops in the new equilibrium position. This means that force of air pressure on the piston increases and again balances the increased weight of the piston with the load. And since the area of the piston remains unchanged, we come to an important conclusion.
When the volume of a gas decreases, its pressure increases.
Let us remember at the same time that the mass of the gas and its temperature during the experiment remained unchanged. The dependence of pressure on volume can be explained as follows. As the volume of a gas increases, the distance between its molecules increases. Each molecule now needs to travel a greater distance from one impact with the vessel wall to the next. The average velocity of the molecules remains unchanged. Consequently, gas molecules hit the walls of the vessel less often, and this leads to a decrease in gas pressure. Conversely, when the gas volume decreases, its molecules more often hit the walls of the vessel, and the gas pressure increases. As the volume of a gas decreases, the distance between its molecules decreases.
Dependence of gas pressure on temperature
In previous experiments, the temperature of the gas remained unchanged, and we studied the change in pressure due to a change in the volume of the gas. Now consider the case when the volume of the gas remains constant and the temperature of the gas changes. The mass also remains unchanged. You can create such conditions by placing a certain amount of gas in a cylinder with a piston and fixing the piston
The change in temperature of a given mass of gas at a constant volume
The higher the temperature, the faster the gas molecules move.
Therefore,
First, the impact of molecules on the walls of the vessel occurs more often;
Secondly, the average impact force of each molecule on the wall becomes larger. This brings us to another important conclusion. As the temperature of a gas increases, its pressure increases. Let us remember that this statement is true if the mass and volume of the gas remain unchanged during the change in its temperature.
Storage and transportation of gases.
The dependence of gas pressure on volume and temperature is often used in engineering and in everyday life. If it is necessary to transport a significant amount of gas from one place to another, or when gases need to be stored for a long time, they are placed in special strong metal vessels. These vessels withstand high pressures, therefore, with the help of special pumps, significant masses of gas can be pumped into them, which under normal conditions would occupy hundreds of times more volume. Since the pressure of the gases in the cylinders is very high even at room temperature, they should never be heated or attempted to make a hole in them in any way, even after use.
Gas laws of physics.
The physics of the real world in calculations is often reduced to somewhat simplified models. This approach is most applicable to describing the behavior of gases. The rules established experimentally were reduced by various researchers to the gas laws of physics and served as the emergence of the concept of "isoprocess". This is such a passage of the experiment, in which one parameter retains a constant value. The gas laws of physics operate with the main parameters of a gas, more precisely, its physical state. Temperature, volume and pressure. All processes that relate to a change in one or more parameters are called thermodynamic. The concept of an isostatic process is reduced to the statement that during any change in state, one of the parameters remains unchanged. This is the behavior of the so-called "ideal gas", which, with some reservations, can be applied to real matter. As noted above, the reality is somewhat more complicated. However, with high certainty, the behavior of a gas at a constant temperature is characterized using the Boyle-Mariotte law, which states:
The product of volume and gas pressure is a constant value. This statement is considered true if the temperature does not change.
This process is called isothermal. In this case, two of the three studied parameters change. Physically, everything looks simple. Squeeze the inflated balloon. The temperature can be considered unchanged. And as a result, the pressure inside the ball will increase with a decrease in volume. The value of the product of the two parameters will remain unchanged. Knowing the initial value of at least one of them, you can easily find out the indicators of the second. Another rule in the list of "gas laws of physics" is the change in the volume of a gas and its temperature at the same pressure. This is called the "isobaric process" and is described using Gay-Lusac's law. The ratio of volume and temperature of the gas is unchanged. This is true under the condition of a constant value of pressure in a given mass of matter. Physically, too, everything is simple. If you have ever charged a gas lighter or used a carbon dioxide fire extinguisher, you have seen the effect of this law “live”. Gas escaping from a fire extinguisher canister or bell expands rapidly. His temperature plummets. You can freeze your skin. In the case of a fire extinguisher, whole flakes of carbon dioxide snow are formed when the gas, under the influence of low temperature, quickly turns into a solid state from a gaseous one. Thanks to the Gay-Lusac law, one can easily find out the temperature of a gas, knowing its volume at any given time. The gas laws of physics also describe behavior under the condition of a constant occupied volume. Such a process is called isochoric and is described by Charles' law, which states: With a constant volume occupied, the ratio of pressure to temperature of the gas remains unchanged at any given time. In reality, everyone knows the rule: you can not heat air fresheners and other vessels containing gas under pressure. The case ends with an explosion. What happens is exactly what Charles's law describes. The temperature is rising. At the same time, the pressure increases as the volume does not change. There is a destruction of the cylinder at the moment when the indicators exceed the allowable. So, knowing the volume occupied and one of the parameters, you can easily set the value of the second. Although the gas laws of physics describe the behavior of some ideal model, they can be easily applied to predict the behavior of gas in real systems. Especially in everyday life, isoprocesses can easily explain how a refrigerator works, why a cold stream of air flies out of a can of air freshener, which causes a chamber or a ball to burst, how a sprinkler works, and so on.
