The relationship between pressure, temperature, volume and the number of moles of gas ("mass" of the gas). Universal (molar) gas constant R. Cliperon-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 are prone to significant dissociation or decomposition as the temperature rises. |
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Let's solve a couple of problems regarding gas volumetric and mass flow rates under the assumption that the gas composition does not change (the gas does not dissociate) - which is true for most of the gases in the above.
This task 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, T 1 and T 2 let it go T 1< T 2... Then we know that:
Naturally, V 1< V 2
- the indicators of the volumetric gas meter are the more "weighty", the lower the temperature
- it is profitable to supply "warm" gas
- it is profitable to buy "cold" gas
How to deal with this? At least a simple temperature compensation is required, i.e. information from an additional temperature sensor must be fed to the reading device.
This task 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 volumetric accumulated costs V 1 and V 2, at pressures, respectively, P 1 and P 2 let it go P 1< P 2... Then we know that:
Naturally, V 1>V 2 for the same amount of gas under the given conditions. Let's try to formulate several conclusions that are important in practice for this case:
- the indicators of the volumetric gas meter are the more "weighty", the higher the pressure
- profitable to supply low pressure gas
- it is profitable to buy high pressure gas
How to deal with this? At least a simple pressure compensation is required, i.e. information from an additional pressure sensor must be supplied to the reading device.
In conclusion, I would like to note that, in theory, every gas meter should have both temperature compensation and pressure compensation. Practically the same ......
2. Isochoric process... V is constant. P and T change. Gas obeys Charles's 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 a constant temperature is inversely proportional to the volume of the gas.
4. From a large number of processes in 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 and volume divided by the absolute temperature is a constant value.
This law is applicable for a large number of processes in gas, when the gas parameters do not change very quickly.
All the listed laws for real gases are approximate. Errors increase with increasing gas pressure and density.
Work order:
1.part of work.
1. The hose of the glass ball is immersed in a vessel with water at room temperature (Fig. 1 in the appendix). Then we heat the ball (with our hands, warm water) Considering the gas pressure constant, write how the gas volume depends on the temperature
Output:………………..
2. Connect a hose to a cylindrical vessel with a millimanometer (Fig. 2). We heat a metal vessel and the air in it with a lighter. Considering the volume of gas constant, write how the gas pressure depends on temperature.
Output:………………..
3. We squeeze the cylindrical vessel connected to the millimometer with our hands, reducing its volume (Fig. 3). Considering the gas temperature constant, write how the gas pressure depends on the volume.
Output:……………….
4. Connect the pump to the ball chamber and pump in several portions of air (Fig. 4). How did the pressure, volume and temperature of the air pumped into the chamber change?
Output:………………..
5. Pour about 2 cm 3 of alcohol into the bottle, close it with a stopper with a hose (Fig. 5) attached to the injection pump. Let's do a few strokes until the cork comes out of the bottle. How do the pressure, volume and temperature of air (and alcohol vapors) change after the cork blows out?
Output:………………..
Part of work.
Checking the Gay - Lussac law.
1. Remove 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
Tube and air column length (at the beginning of the experiment)
The volume of warm air in the tube,
Cross-sectional area of the tube.
The height of the water column that entered the tube when the air in the tube cools.
Cold air column length 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 the bucket
Room temperature
We need to check equation (3) and therefore the Gay-Lussac law.
5. Let's calculate
6. Find the relative measurement error when measuring the length, taking Dl = 0.5 cm.
7. Find the absolute error of the ratio
=……………………..
8. We write down the result of the reading
………..…..
9. Find the relative measurement error T, taking
10. Find the absolute calculation error
11. We 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.
Output:……………………………………………………………………………………………………
Reporting 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. Build isoprocess graphs (for experiments 1, 2, 3) in the axes:; ; ...
7. Solve tasks:
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.
Application
Since P is constant during an isobaric process, after cancellation by P the formula takes the form
V 1 / T 1 = V 2 / T 2,
V 1 / V 2 = T 1 / T 2.
The formula is a mathematical expression of the Gay-Lussac law: with a constant mass of gas and constant pressure, the volume of a gas is directly proportional to its absolute temperature.
Isothermal process
A process in a gas that occurs at a constant temperature is called isothermal. The isothermal process in gas was studied by the English scientist R. Boyle and the French scientist E. Mariot. The connection established by them empirically is obtained directly from the formula by reducing by T:
p 1 V 1 = p 2 V 2,
p 1 / p 2 = V 1 / V 2.
