Ideal gas equation in physics. Ideal gas equation of state
The molecular kinetic concepts developed above and the equations obtained on their basis make it possible to find those relationships that connect the quantities that determine the state of the gas. These quantities are: the pressure under which the gas is located, its temperature and the volume V occupied by a certain mass of gas. These are called state parameters.
The three quantities listed are not independent. Each of them is a function of the other two. The equation connecting all three quantities - pressure, volume and temperature of a gas for a given mass is called the equation of state and can be generally written as follows:
This means that the state of a gas is determined by only two parameters (for example, pressure and volume, pressure and temperature, or, finally, volume and temperature), the third parameter is uniquely determined by the other two. If the equation of state is known explicitly, then any parameter can be calculated by knowing the other two.
To study various processes in gases (and not only in gases), it is convenient to use a graphical representation of the equation of state in the form of curves of the dependence of one of the parameters on another at a given constant third. For example, at a given constant temperature, the dependence of gas pressure on its volume
has the form shown in Fig. 4, where different curves correspond to different temperature values: the higher the temperature, the higher the curve lies on the graph. The state of the gas on such a diagram is represented by a dot. The curve of the dependence of one parameter on another shows a change in state, called a process in a gas. For example, the curves in Fig. 4 depict the process of expansion or compression of a gas at a given constant temperature.
In the future, we will widely use such graphs when studying various processes in molecular systems.
For ideal gases, the equation of state can be easily obtained from the basic equations of kinetic theory (2.4) and (3.1).
In fact, substituting into equation (2.4) instead of the average kinetic energy of molecules its expression from equation (3.1), we obtain:
If volume V contains particles, then substituting this expression into (4.1), we have:
This equation, which includes all three parameters of state, is the equation of state of ideal gases.
However, it is useful to transform it so that, instead of the number of particles inaccessible to direct measurement, it includes an easily measurable mass of gas. For such a transformation, we will use the concept of a gram molecule, or mol. Let us recall that a mole of a substance is a quantity of it whose mass, expressed in grams, is equal to the relative molecular mass of the substance (sometimes called molecular weight). This unique unit of quantity of a substance is remarkable, as is known, in that a mole of any substance contains the same number of molecules. In fact, if we denote the relative masses of two substances by and and the masses of the molecules of these substances, then we can write such obvious equalities;
where is the number of particles in a mole of these substances. Since from the very definition of relative mass it follows that
dividing the first of equalities (4.3) by the second, we obtain that a mole of any substance contains the same number of molecules.
The number of particles in a mole, the same for all substances, is called Avogadro's number. We will denote it by We can thus define the mole as a unit of a special quantity - the amount of a substance:
1 mole is an amount of substance containing a number of molecules or other particles (for example, atoms, if the substance is made of atoms) equal to Avogadro's number.
If we divide the number of molecules in a given mass of gas by Avogadro's number, then we get the number of moles in this mass of gas. But the same value can be obtained by dividing the mass of a gas by its relative mass so that
Let's substitute this expression for into formula (4.2). Then the equation of state will take the form:
This equation includes two universal constants: Avogadro’s number and Boltzmann’s constant. Knowing one of them, for example Boltzmann’s constant, the other (Avogadro’s number) can be determined by simple experiments using equation (4.4) itself. To do this, you should take some gas with a known relative mass, fill it with a vessel of known volume V, measure the pressure of this gas and its temperature and determine its mass by weighing the empty (evacuated) vessel and the vessel filled with gas. Avogadro's number turned out to be equal to moles.
Physical chemistry: lecture notes Berezovchuk A V
2. Equation of state of an ideal gas
Study of empirical gas laws (R. Boyle, J. Gay-Lussac) gradually led to the idea of an ideal gas, since it was discovered that the pressure of a given mass of any gas at a constant temperature is inversely proportional to the volume occupied by this gas, and the thermal coefficients of pressure and volume coincide with high accuracy for various gases, amounting, according to modern data, 1/ 273 deg –1. Having come up with a way to graphically represent the state of a gas in pressure-volume coordinates, B. Clapeyron received a unified gas law connecting all three parameters:
PV = BT,
where is the coefficient IN depends on the type of gas and its mass.
Only forty years later D. I. Mendeleev gave this equation a simpler form, writing it not for mass, but for a unit amount of a substance, i.e. 1 kmole.
PV = RT, (1)
Where R– universal gas constant.
