The ratio of volume to pressure. Basic laws of the gas state. Limitations of practical applicability
Consider how the pressure of a gas depends on temperature when its mass and volume remain constant.
Let's take a closed vessel with gas and we will heat it (Fig. 4.2). We will determine the gas temperature with a thermometer, and the pressure with a manometer M.
First, we place the vessel in melting snow and denote the gas pressure at 0 °C, and then we will gradually heat the outer vessel and record the values for the gas. It turns out that the graph of dependence on built on the basis of such experience has the form of a straight line (Fig. 4.3, a). If we continue this graph to the left, then it will intersect with the abscissa axis at point A, corresponding to zero gas pressure.
From the similarity of triangles in Fig. 4.3, or you can write:
If we denote the constant by y, we get
In the meaning of the coefficient of proportionality y in the described experiments should express the dependence of the change in gas pressure on its kind.
The value characterizing the dependence of the change in gas pressure on its kind in the process of changing temperature at a constant volume and constant mass of gas is called the temperature coefficient of pressure. The temperature coefficient of pressure shows by what part of the pressure of a gas taken at 0 ° C, its pressure changes when heated by
We derive the unit of the temperature coefficient y in SI:
By repeating the described experiment for various gases at various masses, it can be established that, within the experimental errors, point A for all graphs is obtained in the same place (Fig. 4.3, b). In this case, the length of the segment OA turns out to be equal to Thus, for all cases, the temperature at which the gas pressure must vanish is the same and equal to and the temperature coefficient of pressure Note that the exact value of y is
From experiments, the value of y was first determined by the French physicist J. Charles, who in 1787 established the following law: the temperature coefficient of pressure does not depend on the type of gas and is equal. Note that this is true only for gases that have a low density and with small temperature changes ; at high pressures or low temperatures, y depends on the type of gas. Only an ideal gas obeys Charles' law exactly.
Introduction
The state of an ideal gas is fully described by the measured quantities: pressure, temperature, volume. The ratio between these three quantities is determined by the basic gas law:
Objective
Verification of the Boyle-Mariotte law.
Tasks to be solved
Measurement of air pressure in a syringe while changing volume, given that the temperature of the gas is constant.
Experimental setup
Instruments and accessories
pressure gauge
Manual vacuum pump
In this experiment, the Boyle-Mariotte law is confirmed using the setup shown in Figure 1. The volume of air in the syringe is determined as follows:
where p 0 is atmospheric pressure, and p is the pressure measured with a pressure gauge.
Work order
Set the plunger of the syringe to the 50 ml mark.
Push the free end of the connecting hose of the hand vacuum pump tightly onto the outlet of the syringe.
While extending the piston, increase the volume in 5 ml increments, record the pressure gauge readings on the black scale.
To determine the pressure under the piston, it is necessary to subtract the readings of the monometer, expressed in pascals, from the atmospheric pressure. Atmospheric pressure is approximately 1 bar, which corresponds to 100,000 Pa.
To process the measurement results, the presence of air in the connecting hose must be taken into account. To do this, measure the volume of the connecting hose by measuring the length of the hose with a tape measure, and the diameter of the hose with a caliper, given that the wall thickness is 1.5 mm.
Plot the measured air volume versus pressure.
Calculate the dependence of volume on pressure at constant temperature using the Boyle-Mariotte law and plot.
Compare theoretical and experimental dependences.
2133. Dependence of gas pressure on temperature at constant volume (Charles law)
Introduction
Consider the dependence of gas pressure on temperature under the condition of a constant volume of a certain mass of gas. These studies were first made in 1787 by Jacques Alexandre Cesar Charles (1746-1823). The gas was heated in a large flask connected to a mercury manometer in the form of a narrow curved tube. Neglecting a negligible increase in the volume of the flask when heated and a slight change in volume when the mercury is displaced in a narrow manometric tube. Thus, the volume of gas can be considered unchanged. By heating the water in the vessel surrounding the flask, the temperature of the gas was measured using a thermometer T, and the corresponding pressure R- by manometer. By filling the vessel with melting ice, the pressure was determined R about, and the corresponding temperature T about. It was found that if at 0 C the pressure R about , then when heated by 1 C, the pressure increment will be in R about. The value of has the same value (more precisely, almost the same) for all gases, namely 1/273 C -1. The value of is called the temperature coefficient of pressure.
