If we denote the mass of a molecule of a body and the speed of its translational motion, then the kinetic energy of the translational motion of the molecule will be equal to
Molecules of a body can have different speeds and sizes; therefore, to characterize the state of the body, it is used average energy forward movement
where is the total number of molecules in the body. If all molecules are the same, then
Here denotes the root mean square speed of the chaotic movement of molecules:
Since there are interaction forces between molecules, the molecules of the body, in addition to kinetic energy, have potential energy. We will assume that the potential energy of a solitary molecule that does not interact with other molecules is equal to zero. Then, during the interaction of two molecules, the potential energy caused by the repulsive forces will be positive, and the attractive forces will be negative (Fig. 2.1, b), since when the molecules come together, a certain amount of work must be done to overcome the repulsive forces, and the attractive forces, on the contrary, do the work themselves . In Fig. 2.1, b shows the change graph potential energy interactions of two molecules depending on the distance between them. The part of the potential energy graph near its lowest value is called the potential well, and the value of the lowest energy value is called the depth of the potential well.
In the absence of kinetic energy, the molecules would be located at a distance that corresponds to their stable equilibrium, since the resultant of molecular forces in this case is zero (Fig. 2.1, a), and the potential energy is minimal. To remove molecules from each other, you need to do work to overcome the forces of interaction between molecules,
equal in size (in other words, the molecules must overcome a potential barrier of height
Since in reality molecules always have kinetic energy, the distance between them continuously changes and can be either greater or less. If the kinetic energy of molecule B is less, for example in Fig. then the molecule will move within the potential well. Overcoming the counteraction of the forces of attraction (or repulsion), molecule B can move away from A (or approach) to distances at which all its kinetic energy is converted into potential energy of interaction. These extreme positions of the molecule are determined by points on the potential curve at a level from the bottom of the potential well (Fig. 2.1, b). Attractive (or repulsive) forces then push molecule B away from these extreme positions. Thus, interaction forces keep molecules near each other at a certain average distance.
If the kinetic energy of molecule B is greater than Yamiv (Epost" in Fig. 2.1, b), then it will overcome the potential barrier and the distance between the molecules can increase without limit.
When a molecule moves within a potential well, the greater its kinetic energy (in Fig. 2.1, b), i.e., the higher the temperature of the body, the greater the average distance between the molecules becomes. This explains the expansion of solids and liquids when heated.
The increase in the average distance between molecules is explained by the fact that the potential energy graph to the left of the graph rises much steeper than to the right. This asymmetry of the graph is due to the fact that the repulsive forces decrease with increase much faster than the attractive forces (Fig. 2.1, a).
Collision molecules we will call the process of their interaction, as a result of which the speeds of molecules change .
The nature of the interaction of molecules can be imagined if we consider the dependence of the potential energy of interaction of molecules on the distance between their centers. This dependence has the form approximately shown in Figure 11.2.
Let's imagine that one molecule is at the origin, and the second is approaching it from “infinity”, having very little kinetic energy. At distances exceeding , the interaction of molecules has the nature of attraction. Indeed, for As the distance between molecules increases, the potential energy increases. This means that its gradient is directed towards increasing the distance between molecules, and the interaction force () is directed towards decreasing the distance between molecules. Therefore, as molecules approach each other, their mutual speed increases: the potential energy of interaction is converted into kinetic energy, and the approaching molecule accelerates.
At distances less than attraction is replaced by rapidly increasing repulsion. The potential energy of interaction increases sharply (kinetic is converted into potential), and when it is equal to the initial kinetic energy, the molecules stop. Then reverse processes occur, the molecules fly apart.
The minimum distance d by which the centers of molecules approach each other during a collision is called effective molecular diameter . The quantity is called effective cross section of the molecule . is equal to the cross-sectional area of the cylinder along the axis of which a given molecule moves, such that if the center of another molecule falls into the volume of the cylinder, then the molecules must collide.
It is clear that with increasing temperature the centers of molecules during collisions will come closer together, therefore effective diameter depends on temperature . It should be borne in mind that the growth of the potential repulsion energy occurs very quickly, so the dependence of the effective diameter on temperature necessarily occurs, but is not very pronounced .