Physical meaning of the universal gas constant. R– work of expansion of 1 kmole of an ideal gas when heated by one degree, if the pressure does not change. In order to understand the physical meaning R, imagine that the gas is in a vessel at constant pressure, and we increase its temperature by? T, Then
PV 1 = RT 1 , (2)
PV 2 = RT 2 . (3)
Subtracting equation (2) from (3), we obtain
P(V 2 – V 1) = R(T 2 – T 1).
If the right side of the equation is equal to one, i.e. we have heated the gas by one degree, then
R = P?V
Because the P=F/S, A? V equal to the area of the vessel S, multiplied by the lifting height of its piston? h, we have
Obviously, on the right we obtain an expression for the work, and this confirms the physical meaning of the gas constant.
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Equation of stateideal gas(Sometimes the equationClapeyron or the equationMendeleev - Clapeyron) - a formula establishing the relationship between pressure, molar volume and absolute temperature of an ideal gas. The equation looks like:
Since , where is the amount of substance, and , where is the mass, is the molar mass, the equation of state can be written:
This form of recording is called the Mendeleev-Clapeyron equation (law).
In the case of constant gas mass, the equation can be written as:
The last equation is called united gas law. From it the laws of Boyle - Mariotte, Charles and Gay-Lussac are obtained:
- Boyle's law - Mariotta.
- Gay-Lussac's Law.
- lawCharles(Gay-Lussac's second law, 1808). And in the form of proportion 
This law is convenient for calculating the transfer of gas from one state to another. From the point of view of a chemist, this law may sound slightly different: The volumes of reacting gases under the same conditions (temperature, pressure) relate to each other and to the volumes of the resulting gaseous compounds as simple integers. For example, 1 volume of hydrogen combines with 1 volume of chlorine, resulting in 2 volumes of hydrogen chloride:
1 A volume of nitrogen combines with 3 volumes of hydrogen to form 2 volumes of ammonia:
- Boyle's law - Mariotta. The Boyle-Mariotte law is named after the Irish physicist, chemist and philosopher Robert Boyle (1627-1691), who discovered it in 1662, and also after the French physicist Edme Mariotte (1620-1684), who discovered this law independently of Boyle in 1677. In some cases (in gas dynamics), it is convenient to write the equation of state of an ideal gas in the form
where is the adiabatic exponent, is the internal energy per unit mass of a substance. Emil Amaga discovered that at high pressures the behavior of gases deviates from the Boyle-Mariotte law. And this circumstance can be clarified 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 into 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. At not too high pressures, the second circumstance is more significant and the product decreases slightly. At very high pressures, the first circumstance plays a major role and the product increases.
5. Basic equation of the molecular kinetic theory of ideal gases
To derive the basic equation of molecular kinetic theory, consider a monatomic ideal gas. Let us assume that gas molecules move chaotically, the number of mutual collisions between gas molecules is negligible compared to the number of impacts on the walls of the vessel, and the collisions of molecules with the walls of the vessel are absolutely elastic. Let us select some elementary area DS on the wall of the vessel and calculate the pressure exerted on this area. With each collision, a molecule moving perpendicular to the platform transfers momentum to it m 0 v-(-m 0 v)=2m 0 v, Where T 0 - the mass of the molecule, v - its speed.
During the time Dt of the site DS, only those molecules that are enclosed in the volume of a cylinder with a base DS and height v D t .The number of these molecules is equal n D Sv D t (n- concentration of molecules).
It is necessary, however, to take into account that in reality the molecules move towards the site
DS at different angles and have different speeds, and the speed of the molecules changes with each collision. To simplify calculations, the chaotic movement of molecules is replaced by movement along three mutually perpendicular directions, so that at any moment of time 1/3 of the molecules move along each of them, with half of the molecules (1/6) moving along a given direction in one direction, half in the opposite direction . Then the number of impacts of molecules moving in a given direction on the DS pad will be 1/6 nDSvDt. When colliding with the platform, these molecules will transfer momentum to it
D R = 2m 0 v 1 / 6 n D Sv D t= 1 / 3 n m 0 v 2D S D t.
Then the gas pressure exerted by it on the wall of the vessel is
p=DP/(DtDS)= 1 / 3 nm 0 v 2 . (3.1)
If the gas volume V contains N molecules,
moving at speeds v 1 , v 2 , ..., v N, That
it is advisable to consider root mean square speed
characterizing the entire set of gas molecules.
Equation (3.1), taking into account (3.2), will take the form
p =
1
/
3
Fri 0
Expression (3.3) is called the basic equation of the molecular kinetic theory of ideal gases. Accurate calculation taking into account the movement of molecules throughout
possible directions is given by the same formula.