Charles' law allows you to calculate the pressure of a gas at any temperature if its pressure at a temperature of 0 C is known. Let the pressure of a given mass of gas at 0 C in a given volume p o, and the pressure of the same gas at temperature tp. The temperature changes to t, and the pressure changes to R about t, then the pressure R equals:
At very low temperatures, when the gas approaches the state of liquefaction, and also in the case of highly compressed gases, Charles's law is not applicable. The coincidence of the coefficients and included in Charles's law and Gay-Lussac's law is not accidental. Since gases obey the Boyle-Mariotte law at constant temperature, then and must be equal to each other.
Let us substitute the value of the temperature coefficient of pressure into the formula for the temperature dependence of pressure:
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Value ( 273+ t) can be considered as a temperature value measured on a new temperature scale, the unit of which is the same as that of the Celsius scale, and the point lying 273 below the point taken as zero of the Celsius scale, i.e., the melting point of ice . The zero of this new scale is called absolute zero. This new scale is called the thermodynamic temperature scale, where T t+273 .
Then, at a constant volume, Charles's law is valid:
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Objective
Checking Charles' Law
Tasks to be solved
Determination of the dependence of gas pressure on temperature at constant volume
Determination of the absolute temperature scale by extrapolation towards low temperatures
Safety
Attention: glass is used in the work.
Be extremely careful when working with a gas thermometer; glass jar and measuring cup.
Be extremely careful when working with hot water.
Experimental setup
Instruments and accessories
gas thermometer
Mobile CASSY Lab
Thermocouple
Electric hot plate
glass measuring cup
glass vessel
Manual vacuum pump
When air is pumped out at room temperature using a hand pump, pressure is created on the air column р0 + р, where R 0 - external pressure. A drop of mercury also exerts pressure on a column of air:
In this experiment, this law is confirmed using a gas thermometer. The thermometer is placed in water at a temperature of about 90°C and this system is gradually cooled. By evacuating the gas thermometer with a hand-held vacuum pump, a constant volume of air is maintained during cooling.
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Work order
Open the cap of the gas thermometer, connect a hand vacuum pump to the thermometer.
Turn the thermometer carefully as shown on the left in fig. 2 and evacuate air from it using a pump so that a drop of mercury is at point a) (see fig. 2).
After the drop of mercury has collected at point a) turn the thermometer with the hole upwards and bleed the forced air with the handle b) on the pump (see Fig. 2) carefully so that the mercury does not separate into several droplets.
Heat water in a glass vessel on a hot plate to 90°C.
Pour hot water into a glass vessel.
Place a gas thermometer in the vessel, fixing it on a tripod.
Place the thermocouple in water, this system gradually cools down. By evacuating the air from the gas thermometer using a handheld vacuum pump, maintain a constant volume of air column throughout the entire cooling process.
Record the pressure gauge reading R and temperature T.
Plot the dependence of the total gas pressure p 0 +p+p Hg from temperature in about C.
Continue the graph until it intersects with the x-axis. Determine the intersection temperature, explain the results.
Determine the temperature coefficient of pressure from the tangent of the slope.
Calculate the dependence of pressure on temperature at constant volume according to Charles's law and plot it. Compare theoretical and experimental dependences.
Ideal gas isoprocesses- processes in which one of the parameters remains unchanged.
1. Isochoric process . Charles' law. V = const.
Isochoric process called the process that takes place constant volume V. The behavior of the gas in this isochoric process obeys Charles law :
With a constant volume and constant values of the gas mass and its molar mass, the ratio of gas pressure to its absolute temperature remains constant: P / T= const.
Graph of the isochoric process on PV-diagram called isochore . It is useful to know the graph of the isochoric process on RT- and VT-diagrams (Fig. 1.6). Isochore equation:
Where Р 0 - pressure at 0 ° С, α - temperature coefficient of gas pressure equal to 1/273 deg -1. The graph of such a dependence on Pt-diagram has the form shown in Figure 1.7.