Fundamentals of MKT.
Molecular-kinetic theory of matter- way of explaining thermal phenomena, which connects the course of thermal phenomena and processes with the features of the internal structure of matter and studies the causes that determine thermal motion. This theory was recognized only in the 20th century, although it comes from the ancient Greek atomic theory of the structure of matter.
explains thermal phenomena by the peculiarities of motion and interaction of microparticles of matter
It is based on the laws of classical mechanics of I. Newton, which allow deriving the equation of motion of microparticles. Nevertheless, due to their huge number (there are about 10 23 molecules in 1 cm 3 of a substance), it is impossible to uniquely describe the movement of each molecule or atom every second using the laws of classical mechanics. Therefore, to build a modern theory of heat, methods of mathematical statistics are used, which explain the course of thermal phenomena based on the laws of behavior of a significant number of microparticles.
Molecular Kinetic Theory built on the basis of generalized equations of motion of a huge number of molecules.
Molecular Kinetic Theory explains thermal phenomena from the standpoint of ideas about the internal structure of matter, that is, clarifies their nature. This is a deeper, although more complex theory, which explains the essence of thermal phenomena and determines the laws of thermodynamics.
Both existing approaches are thermodynamic approach and molecular kinetic theory- are scientifically proven and mutually complement each other, and do not contradict each other. In this regard, the study of thermal phenomena and processes is usually considered from the positions of either molecular physics or thermodynamics, depending on how the material is presented in a simpler way.
Thermodynamic and molecular-kinetic approaches complement each other in explaining thermal phenomena and processes.
Ideal gas equation of state determines the relationship between temperature, volume and pressure of bodies.
- Allows you to determine one of the quantities characterizing the state of the gas, according to the other two (used in thermometers);
- Determine how processes proceed under certain external conditions;
- Determine how the state of the system changes if it does work or receives heat from external bodies.
Mendeleev-Clapeyron equation (ideal gas equation of state)
- universal gas constant, R = kN A
Clapeyron's equation (combined gas law)
Particular cases of the equation are gas laws that describe isoprocesses in ideal gases, i.e. processes in which one of the macro parameters (T, P, V) is constant in a closed isolated system.
Quantitative dependences between two parameters of a gas of the same mass with a constant value of the third parameter are called gas laws.
Gas laws
Boyle's law - Mariotte
The first gas law was discovered by the English scientist R. Boyle (1627-1691) in 1660. Boyle's work was called "New Experiments Concerning the Air Spring". Indeed, the gas behaves like a compressed spring, as you can see by compressing the air in a conventional bicycle pump.
Boyle studied the change in gas pressure as a function of volume at a constant temperature. The process of changing the state of a thermodynamic system at a constant temperature is called isothermal (from the Greek words isos - equal, therme - heat).
Regardless of Boyle, a little later, the French scientist E. Mariotte (1620-1684) came to the same conclusions. Therefore, the law found was called the Boyle-Mariotte law.
The product of the pressure of a gas of a given mass and its volume is constant if the temperature does not change
pV = const
Gay-Lussac's law
The announcement of the discovery of another gas law was published only in 1802, almost 150 years after the discovery of the Boyle-Mariotte law. The law that determines the dependence of gas volume on temperature at constant pressure (and constant mass) was established by the French scientist Gay-Lussac (1778-1850).
The relative change in the volume of a gas of a given mass at constant pressure is directly proportional to the change in temperature
V = V 0 αT
Charles' law
The dependence of gas pressure on temperature at constant volume was experimentally established by the French physicist J. Charles (1746-1823) in 1787.
J. Charles in 1787, i.e., earlier than Gay-Lussac, also established the dependence of volume on temperature at constant pressure, but he did not publish his work in time.