Considering that n = N/V we get
Where E - the total kinetic energy of the translational motion of all gas molecules.
Since the mass of gas m =Nm 0 , then equation (3.4) can be rewritten as
pV= 1 / 3 m
For one mole of gas t = M (M - molar mass), so
pV m = 1 / 3 M
Where V m - molar volume. On the other hand, according to the Clapeyron-Mendeleev equation, pV m =RT. Thus,
RT= 1 / 3 M
Since M = m 0 N A, where m 0 is the mass of one molecule, and N A is Avogadro’s constant, it follows from equation (3.6) that
Where k = R/N A- Boltzmann constant. From here we find that at room temperature, oxygen molecules have a mean square speed of 480 m/s, hydrogen molecules - 1900 m/s. At the temperature of liquid helium, the same speeds will be 40 and 160 m/s, respectively.
Average kinetic energy of translational motion of one ideal gas molecule
(we used formulas (3.5) and (3.7)) is proportional to the thermodynamic temperature and depends only on it. From this equation it follows that at T=0
« Physics - 10th grade"
This chapter will discuss the implications that can be drawn from the concept of temperature and other macroscopic parameters. The basic equation of the molecular kinetic theory of gases has brought us very close to establishing connections between these parameters.
We examined in detail the behavior of an ideal gas from the point of view of molecular kinetic theory. The dependence of gas pressure on the concentration of its molecules and temperature was determined (see formula (9.17)).
Based on this dependence, it is possible to obtain an equation connecting all three macroscopic parameters p, V and T, characterizing the state of an ideal gas of a given mass.
Formula (9.17) can only be used up to a pressure of the order of 10 atm.
The equation relating three macroscopic parameters p, V and T is called ideal gas equation of state.
Let us substitute the expression for the concentration of gas molecules into the equation p = nkT. Taking into account formula (8.8), the gas concentration can be written as follows:
where N A is Avogadro's constant, m is the mass of the gas, M is its molar mass. After substituting formula (10.1) into expression (9.17) we will have
The product of Boltzmann's constant k and Avogadro's constant N A is called the universal (molar) gas constant and is denoted by the letter R:
R = kN A = 1.38 10 -23 J/K 6.02 10 23 1/mol = 8.31 J/(mol K). (10.3)
Substituting the universal gas constant R into equation (10.2) instead of kN A, we obtain the equation of state of an ideal gas of arbitrary mass
The only quantity in this equation that depends on the type of gas is its molar mass.
The equation of state implies a relationship between the pressure, volume and temperature of an ideal gas, which can be in any two states.
If index 1 denotes the parameters related to the first state, and index 2 denotes the parameters related to the second state, then according to equation (10.4) for a gas of a given mass
The right-hand sides of these equations are the same, therefore, their left-hand sides must also be equal:
It is known that one mole of any gas under normal conditions (p 0 = 1 atm = 1.013 10 5 Pa, t = 0 °C or T = 273 K) occupies a volume of 22.4 liters. For one mole of gas, according to relation (10.5), we write:
We have obtained the value of the universal gas constant R.
Thus, for one mole of any gas
The equation of state in the form (10.4) was first obtained by the great Russian scientist D.I. Mendeleev. He is called Mendeleev-Clapeyron equation.
The equation of state in the form (10.5) is called Clapeyron equation and is one of the forms of writing the equation of state.
B. Clapeyron worked in Russia for 10 years as a professor at the Institute of Railways. Returning to France, he participated in the construction of many railways and drew up many projects for the construction of bridges and roads.
His name is included in the list of the greatest scientists of France, placed on the first floor of the Eiffel Tower.
The equation of state does not need to be derived every time, it must be remembered. It would be nice to remember the value of the universal gas constant:
R = 8.31 J/(mol K).
So far we have talked about the pressure of an ideal gas. But in nature and in technology, we very often deal with a mixture of several gases, which under certain conditions can be considered ideal.
The most important example of a mixture of gases is air, which is a mixture of nitrogen, oxygen, argon, carbon dioxide and other gases. What is the pressure of the gas mixture?
Dalton's law is valid for a mixture of gases.
Dalton's law
The pressure of a mixture of chemically non-interacting gases is equal to the sum of their partial pressures
p = p 1 + p 2 + ... + p i + ... .
where p i is the partial pressure of the i-th component of the mixture.
