The law is formulated as follows: the product of the volume of a given mass of gas and its pressure at a constant temperature is a constant value. Mathematically, this law can be written like this:
P 1 V 1 = P 2 V 2 or PV = const (1)
The following consequences follow from the Boyle-Marriott law: the density and concentration of a gas at a constant temperature is directly proportional to the pressure under which the gas is located:
(2);
(3) ,
Where d 1 – density, C 1 – gas concentration under pressure P 1; d 2 and C 2 are the corresponding values under pressure P 2 .
Example 1. A gas cylinder with a capacity of 0.02 m 3 contains gas under a pressure of 20 atm. What volume will the gas occupy if the cylinder valve is opened without changing its temperature? Final pressure 1 atm.
Example 2. Compressed air is supplied to a gas holder (gas collection tank) with a volume of 10 m3. How long will it take to pump it up to a pressure of 15 atm if the compressor sucks in 5.5 m 3 of atmospheric air per minute at a pressure of 1 atm? The temperature is assumed to be constant.
Example 3. 112 g of nitrogen under a pressure of 4 atm occupy a volume of 20 liters. What pressure must be applied so that the nitrogen concentration becomes 0.5 mol/l, provided that the temperature remains unchanged?
1.1.2 Gay-Lussac's and Charles's laws
Gay-Lussac found that at constant pressure, with an increase in temperature of 1°C, the volume of a given mass of gas increases by 1/273 of its volume at 0°C.
Mathematically, this law is written:
(4)
,
Where V- volume of gas at temperature t°С, a V 0 – volume of gas at 0°C.
Charles showed that the pressure of a given mass of gas, when heated by 1°C at a constant volume, increases by 1/273 of the pressure that the gas has at 0°C. Mathematically, this law is written as follows:
(5)
,
where P 0 and P are gas pressures, respectively, at temperatures 0С and tС.
When replacing the Celsius scale with the Kelvin scale, the connection between them is established by the relation T = 273 + t, the formulas of Gay-Lussac's and Charles's laws are significantly simplified.
Gay-Lussac's Law: at constant pressure, the volume of a given mass of gas is directly proportional to its absolute temperature:
(6)
.
Charles's Law: at constant volume, the pressure of a given mass of gas is directly proportional to its absolute temperature:
(7)
.
From the laws of Gay-Lussac and Charles it follows that at constant pressure the density and concentration of a gas are inversely proportional to its absolute temperature:
(8)
,
(9) .
Where d 1 and C 1 - density and concentration of gas at absolute temperature T 1, d 2 and C 2 are the corresponding values at absolute temperature T 2 .
Example 4. At 20ºC the volume of gas is 20.4 ml. What volume will the gas occupy when it is cooled to 0°C if the pressure remains constant?
Primep 5. At 9°C, the pressure inside the oxygen cylinder was 94 atm. Calculate how much the pressure in the cylinder increased if the temperature rose to 27ºC?
Example 6. Density of chlorine gas at 0ºС and pressure 760 mm Hg. Art. equal to 3.220 g/l. Find the density of chlorine, taking it as an ideal gas, at 27ºС at the same pressure.
Example 7. Under normal conditions, the concentration of carbon monoxide is 0.03 kmol/m3. Calculate at what temperature the mass of 10 m 3 of carbon monoxide will be equal to 7 kg?
Combined Boyle-Mariotte-Charles-Gay-Lussac law.
The formulation of this law: for a given mass of gas, the product of pressure and volume divided by absolute temperature is constant for all changes occurring in the gas. Mathematical notation:
(10)
where V 1 is the volume and P 1 is the pressure of a given mass of gas at absolute temperature T 1 , V 2 - volume and P 2 - pressure of the same mass of gas at absolute temperature T 2.
One of the most important applications of the unified gas law is “bringing the volume of gas to normal conditions.”
Example 8. Gas at 15°C and pressure 760 mmHg. Art. occupies a volume of 2 liters. Bring the volume of gas to normal conditions.
To facilitate such calculations, you can use the conversion factors given in the tables.
Example 9. In the gasometer above the water there is 7.4 liters of oxygen at a temperature of 23°C and a pressure of 781 mm Hg. Art. The water vapor pressure at this temperature is 21 mm Hg. Art. What volume will the oxygen in the gasometer occupy under normal conditions?
The Boyle-Mariotte law is one of the fundamental laws of physics and chemistry, which relates changes in pressure and volume of gaseous substances. Using our calculator it is easy to solve simple problems in physics or chemistry.