Rice. 1.7
2. isobaric process. Gay-Lussac's law. R= const.
An isobaric process is a process that occurs at a constant pressure P . The behavior of a gas in an isobaric process obeys Gay-Lussac's law:
At constant pressure and constant values of the mass of both the gas and its molar mass, the ratio of the volume of the gas to its absolute temperature remains constant: V/T= const.
Graph of the isobaric process on VT-diagram called isobar . It is useful to know the graphs of the isobaric process on PV- and RT-diagrams (Fig. 1.8).
Rice. 1.8
Isobar equation:
Where α \u003d 1/273 deg -1 - temperature coefficient of volume expansion. The graph of such a dependence on Vt the diagram has the form shown in Figure 1.9.
Rice. 1.9
3. isothermal process. Boyle's Law - Mariotte. T= const.
Isothermal process is a process that takes place when constant temperature T.
The behavior of an ideal gas in an isothermal process obeys Boyle-Mariotte law:
At a constant temperature and constant values of the gas mass and its molar mass, the product of the gas volume and its pressure remains constant: PV= const.
Isothermal process diagram PV-diagram called isotherm . It is useful to know the graphs of the isothermal process on VT- and RT-diagrams (Fig. 1.10).
Rice. 1.10
Isotherm equation:
| (1.4.5) |
4. adiabatic process(isoentropic):
An adiabatic process is a thermodynamic process that occurs without heat exchange with the environment.
5. polytropic process. A process in which the heat capacity of a gas remains constant. A polytropic process is a general case of all the processes listed above.
6. Avogadro's law. At the same pressures and the same temperatures, equal volumes of different ideal gases contain the same number of molecules. One mole of various substances contains N A\u003d 6.02 10 23 molecules (Avogadro number).
7. Dalton's Law. The pressure of a mixture of ideal gases is equal to the sum of the partial pressures P of the gases included in it:
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(1.4.6) |
The partial pressure Pn is the pressure that a given gas would exert if it alone occupied the entire volume.
At 
, the pressure of the mixture of gases.
DEFINITION
Processes in which one of the parameters of the state of the gas remains constant are called isoprocesses.
DEFINITION
Gas laws are the laws describing isoprocesses in an ideal gas.
The gas laws were discovered experimentally, but they can all be derived from the Mendeleev-Clapeyron equation.
Let's consider each of them.
Boyle-Mariotte's law (isothermal process)
Isothermal process A change in the state of a gas so that its temperature remains constant is called.
For a constant mass of gas at a constant temperature, the product of gas pressure and volume is a constant value:
The same law can be rewritten in another form (for two states of an ideal gas):
This law follows from the Mendeleev-Clapeyron equation:
Obviously, at a constant mass of gas and at a constant temperature, the right side of the equation remains constant.
Graphs of dependence of gas parameters at constant temperature are called isotherms.
Denoting the constant by the letter , we write down the functional dependence of pressure on volume in an isothermal process:
It can be seen that the pressure of a gas is inversely proportional to its volume. Inversely proportional graph, and, consequently, the graph of the isotherm in coordinates is a hyperbola(Fig. 1, a). Figure 1 b) and c) shows isotherms in coordinates and respectively.
Fig.1. Graphs of isothermal processes in various coordinates
Gay-Lussac's law (isobaric process)
isobaric process A change in the state of a gas so that its pressure remains constant is called.
For a constant mass of gas at constant pressure, the ratio of gas volume to temperature is a constant value:
This law also follows from the Mendeleev-Clapeyron equation:
isobars.
Consider two isobaric processes with pressures and title="(!LANG:Rendered by QuickLaTeX.com" height="18" width="95" style="vertical-align: -4px;">. В координатах и изобары будут иметь вид прямых линий, перпендикулярных оси (рис.2 а,б).!}
Let's determine the type of graph in coordinates. Denoting the constant with the letter, we write down the functional dependence of the volume on temperature during the isobaric process:
It can be seen that at constant pressure, the volume of a gas is directly proportional to its temperature. Direct proportionality graph, and, consequently, the graph of the isobar in coordinates is a straight line passing through the origin(Fig. 2, c). In reality, at sufficiently low temperatures, all gases turn into liquids, to which gas laws are no longer applicable. Therefore, near the origin, the isobars in Fig. 2, c) are shown by dotted lines.
