The ideal gas equation of state is valid for. Equation of state of gases. The volume remains constant
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1. An ideal gas is a gas in which there are no intermolecular interaction forces. With a sufficient degree of accuracy, gases can be considered ideal in cases where their states are considered that are far from the regions of phase transformations.
2. The following laws are valid for ideal gases:
a) Boyle’s Law - Mapuomma: at constant temperature and mass, the product of the numerical values of pressure and volume of a gas is constant:
pV = const
Graphically, this law in PV coordinates is depicted by a line called an isotherm (Fig. 1).
b) Gay-Lussac's law: at constant pressure, the volume of a given mass of gas is directly proportional to its absolute temperature:
V = V0(1 + at)
where V is the volume of gas at temperature t, °C; V0 is its volume at 0°C. The quantity a is called the temperature coefficient of volumetric expansion. For all gases a = (1/273°С-1). Hence,
V = V0(1 +(1/273)t) Graphically, the dependence of volume on temperature is depicted by a straight line - an isobar (Fig. 2). At very low temperatures
ah (close to -273°C), Gay-Lussac’s law is not satisfied, so the solid line on the graph is replaced by a dotted line.
c) Charles’s law: at constant volume, the pressure of a given mass of gas is directly proportional to its absolute temperature:
p = p0(1+gt)
where p0 is the gas pressure at temperature t = 273.15 K.
The quantity g is called the temperature coefficient of pressure. Its value does not depend on the nature of the gas; for all gases = 1/273 °C-1. Thus,
p = p0(1 +(1/273)t)
The graphical dependence of pressure on temperature is depicted by a straight line - an isochore (Fig. 3).
d) Avogadro's law: at the same pressures and the same temperatures and equal volumes of different ideal gases, the same number of molecules is contained; or, what is the same: at the same pressures and the same temperatures, the gram molecules of different ideal gases occupy the same volumes.
So, for example, under normal conditions (t = 0°C and p = 1 atm = 760 mm Hg), gram molecules of all ideal gases occupy a volume Vm = 22.414 liters. The number of molecules located in 1 cm3 of an ideal gas at under normal conditions, is called the Loschmidt number; it is equal to 2.687*1019> 1/cm3
3. The equation of state of an ideal gas has the form:
where p, Vm and T are the pressure, molar volume and absolute temperature of the gas, and R is the universal gas constant, numerically equal to the work done by 1 mole of an ideal gas when heated isobarically by one degree:
R = 8.31*103 J/(kmol*deg)
For an arbitrary mass M of gas, the volume will be V = (M/m)*Vm and the equation of state has the form:
pV = (M/m)RT
This equation is called the Mendeleev-Clapeyron equation.
4. From the Mendeleev-Clapeyron equation it follows that the number n0 of molecules contained in a unit volume of an ideal gas is equal to
n0 = NA/Vm = p*NA /(R*T) = p/(kT)
where k = R/NA = 1/38*1023 J/deg - Boltzmann's constant, NA - Avogadro's number.
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 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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The Mendeleev-Clapeyron equation is an equation of state for an ideal gas, referred to 1 mole of gas. In 1874, D.I. Mendeleev, based on the Clapeyron equation, combining it with Avogadro’s law, using the molar volume V m and relating it to 1 mole, derived the equation of state for 1 mole of an ideal gas:
pV = RT, Where R- universal gas constant,
R = 8.31 J/(mol. K)
The Clapeyron-Mendeleev equation shows that for a given mass of gas it is possible to simultaneously change three parameters characterizing the state of an ideal gas. For an arbitrary gas mass M, molar mass which m: pV = (M/m) . RT. or pV = N A kT,
where N A is Avogadro's number, k is Boltzmann's constant.
Derivation of the equation:
Using the equation of state of an ideal gas, one can study processes in which the mass of the gas and one of the parameters - pressure, volume or temperature - remain constant, and only the other two change, and theoretically obtain gas laws for these conditions of change in the state of the gas.
Such processes are called isoprocesses.
The laws describing isoprocesses were discovered long before the theoretical derivation of the equation of state of an ideal gas.
Isothermal process - the process of changing the state of a system at a constant temperature. For a given mass of gas, the product of the gas pressure and its volume is constant if the gas temperature does not change . Thisdifferent isotherms correspond.
Isobaric process - the process of changing the state of a system at constant pressure. For a gas of a given mass, the ratio of gas volume to its temperature remains constant if the gas pressure does not change. This Gay-Lussac's law. According to Gay-Lussac's law, the volume of a gas is directly proportional to its temperature: V/T=const. Graphically, this dependence is V-T coordinates is depicted as a straight line coming from the point T=0. This straight line is called an isobar. Different pressuresdifferent isobars correspond. Gay-Lussac's law is not observed in the region of low temperatures close to the temperature of liquefaction (condensation) of gases.
- the process of changing the state of the system at a constant volume. For a given mass of gas, the ratio of gas pressure to its temperature remains constant if the volume of the gas does not change.This is Charles' gas law. According to Charles's law, gas pressure is directly proportional to its temperature: P/T=const. 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. Charles's law is not observed in the region of low temperatures close to the temperature of liquefaction (condensation) of gases.
The laws of Boyle - Mariotte, Gay-Lussac and Charles are special cases of the combined gas law: The ratio of the product of gas pressure and volume to temperature for a given mass of gas is a constant value: PV/T=const.