Boyle-Marriott Law
The isothermal gas law was discovered by an Irish scientist Robert Boyle, who conducted experiments on gases under pressure. Using a U-shaped tube and ordinary mercury, Boyle established a simple principle that at any given time the product of pressure and volume of a gas is constant. Speaking in dry mathematical language, the Boyle-Mariotte law states that at constant temperature the product of pressure and volume is constant:
To maintain a constant ratio, quantities must change in different directions: by how many times one quantity decreases, by the same number of times another increases. Consequently, the pressure and volume of a gas are inversely proportional and the law can be rewritten as follows:
P1×V1 = P2×V2,
where P1 and V1 are the initial values of pressure and volume, respectively, and P2 and V2 are the final values.
Application of the Boyle-Mariotte law
The best illustration of the manifestation of the law discovered by Boyle is the immersion of a plastic bottle under water. It is known that if a gas is placed in a cylinder, then the pressure on the substance will be determined only by the walls of the cylinder. It's another matter when it is a plastic bottle that easily changes its shape. On the surface of the water (pressure 1 atmosphere), a closed bottle will retain its shape, but when immersed to a depth of 10 m, a pressure of 2 atmospheres will act on the walls of the vessel, the bottle will begin to shrink, and the volume of air will decrease by half. The deeper the plastic container is immersed, the less volume the air inside it will occupy.
This simple demonstration of the gas law illustrates an important point for many divers. If on the surface of the water an air cylinder has a capacity of 20 liters, then when diving to a depth of 30 m, the air inside will be compressed three times, therefore, the air for breathing at such a depth will be three times less than on the surface.
Beyond the diving theme, the Boyle-Marriott law in action can be observed in the process of compressing air in a compressor or in the expansion of gases when using a pump.
Our program is an online tool that makes it easy to calculate the proportion for any gas isothermal process. To use the tool, you need to know any three quantities, and the calculator will automatically calculate the required one.
Examples of how the calculator works
School task
Let's consider a simple school problem in which you need to find the initial volume of a gas if the pressure changes from 1 to 3 atmospheres and the volume decreases to 10 liters. So, we have all the data for the calculation that needs to be entered into the appropriate cells of the calculator. As a result, we find that the initial volume of gas was 30 liters.
More about diving
Let's remember a plastic bottle. Let's imagine that we immersed a bottle filled with 19 liters of air to a depth of 40 m. How will the volume of air on the surface change? This is a more difficult problem, but only because we need to convert depth into pressure. We know that at the surface of water the atmospheric pressure is 1 bar, and when immersed in water the pressure increases by 1 bar every 10 m. This means that at a depth of 40 m the bottle will be under a pressure of approximately 5 atmospheres. We have all the data for the calculation, and as a result we will see that the volume of air on the surface will increase to 95 liters.
Conclusion
The Boyle-Marriott law occurs quite often in our lives, so you will undoubtedly need a calculator that automates calculations using this simple proportion.
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 exactly does the pressure of a gas increase with a decrease in volume 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 Ah, 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 Ah, can be counted by tube divisions A. The pressure of this air will differ from atmospheric pressure by the pressure of a column of mercury, 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; air volume in A
increases accordingly. Comparing the values obtained in this way for the pressure and volume of air locked in tube A, 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. The air temperature in the tube can be considered constant in our experiments. 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 satisfied to a high degree
accuracy. For highly compressed or cooled gases, noticeable deviations from this law are found. Formula expressing the Boyle-Mariotte law.
Boyle-Mariotte Law (Isotherm), one of the basic gas laws that describes isothermal processes in ideal gases. It was established by scientists R. Boyle in 1662 and E. Marriott in 1676 independently of each other during an experimental study of the dependence of gas pressure on its volume at a constant temperature.
According to the Boyle-Mariotte law at constant temperature (T=const), the Volume (V) of a given mass (m) of an ideal gas is inversely proportional to its pressure (p):
pV = const = С at T=const and m=const
The constant C is proportional to the mass of the gas (number of moles) and its absolute temperature. In other words: the product of the volume of a given mass of an ideal gas and its pressure is constant at a constant temperature. Boyle-Mariotte's law holds strictly for an ideal gas. For real gases, the Boyle-Mariotte law is satisfied approximately. Almost all gases behave as ideal gases at not too high pressures and not too low temperatures.