In a second, a molecule travels an average distance equal to its average speed. If in a second she undergoes collisions, then the average free run length molecules
For the calculation, we assume that all molecules, except this one, are at rest in their places. Having struck one of the stationary molecules, this one will fly in a straight line until it collides with the other. The next collision will occur if the center motionless molecules will be from the straight line along which it flies given molecule at a distance less than its effective diameter. In a second, a molecule will collide with all molecules whose centers fall within the volume of a cranked cylinder with a base and a length equal to the average speed. If the concentration of molecules is n, then the number of collisions along this path
It is necessary to take into account that in fact all molecules move, and in (11.9) it is necessary to take into account not , but the average relative the speed of movement of molecules, which is several times greater. Then for the mean free path l we can write:
Of interest quantification of l And . We will assume that in a liquid the molecules are at small distances from each other. Then the third root of the volume per molecule will give us an estimate of the size of the molecule. One mole of water occupies a volume of 18 * 10 -10 m3 and contains Avogadro's number 6 * 10 23 molecules. Then there is “30*10 -30 m3 per molecule, and the diameter of the molecule is” 3*10 -10 m. Under conditions close to normal, one mole of gas occupies a volume of . Then the concentration of molecules under normal conditions can be estimated by the formula, and average length free path in accordance with formula (11.10)
When studying the behavior of a large collection of molecules, it is more convenient to use potential energy instead of the interaction force between molecules.
It is necessary to calculate the average characteristics of the system, and the concept of the average force of interaction between molecules is meaningless, since the sum of all forces acting between molecules, in accordance with Newton’s third law, is zero. The average potential energy largely determines the state and properties of a substance.
Dependence of potential energy on the distance between molecules
Since the change in potential energy is determined by the work of force, then from the known dependence of force on distance one can find the dependence of potential energy on distance. But it is enough for us to know only the approximate shape of the potential curve E R (r). First of all, let us remember that potential energy is determined up to an arbitrary constant, because the direct meaning is not the potential energy itself, but the difference between the potential energies at two points, equal to the work taken with the opposite sign. We will assume, as is customary in physics, E = 0 at r→ ∞. The potential energy of a system can be considered as the work that the system can do, and the potential energy is determined by the location of the bodies, but not by their velocities. The greater the distance between molecules, the great job will create attractive forces between them as they approach. Therefore, when decreasing r, starting from very large values, the potential energy will decrease. We accepted that r→ ∞ potential energy is zero, therefore, with decreasing r potential energy becomes negative (Fig. 2.12).
At the point r = r 0 the force is zero (see Fig. 2.10). Therefore, if the molecules are located at this distance, then they will be at rest, and the system cannot perform any work. This means that when r = r 0 potential energy has a minimum. We could have this potential energy value E p Take 0 as the origin of potential energy. Then it would be positive everywhere (Fig. 2.13). Both curves (see Fig. 2.12 and 2.13) equally characterize the interaction of molecules. Difference of values E R for two points is the same for both curves, and only it makes sense.
At r < r 0 rapidly growing repulsive forces appear. They can also do work. Therefore, the potential energy increases with further approach of molecules, and very quickly.
The potential curve will have the shape shown in Figure 2.12 if the molecules approach each other in a plane A along the line connecting their centers (Fig. 2.14). If the molecules approach each other in a plane IN or in the plane WITH, then the potential curve will have the form shown in Figure 2.15, respectively, A and 2.15, b.
the main task
Much can be explained and understood based on certain ideas about the nature of the interaction of molecules in a substance. We will focus only on one very general question: how does knowledge of the dependence of potential energy on the distance between molecules allow us to establish a quantitative criterion for the difference between gases, liquids and solids from the point of view of molecular kinetic theory.
Let us first consider the movement of molecules from an energy point of view.
Chemical bond. Dependence of potential energy on internuclear distance in a diatomic molecule. Types of chemical bonds. Main characteristics of a chemical bond: length, energy, bond multiplicity, bond angle. Types of chemical bonds. Ionic bond. Metal connection. Intermolecular interactions. Hydrogen bond.
The formation of chemical compounds is due to the emergence of chemical bonds between atoms in molecules and crystals.
Chemical bond– a set of interactions of atoms, leading to the formation of stable systems (molecules, complexes, crystals, etc.). A chemical bond occurs if, as a result of overlapping electron clouds of atoms, a decrease occurs total energy systems.
A measure of the strength of the chemical bond between atoms A and B is binding energy E A-B, which is determined by the work required to destroy a given connection. Thus, to atomize 1 mole of hydrogen gas, it is necessary to expend energy E = 436 kJ, therefore, the formation of an H 2 molecule from atoms
H+H=H 2 is accompanied by the release of the same amount of energy, i.e. E H-H = 436 kJ/mol.
Important characteristic communication is hers length, i.e. the distance between the centers of atoms A and B in a molecule. The energy and length of bonds depend on the nature of the distribution of electron density between atoms. The distribution of electron density is influenced by spatial chemical orientation communications. If diatomic molecules are always linear, then the shapes of polyatomic molecules can be different. So a triatomic molecule can have a linear or angular shape. Corner between imaginary lines that can be drawn through the centers of bonded atoms is called valence
The distribution of electron density between atoms also depends on the size of the atoms and their electronegativity– the ability of atoms to attract the electron density of partners. In homoatomic (i.e., consisting of identical atoms) molecules, the electron density is distributed evenly between the atoms. In heteroatomic (consisting of atoms of different elements) molecules, the electron density shifts in the direction that helps to reduce the energy of the system (towards a more electronegative element). The electron density increases near the nucleus of an atom of a more electronegative element. Bonding in heteroatomic molecules is always to some extent polar, since the electron density in them is distributed asymmetrically.