Fig.2. Graphs of isobaric processes in various coordinates
Charles' law (isochoric process)
Isochoric process A change in the state of a gas so that its volume remains constant is called.
For a constant mass of gas at a constant volume, the ratio of gas pressure to its temperature is a constant value:
For two states of a gas, this law can be written as:
This law can also be obtained from the Mendeleev-Clapeyron equation:
Graphs of dependence of gas parameters at constant pressure are called isochores.
Consider two isochoric processes with volumes and title="(!LANG:Rendered by QuickLaTeX.com" height="18" width="98" style="vertical-align: -4px;">. В координатах и графиками изохор будут прямые, перпендикулярные оси (рис.3 а, б).!}
To determine the type of graph of the isochoric process in coordinates, we denote the constant in Charles's law by the letter , we get:
Thus, the functional dependence of pressure on temperature at constant volume is a direct proportionality, the graph of such a dependence is a straight line passing through the origin (Fig. 3, c).
Fig.3. Graphs of isochoric processes in various coordinates
Examples of problem solving
EXAMPLE 1
| Exercise | To what temperature must a certain mass of gas with an initial temperature be cooled isobarically so that the volume of the gas decreases by one quarter? |
| Solution | The isobaric process is described by the Gay-Lussac law: According to the condition of the problem, the volume of gas due to isobaric cooling decreases by one quarter, therefore: whence the final temperature of the gas: Let's convert the units to the SI system: initial gas temperature. Let's calculate:
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| Answer | The gas must be cooled to a temperature |
EXAMPLE 2
| Exercise | A closed vessel contains a gas at a pressure of 200 kPa. What will be the pressure of the gas if the temperature is increased by 30%? |
| Solution | Since the gas container is closed, the volume of the gas does not change. The isochoric process is described by Charles' law: According to the condition of the problem, the gas temperature increased by 30%, so we can write: Substituting the last relation into Charles's law, we get: Let's convert the units to the SI system: the initial gas pressure kPa \u003d Pa. Let's calculate: |
| Answer | The gas pressure will become equal to 260 kPa. |
EXAMPLE 3
| Exercise | The oxygen system that the aircraft is equipped with has  oxygen at a pressure of Pa. At the maximum lifting height, the pilot connects this system with an empty cylinder with a crane using a crane. What pressure will be established in it? The process of gas expansion occurs at a constant temperature. |
| Solution | The isothermal process is described by the Boyle-Mariotte law: |
Annotation: traditional presentation of the topic, supplemented by a demonstration on a computer model.
Of the three aggregate states of matter, the simplest is the gaseous state. In gases, the forces acting between molecules are small and under certain conditions they can be neglected.
The gas is called perfect , if:
The size of molecules can be neglected, i.e. molecules can be considered material points;
We can neglect the forces of interaction between molecules (the potential energy of interaction of molecules is much less than their kinetic energy);
The collisions of molecules with each other and with the walls of the vessel can be considered absolutely elastic.
Real gases are close in properties to the ideal at:
Conditions close to normal conditions (t = 0 0 C, p = 1.03 10 5 Pa);
At high temperatures.
The laws that govern the behavior of ideal gases were discovered experimentally quite a long time ago. So, Boyle's law - Mariotte was established in the 17th century. We give the formulations of these laws.
Boyle's Law - Mariotte. Let the gas be under conditions where its temperature is kept constant (such conditions are called isothermal ). Then for a given mass of gas, the product of pressure and volume is a constant value:
This formula is called isotherm equation. Graphically, the dependence of p on V for various temperatures is shown in the figure.
The property of a body to change pressure with a change in volume is called compressibility. If the change in volume occurs at T=const, then the compressibility is characterized by isothermal compressibility factor which is defined as the relative change in volume that causes a change in pressure per unit.