The Boyle-Mariotte law follows from the kinetic theory of gases, when the assumption is made that the sizes of molecules are negligible compared to the distance between them and there is no intermolecular interaction. At high pressures, it is necessary to introduce corrections for the forces of attraction between molecules and for the volume of the molecules themselves. Like the Clayperon equation, the Boyle-Mariotte law describes the limiting case of the behavior of a real gas, more accurately described by the van der Waals equation. The application of the law can be approximately observed in the process of compressing air by a compressor or as a result of the expansion of gas under the piston of a pump when pumping it out of a vessel.
A thermodynamic process that occurs at a constant temperature is called isothermal. Its image on the graph (Fig. 1) is called an isotherm.
Fig.1
Gay-Lussac's law. Isobar
In 1802, the French scientist J. Gay-Lussac experimentally discovered the dependence of gas volume on temperature at constant pressure. The data is the basis of Gay-Lussac's gas law.
The formulation of Gay-Lussac's law is as follows: for a given mass of gas, the ratio of the volume of the gas to its temperature is constant if the pressure of the gas does not change. This relationship is written mathematically as follows:
V/T=const, if P=const and m=const
This law can be approximately observed when gas expands when it is heated in a cylinder with a movable piston. Constant pressure in the cylinder is ensured by atmospheric pressure on the outer surface of the piston. Another manifestation of Gay-Lussac's law in action is the balloon. Gay-Lussac's law is not observed in the region of low temperatures close to the temperature of liquefaction (condensation) of gases.
The law is valid for an ideal gas. It works well for rarefied gases, which are close to ideal in their properties. The gas temperature must be high enough.
Graphically, this dependence in V-T coordinates is depicted as a straight line extending from the point T=0. This straight line is called an isobar. Different pressures correspond to different isobars. The process of changing the state of a thermodynamic system at constant pressure is called isobaric (Fig. 2 graph of an isobaric process).
Fig.2
Charles's law. Isochora
In 1787, the French scientist J. Charles experimentally discovered the dependence of gas pressure on temperature at constant volume. The data is the basis of Charles' gas law.
The formulation of Charles's law is as follows: for a given mass of gas, the ratio of gas pressure to its temperature is constant if the volume of the gas does not change. This relationship is written mathematically as follows:
P/T=const, if V=const and m=const
This law can be approximately observed when gas pressure increases in any container or in an electric light bulb when heated. The isochoric process is used in constant-volume gas thermometers. Charles's law is not observed in the region of low temperatures close to the temperature of liquefaction (condensation) of gases.
The law is valid for an ideal gas. It works well for rarefied gases, which are close to ideal in their properties. The gas temperature must be high enough. The process must be very slow
Graphically, this dependence in P-T coordinates is depicted as a straight line extending from the point T=0. This straight line is called an isochore. Different isochores correspond to different volumes. The process of changing the state of a thermodynamic system at a constant volume is called isochoric. Fig. 3 (graph of an isochoric process).
22. Boyle-Mariotte Law
One of the ideal gas laws is Boyle-Marriott law, which reads: product of pressure P per volume V gas at constant gas mass and temperature constantly. This equality is called isotherm equations. The isotherm is depicted on the PV diagram of the gas state in the form of a hyperbola and, depending on the temperature of the gas, occupies one position or another. The process going on T= const, called isothermal. Gas at T= const has constant internal energy U. If a gas expands isothermally, then all the heat goes to doing work. The work that a gas does when expanding isothermally is equal to the amount of heat that needs to be imparted to the gas to perform it:
dA= dQ= PdV,
where d A– basic work;
dV- elementary volume;
P- pressure. If V 1 > V 2 and P 1< P 2 , то газ сжимается, и работа принимает отрицательное значение. Для того чтобы условие T= const was fulfilled, it is necessary to assume that changes in pressure and volume are infinitely slow. There is also a requirement for the environment in which the gas is located: it must have a sufficiently high heat capacity. The calculation formulas are also suitable in the case of supplying thermal energy to the system. Compressibility The property of a gas to change in volume when pressure changes is called. Each substance has compressibility factor, and it is equal to:
c = 1 / V O(dV/CP)T,
here the derivative is taken at T= const.
The compressibility coefficient is introduced to characterize the change in volume with a change in pressure. For an ideal gas it is equal to:
c = -1 / P.
In SI, the compressibility coefficient has the following dimension: [c] = m 2 /N.
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