The formula is a mathematical expression Boyle's law: at constant gas mass and constant temperature, gas pressure is inversely proportional to its volume. In other words, under these conditions, the product of the gas volume and the corresponding pressure is a constant value:
The plot of p versus V for an isothermal process in a gas is a hyperbola and is called an isotherm. Figure 3 shows isotherms for the same mass of gas, but at different temperatures T. In an isothermal process, the gas density changes in direct proportion to the pressure:
ρ 1 / ρ 2 = p 1 / p 2
Dependence of gas pressure on temperature at constant volume
Let us consider how the gas pressure depends on temperature when its mass and volume remain constant. Take a closed vessel with gas and heat it up (Figure 4). The gas temperature t will be determined with a thermometer, and the pressure with a manometer M.
First, we will place the vessel in melting snow and the gas pressure at 0 0 C will be denoted by p 0, and then we will gradually heat the outer vessel and record the values of p and t for the gas.
It turns out that the plot of p and t dependence, built on the basis of such experience, looks like a straight line (Figure 5).
If we continue this graph to the left, then it will intersect with the abscissa at point A, corresponding to zero gas pressure. From the similarity of the triangles in Figure 5, but you can write:
P 0 / OA = Δp / Δt,
l / OA = Δp / (p 0 Δt).
If we denote the constant l / ОА by α, then we get
α = Δp // (p 0 Δt),
Δp = α p 0 Δt.
In terms of its meaning, the coefficient of proportionality α in the described experiments should express the dependence of the change in gas pressure on its kind.
The magnitude γ, characterizing the dependence of the change in gas pressure on its kind in the process of temperature change at a constant volume and constant mass of gas, is called the temperature coefficient of pressure. The temperature coefficient of pressure shows how much of the gas pressure taken at 0 0 С changes by 1 0 С when heated. Let's derive the unit of the temperature coefficient α in SI:
α = l ΠA / (l ΠA * l 0 C) = l 0 C -1
In this case, the length of the segment ОА turns out to be equal to 273 0 С. Thus, for all cases, the temperature at which the gas pressure should vanish is the same and equal to - 273 0 С, and the temperature coefficient of pressure α = 1 / ОА = (1/273 ) 0 С -1.
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When solving problems, they usually use an approximate value of α equal to α = 1 / ОА = (1/273) 0 С -1. From experiments, the value of α 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 to (1 / 273.15) 0 С -1. Note that this is true only for low-density gases and for small temperature changes; at high pressures or low temperatures, α depends on the type of gas. Only ideal gas exactly obeys Charles's law. Let us find out how you can determine the pressure of any gas p, at an arbitrary temperature t.
Substituting these values Δр and Δt in the formula, we obtain
p 1 -p 0 = αp 0 t,
p 1 = p 0 (1 + αt).
Since α ~ 273 0 С, when solving problems, the formula can be used in the following form:
p 1 = p 0
The unified gas law is applicable to any isoprocess, taking into account that one of the parameters remains constant. In an isochoric process, the volume V remains constant, the formula after reduction by V takes the form
Ideal gas equation of state defines the relationship of temperature, volume and pressure of bodies.
- Allows you to determine one of the quantities characterizing the state of the gas, by the other two (used in thermometers);
- Determine how the 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 describing isoprocesses in ideal gases, i.e. processes in which one of the macroparameters (T, P, V) in a closed isolated system is constant.
The quantitative relationships 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, gas behaves like a compressed spring, as can be seen by compressing 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).
Independently of Boyle, a little later the French scientist E. Mariotte (1620-1684) came to the same conclusions. Therefore, the found law was called the Boyle-Mariotte law.
The product of the gas pressure of a given mass by its volume is constant, if the temperature does not change
pV = const
Gay Lussac's Law
The message about the discovery of another gas law was published only in 1802, almost 150 years after the discovery of the Boyle-Mariotte law. The law determining the dependence of the volume of gas 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 a constant volume was experimentally established by the French physicist J. Charles (1746-1823) in 1787.
J. Charles in 1787, that is, earlier than Gay-Lussac, also established the dependence of volume on temperature at constant pressure, but he did not publish his works in a timely manner.