An ideal gas, the equation of state of an ideal gas, its temperature and pressure, volume... the list of parameters and definitions that are used in the corresponding section of physics can be continued for quite a long time. Today we will talk exactly about this topic.
What is considered in molecular physics?
The main object considered in this section is an ideal gas. The ideal gas was obtained taking into account normal environmental conditions, and we will talk about this a little later. Now let's approach this “problem” from afar.
Let's say we have a certain mass of gas. Her condition can be determined using three characters. These are, of course, pressure, volume and temperature. The equation of state of the system in this case will be the formula for the relationship between the corresponding parameters. It looks like this: F (p, V, T) = 0.
Here we are for the first time slowly approaching the emergence of such a concept as an ideal gas. It is a gas in which interactions between molecules are negligible. In general, this does not exist in nature. However, anyone is very close to him. Nitrogen, oxygen and air under normal conditions differ little from ideal. To write down the equation of state of an ideal gas, we can use the combined We get: pV/T = const.
Related Concept #1: Avogadro's Law
He can tell us that if we take the same number of moles of absolutely any random gas and put them in the same conditions, including temperature and pressure, then the gases will occupy the same volume. In particular, the experiment was carried out under normal conditions. This means that the temperature was equal to 273.15 Kelvin, the pressure was one atmosphere (760 millimeters of mercury or 101325 Pascals). With these parameters, the gas occupied a volume of 22.4 liters. Consequently, we can say that for one mole of any gas the ratio of numerical parameters will be a constant value. That is why it was decided to designate this number with the letter R and call it the universal gas constant. Thus, it is equal to 8.31. Dimension J/mol*K.
Ideal gas. Equation of state of an ideal gas and manipulation with it
Let's try to rewrite the formula. To do this, we write it in this form: pV = RT. Next, let's perform a simple action: multiply both sides of the equation by an arbitrary number of moles. We get pVu = uRT. Let's take into account the fact that the product of the molar volume and the amount of substance is simply volume. But the number of moles will simultaneously be equal to the quotient of mass and molar mass. This is exactly what it looks like. It gives a clear idea of what kind of system an ideal gas forms. The equation of state of an ideal gas will take the form: pV = mRT/M.
Let's derive the formula for pressure
Let's do some more manipulations with the resulting expressions. To do this, multiply the right side of the Mendeleev-Clapeyron equation and divide it by Avogadro's number. Now we carefully look at the product of the amount of substance by This is nothing more than the total number of molecules in the gas. But at the same time, the ratio of the universal gas constant to the Avogadro number will be equal to the Boltzmann constant. Therefore, the formulas for pressure can be written as follows: p = NkT/V or p = nkT. Here the designation n is the concentration of particles.
Ideal gas processes
In molecular physics there is such a thing as isoprocesses. These are those that take place in the system under one of the constant parameters. In this case, the mass of the substance must also remain constant. Let's look at them more specifically. So, the laws of ideal gas.
The pressure remains constant
This is Gay-Lussac's law. It looks like this: V/T = const. It can be rewritten in another way: V = Vo (1+at). Here a is equal to 1/273.15 K^-1 and is called the “volume expansion coefficient”. We can substitute the temperature on both the Celsius and Kelvin scales. In the latter case we obtain the formula V = Voat.
The volume remains constant
This is Gay-Lussac's second law, more commonly called Charles's law. It looks like this: p/T = const. There is another formulation: p = po (1 + at). Conversions can be carried out in accordance with the previous example. As you can see, the laws of an ideal gas are sometimes quite similar to each other.
The temperature remains constant
If the temperature of an ideal gas remains constant, then we can obtain the Boyle-Mariotte law. It can be written this way: pV = const.
Related Concept #2: Partial Pressure
Let's say we have a vessel with gases. It will be a mixture. The system is in a state of thermal equilibrium, and the gases themselves do not react with each other. Here N will denote the total number of molecules. N1, N2 and so on, respectively, the number of molecules in each of the components of the existing mixture. Let's take the pressure formula p = nkT = NkT/V. It can be opened for a specific case. For a two-component mixture, the formula will take the form: p = (N1 + N2) kT/V. But then it turns out that the total pressure will be summed up from the partial pressures of each mixture. This means that it will look like p1 + p2 and so on. These will be the partial pressures.
What is it for?
The formula we obtained indicates that pressure in the system is exerted by each group of molecules. By the way, it does not depend on others. Dalton took advantage of this when formulating the law that was later named after him: in a mixture where gases do not react chemically with each other, the total pressure will be equal to the sum of the partial pressures.
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