So, from the law pV = (M/m). RT derives the following laws: =
T=>
const =
TPV
- Boyle's law - Mariotta. p = const => V/T = const
- Charles's law
If an ideal gas is a mixture of several gases, then according to Dalton’s law, the pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the gases entering it. Partial pressure is the pressure that a gas would produce if it alone occupied the entire volume equal to the volume of the mixture. Some may be interested in the question of how it was possible to determine Avogadro’s constant N A = 6.02·10 23? The value of Avogadro's number was experimentally established only in late XIX
– beginning of the 20th century. Let us describe one of these experiments.
A sample of the radium element weighing 0.5 g was placed in a vessel with a volume V = 30 ml, evacuated to a deep vacuum and kept there for one year. It was known that 1 g of radium emits 3.7 x 10 10 alpha particles per second. These particles are helium nuclei, which immediately accept electrons from the walls of the vessel and turn into helium atoms. Over the course of a year, the pressure in the vessel increased to 7.95·10 -4 atm (at a temperature of 27 o C). The change in the mass of radium over a year can be neglected. So, what is N A equal to?
First, let's find how many alpha particles (that is, helium atoms) were formed in one year. Let's denote this number as N atoms:
N = 3.7 10 10 0.5 g 60 sec 60 min 24 hours 365 days = 5.83 10 17 atoms. Let us write the Clapeyron-Mendeleev equation PV = n Let us write the Clapeyron-Mendeleev equation PV = RT and note that the number of moles of helium
= N/N A . From here: N A = = 5,83 . 10 17 . 0,0821 . 300 = 6,02 . 10 23
NRT
At the beginning of the 20th century, this method of determining Avogadro's constant was the most accurate. But why did the experiment last so long (a year)? The fact is that radium is very difficult to obtain. With a small amount (0.5 g) radioactive decay This element produces very little helium. And the less gas in a closed vessel, the less pressure it will create and the greater the measurement error will be. It is clear that a noticeable amount of helium can be formed from radium only over a sufficiently long time.
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, can be neglected.
Gas is called perfect , If:
The sizes of the molecules can be neglected, i.e. molecules can be considered material points;
The forces of interaction between molecules can be neglected (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 ideal gases when:
Conditions close to normal conditions (t = 0 0 C, p = 1.03·10 5 Pa);
At high temperatures.
The laws governing the behavior of ideal gases were discovered experimentally quite a long time ago. Thus, the Boyle-Mariotte law was established back in the 17th century. Let us give the formulations of these laws.
Boyle's Law - Mariotte. Let the gas be in conditions where its temperature is maintained constant (such conditions are called isothermal ).Then for a given mass of gas, the product of pressure and volume is a constant:
This formula is called isotherm equation. Graphically the dependence of p on V for different temperatures shown in the figure.
The property of a body to change pressure when volume changes is called compressibility. If the volume change occurs at T=const, then the compressibility is characterized isothermal compressibility coefficient which is defined as the relative change in volume causing a unit change in pressure.
For an ideal gas it is easy to calculate its value. From the isotherm equation we obtain:
The minus sign indicates that as volume increases, pressure decreases. Thus, the isothermal compressibility coefficient of an ideal gas is equal to the reciprocal of its pressure. As pressure increases, it decreases, because The higher the pressure, the less opportunity the gas has for further compression.
Gay-Lussac's law. Let the gas be in conditions where its pressure is maintained constant (such conditions are called isobaric ). They can be achieved by placing gas in a cylinder closed by a movable piston. Then a change in gas temperature will lead to movement of the piston and a change in volume. The gas pressure will remain constant. In this case, for a given mass of gas, its volume will be proportional to the temperature:
where V 0 is the volume at temperature t = 0 0 C, - volumetric expansion coefficient gases It can be represented in a form similar to the compressibility coefficient:
Graphically, the dependence of V on T for different pressures shown in the figure.
Moving from temperature in Celsius to absolute temperature, Gay-Lussac's law can be written as:
Charles's law. If a gas is in 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 = 0 0 C, - pressure coefficient. It shows the relative increase in gas pressure when it is heated by 1 0:
Charles's 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 is called. Avogadro's constant :
It is easy to calculate the number n0 of particles per 1 m3 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 entering 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.
Clapeyron - Mendeleev equation. From the ideal gas laws we can obtain equation of state , connecting 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 a certain mass of gas occupy a volume V 1, have a pressure p 1 and be 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 occurs in the form of two processes: isothermal (1 - 1") and isochoric (1" - 2).
For these processes, we can write the laws of Boyle - Mariotte and Gay - Lussac:
Eliminating p 1 " from the equations, we obtain
Since states 1 and 2 were chosen arbitrarily, the last equation can be written as:
This equation is called Clapeyron 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 with the same p and T occupies the same volume V m, therefore the constant B will be the same for all gases. This constant common to all gases is denoted by 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, remember that the volume of a gas of mass m is related to the volume of one mole by the formula V = (m/M)V m, where M is molar mass of gas. Then the Clapeyron-Mendeleev equation for a gas of mass m will have the form:
where is the number of moles.
Often the equation of state of an ideal gas is written in terms of Boltzmann constant :
Based on this, the equation of state can be represented as
where is the concentration of molecules. From the last equation it is clear that the pressure of an ideal gas is directly proportional to its temperature and concentration of molecules.
Small demonstration ideal gas laws. After pressing the button "Let's begin" You will see the presenter'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. experiment is in progress), this button is not active. Move on to the next frame only after comprehending the result obtained in the current experiment. (If your perception does not coincide with the presenter’s comments, write!)
You can verify the validity of the ideal gas laws on the existing
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 in general view written like this:
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 different meanings temperatures: 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 inaccessible direct measurement the number of particles included 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 kind of gas from known value relative mass, fill a vessel of known volume V with it, 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.