The pressure of a given mass of gas at a constant volume is directly proportional to the absolute temperature.
p = p 0 γT
| Name | The wording | Charts |
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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 |
Introduction
The state of an ideal gas is fully described by measurable quantities: pressure, temperature, volume. The relationship between these three quantities is determined by the basic gas law:
purpose of work
Boyle-Marriott's Law Test.
Tasks to be solved
Measurement of air pressure in a syringe with a change in volume, taking into account that the gas temperature is constant.
Experimental setup
Devices 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 manometer.
Work order
Set the syringe plunger to the 50 ml mark.
Push the free end of the hand vacuum pump connecting hose tightly onto the syringe outlet.
While extending the piston, increase the volume in 5 ml increments, record the manometer reading 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. The atmospheric pressure is approximately 1 bar, which corresponds to 100,000 Pa.
The presence of air in the connecting hose must be taken into account when evaluating the measurement results. To do this, measure and calculate 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 vernier caliper, taking into account that the wall thickness is 1.5 mm.
Plot the measured air volume versus pressure.
Calculate the dependence of volume on pressure at a constant temperature according to the Boyle-Mariotte law and build a graph.
Compare theoretical and experimental relationships.
2133. Dependence of gas pressure on temperature at constant volume (charles' law)
Introduction
Let us consider the dependence of gas pressure on temperature, provided that the volume of a certain mass of gas remains unchanged. These studies were first carried out in 1787 by Jacques-Alexander 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. Ignoring the negligible increase in the volume of the flask when heated and the slight change in volume when the mercury moves in a narrow gauge tube. Thus, the gas volume can be considered unchanged. Heating the water in the vessel surrounding the flask, the gas temperature was measured by a thermometer T, and the corresponding pressure R- according to the manometer. After filling the vessel with melting ice, the pressure was determined R O, and the corresponding temperature T O... It was found that if at 0 С the pressure R O , then when heating by 1 C, the pressure increment will be in R O... The quantity has the same value (more precisely, almost the same) for all gases, namely 1/273 C -1. The quantity is called the temperature coefficient of pressure.
Charles's law makes it possible to calculate the pressure of a gas at any temperature if its pressure is known at a temperature of 0 C. 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 a temperature tp... The temperature changes by t, and the pressure changes by R O t then the pressure R equals:
At very low temperatures, when the gas approaches the state of liquefaction, as well as 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.
Substitute the value of the temperature coefficient of pressure into the formula for the temperature dependence of pressure:
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The quantity ( 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 a point lying 273 below the point taken as zero of the Celsius scale, i.e., the point of ice melting, is taken as zero ... The zero of this new scale is called absolute zero. This new scale is called the thermodynamic temperature scale, where T t+273 .
Then, with a constant volume, Charles's law is valid:
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purpose of work
Checking Charles's 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 engineering
Attention: glass is used in the work.
Be extremely careful when working with a gas thermometer; glass vessel and measuring beaker.
Be extremely careful when working with hot water.
Experimental setup
Devices and accessories
Gas thermometer
Mobile CASSY Lab
Thermocouple
Electric heating plate
Glass beaker
Glass vessel
Manual vacuum pump
When pumping out air at room temperature using a hand pump, pressure is created on the air column p0 + p, where R 0 - external pressure. A drop of mercury also puts pressure on the air column:
In this experiment, this law is confirmed using a gas thermometer. The thermometer is placed in water with a temperature of about 90 ° C and this system is gradually cooled. By evacuating air from the gas thermometer using a hand-held vacuum pump, a constant air volume is maintained during cooling.
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Work order
Open the plug of the gas thermometer, connect the 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 droplet of mercury is at point a) (see Fig. 2).
After a droplet of mercury has collected at point a), turn the thermometer with the hole upward and release the blown air with the handle b) on the pump (see Fig. 2), carefully so that the mercury does not split into several droplets.
Heat water in a glass vessel on a hotplate to 90 ° C.
Pour hot water into a glass container.
Place the gas thermometer in the vessel, securing it to the tripod.
Place the thermocouple in water, this system gradually cools down. By evacuating air from a gas thermometer using a manual vacuum pump, you maintain a constant volume of the column of air throughout the entire cooling process.
Take a reading of the pressure gauge 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 to the intersection with the abscissa. Determine the intersection temperature, explain the results obtained.
Determine the temperature coefficient of pressure from the slope.
Calculate the dependence of pressure on temperature at constant volume according to Charles's law and build a graph. Compare theoretical and experimental relationships.