The pressure of a given mass of gas at constant volume is directly proportional to the absolute temperature.
p = p 0 γT
| Name | Wording | Graphs |
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Boyle-Mariotte law – isothermal process |
For a given mass of gas, the product of pressure and volume is constant if the temperature does not change
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Gay-Lussac's law - isobaric process |
2. Isochoric process. V is constant. P and T change. Gas obeys Charles' law . Pressure, at constant volume, is directly proportional to absolute temperature
3. Isothermal process. T is constant. P and V change. In this case, the gas obeys the Boyle-Mariotte law . The pressure of a given mass of gas at constant temperature is inversely proportional to the volume of gas.
4. From a large number of processes in a gas, when all parameters change, we single out a process that obeys the unified gas law. For a given mass of gas, the product of pressure times volume divided by absolute temperature is a constant.
This law is applicable to a large number of processes in a gas, when the parameters of the gas do not change very quickly.
All the listed laws for real gases are approximate. The errors increase with increasing pressure and density of the gas.
Work order:
1. part of the work.
1. We lower the hose of the glass ball into a vessel with water at room temperature (Fig. 1 in the appendix). Then we heat the ball (with hands, warm water). Considering the gas pressure to be constant, write how the volume of gas depends on temperature
Conclusion:………………..
2. Connect a cylindrical vessel with a millimanometer with a hose (Fig. 2). Let's heat a metal vessel and the air in it with a lighter. Assuming the volume of a gas to be constant, write how the pressure of a gas depends on temperature.
Conclusion:………………..
3. We squeeze the cylindrical vessel attached to the millimanometer with our hands, reducing its volume (Fig. 3). Assuming the temperature of the gas to be constant, write how the pressure of the gas depends on the volume.
Conclusion:……………….
4. Connect the pump to the chamber from the ball and pump in several portions of air (Fig. 4). How did the pressure, volume and temperature of the air pumped into the chamber change?
Conclusion:………………..
5. Pour about 2 cm 3 of alcohol into the bottle, close the cork with a hose (Fig. 5)attached to the injection pump. Let's make a few strokes until the cork leaves the bottle. How do the pressure, volume, and temperature of air (and alcohol vapor) change after the cork has taken off?
Conclusion:………………..
Part of work.
Verification of Gay-Lussac's law.
1. We take out the heated glass tube from hot water and lower the open end into a small vessel with water.
2. Hold the tube vertically.
3. As the air in the tube cools, water from the vessel enters the tube (Fig. 6).
4. Find and
The length of the tube and air column (at the beginning of the experiment)
The volume of warm air in the tube
The cross-sectional area of the tube.
The height of the column of water entering the tube when the air in the tube cools.
The length of the column of cold air in the tube
The volume of cold air in the tube.
Based on the Gay-Lussac law We have for two states of air
Or (2) (3)
Hot water temperature in bucket
Room temperature
We need to check equation (3) and hence the Gay-Lussac law.
5. Calculate
6. We find the relative measurement error when measuring the length, taking Dl = 0.5 cm.
7. Find the absolute error of the ratio
=……………………..
8. Write down the result of the reading
………..…..
9. We find the relative measurement error T, taking
10. Find the absolute calculation error
11. Write down the result of the calculation
12. If the interval for determining the temperature ratio (at least partially) coincides with the interval for determining the ratio of the lengths of the air columns in the tube, then equation (2) is valid and the air in the tube obeys the Gay-Lussac law.
Conclusion:……………………………………………………………………………………………………
Report requirement:
1. Title and purpose of the work.
2. List of equipment.
3. Draw pictures from the application and draw conclusions for experiments 1, 2, 3, 4.
4. Write the content, purpose, calculations of the second part of the laboratory work.
5. Write a conclusion on the second part of the laboratory work.
6. Plot graphs of isoprocesses (for experiments 1,2,3) in axes: ; ; .
7. Solve problems:
1. Determine the density of oxygen if its pressure is 152 kPa and the mean square velocity of its molecules is -545 m/s.
2. A certain mass of gas at a pressure of 126 kPa and a temperature of 295 K occupies a volume of 500 liters. Find the volume of gas under normal conditions.
3. Find the mass of carbon dioxide in a cylinder with a capacity of 40 liters at a temperature of 288 K and a pressure of 5.07 MPa.
Appendix
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