For an ideal gas, it is easy to calculate its value. From the isotherm equation we get:
The minus sign indicates that as the volume increases, the pressure decreases. Thus, the isothermal compressibility of an ideal gas is equal to the reciprocal of its pressure. With increasing pressure, it decreases, because. the greater the pressure, the less the gas has the ability to further compress.
Gay-Lussac law. Let the gas be under conditions where its pressure is maintained constant (such conditions are called isobaric ). They can be carried out by placing gas in a cylinder closed by a movable piston. Then a change in the temperature of the gas will move the piston and change the volume. The pressure of the gas will remain constant. In this case, for a given mass of gas, its volume will be proportional to the temperature:
where V 0 - volume at temperature t = 0 0 C, - volume expansion coefficient gases. It can be represented in a form similar to the compressibility factor:
Graphically, the dependence of V on T for various pressures is shown in the figure.
Moving from temperature in the Celsius scale to absolute temperature, Gay-Lussac's law can be written as:
Charles' Law. If the gas is under conditions where its volume remains constant ( isochoric conditions), then for a given mass of gas, the pressure will be proportional to the temperature:
where p 0 - pressure at temperature t \u003d 0 0 C, - pressure coefficient. It shows the relative increase in gas pressure when it is heated by 10:
Charles' law can also be written as:
Avogadro's law: One mole of any ideal gas at the same temperature and pressure occupies the same volume. Under normal conditions (t = 0 0 C, p = 1.03 10 5 Pa), this volume is equal to m -3 / mol.
The number of particles contained in 1 mole of various substances, called. Avogadro's constant :
It is easy to calculate the number n 0 particles in 1 m 3 under normal conditions:
This number is called Loschmidt number.
Dalton's law: the pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the gases included in it, i.e.
where - partial pressures- the pressure that the components of the mixture would exert if each of them occupied a volume equal to the volume of the mixture at the same temperature.
Equation of Clapeyron - Mendeleev. From the laws of an ideal gas, one can obtain equation of state , linking T, p and V of an ideal gas in a state of equilibrium. This equation was first obtained by the French physicist and engineer B. Clapeyron and Russian scientists D.I. Mendeleev, therefore bears their name.
Let some mass of gas occupies a volume V 1 , has a pressure p 1 and is at a temperature T 1 . The same mass of gas in a different state is characterized by the parameters V 2 , p 2 , T 2 (see figure). The transition from state 1 to state 2 is carried out in the form of two processes: isothermal (1 - 1") and isochoric (1" - 2).
For these processes, one can write down the laws of Boyle - Mariotte and Gay - Lussac:
Eliminating p 1 " from the equations, we get
Since states 1 and 2 were chosen arbitrarily, the last equation can be written as:
This equation is called Clapeyron's equation , in which B is a constant, different for different masses of gases.
Mendeleev combined Clapeyron's equation with Avogadro's law. According to Avogadro's law, 1 mole of any ideal gas at the same p and T occupies the same volume V m, so the constant B will be the same for all gases. This common constant for all gases is denoted R and is called universal gas constant. Then
This equation is ideal gas equation of state , which is also called Clapeyron - Mendeleev equation .
The numerical value of the universal gas constant can be determined by substituting the values of p, T and V m into the Clapeyron - Mendeleev equation under normal conditions:
The Clapeyron - Mendeleev equation can be written for any mass of gas. To do this, recall that the volume of a gas of mass m is related to the volume of one mole by the formula V \u003d (m / M) V m, where M is molar mass of gas. Then the Clapeyron - Mendeleev equation for a gas of mass m will look like:
where is the number of moles.
The equation of state for an ideal gas is often written in terms of Boltzmann's constant :
Based on this, the equation of state can be represented as
where is the concentration of molecules. From the last equation it can be seen that the pressure of an ideal gas is directly proportional to its temperature and concentration of molecules.
Small demo ideal gas laws. After pressing the button "Let's start" You will see the host's comments on what is happening on the screen (black color) and a description of the computer's actions after you press the button "Further"(Brown color). When the computer is "busy" (i.e., experience is in progress), this button is not active. Move on to the next frame only after understanding the result obtained in the current experiment. (If your perception does not match the host's comments, write!)
You can verify the validity of the ideal gas laws on the existing