The formation of a covalent bond occurs due to the unpaired electrons of each atom, which form a common pair. If one bond (one common pair) has arisen between atoms, such the connection is called single. Example HCl, HBr, NaCl, H 2
If more than one common electron pair appears between atoms, then the bond is called multiples: double (two common pairs), triple (three common pairs). An example of a molecule with a triple bond is the nitrogen molecule. Each nitrogen atom has three unpaired p electrons. Each of them takes part in the formation of the connection. There are three bonds between atoms in the N2 molecule. The presence of a triple bond explains the high chemical stability of the molecule. Example with an O2 double bond. Each oxygen atom has 2 unpaired p-electrons, which are involved in the formation of bonds.
The dependence of potential energy on the distance between atoms in a diatomic molecule is expressed by the following relationship: (page 112 figure and equation)
U = 1∕4πε 0 × (e 2 ∕ r A-B + e 2 ∕ r 12 - e 2 ∕ r A1 - e 2 ∕ r B2 - e 2 ∕ r A2 - e 2 ∕ r B1), where ε 0 - electrical constant. Therefore, potential energy is inversely proportional to the distance between nuclei in a diatomic molecule.
An ionic chemical bond is a bond formed by the electrostatic attraction of cations (positively charged ions) to anions (negatively charged ions).
The most stable electronic configuration of atoms is one in which, at the external electronic level, atoms are like noble gases. There will be 8 electrons (or for the first period 2). During chemical interactions, atoms strive to acquire just such a stable electronic configuration and often achieve this either as a result of gaining valence electrons from other atoms (the reduction process), or as a result of giving up their valence electrons (opening a completed layer) - the process of oxidation. Atoms that have added foreign electrons turn into negative ions - anions. Atoms that donate their electrons turn into positive ions - cations. Electrostatic attraction forces arise between anions and cations, which will hold them near each other, thereby creating an ionic bond. Since cations form mainly metal atoms, and anions form non-metal atoms, this type of bond is typical for compounds of typical metals (elements of the main subgroup of groups 1-2 except Mg, Be) with typical non-metals (elements of the main subgroup of group 7) NaCl. Substances with ionic bonds have an ionic crystal lattice. Ionic compounds are harder, stronger, and more refractory. Solutions and melts of most ionic compounds are electrolytes. This type of bond is characteristic of hydroxides of typical metals and many salts of oxygen-containing acids (soluble). When an ionic bond is formed, a complete (ideal) transfer of electrons does not occur. The interaction of ions does not depend on direction; unlike a covalent bond, it is non-directional. An ionic bond exists in ammonium salts, where the role of the cation is played by NH 4 + - ammonium ion. (NH 4)OH, NH 4 Cl.
The bond in metals and alloys that is performed by relatively free (generalized) electrons between metal ions in a metal crystal lattice is called metallic. Such a bond is non-directional, unsaturated, characterized by a small number of valence electrons (external unpaired) and a large number of free orbitals, which is typical for metal atoms. The presence of a metal bond is due to physical properties metals and alloys: hardness, electrical and thermal conductivity, malleability, ductility, shine. Substances with a metallic bond have a metallic crystal lattice. Its nodes contain atoms or ions, between which electrons (electron gas) move freely (within the crystal). The metallic bond is characteristic only of the condensed state of matter. In the vapor and gaseous state, the atoms of all substances, including metals, are connected to each other only by covalent bonds. The electron density of a metal bond is evenly distributed in all directions. A metallic bond does not exclude some degree of covalency. In its frequent form, metallic bonds are characteristic only of alkali and alkaline earth metals. In transition metals, only a small part of the valence electrons is in a state of sharing. The number of electrons belonging to the entire crystal is small. The remaining electrons carry out directed covalent bonds between neighboring atoms. The formation of bonds can occur not only between atoms, but also between molecules. It causes the condensation of gases and their transformation into liquids and solids. The first formulation of the forces of intermolecular interaction was given in 1871 by Van der Waals. (Van der Waals forces). Polar molecules, due to the electrostatic attraction of the ends of the dipoles, are oriented in space so that the negative ends of the dipoles of some molecules are turned to the positive ends of the dipoles of other molecules. The energy of such interaction is determined by the electrostatic attraction of two dipoles. The larger the dipole, the stronger the intermolecular attraction. Under the influence of the dipole of one molecule, the dipole of another can increase, and a non-polar molecule can become polar. Such a dipole moment that appears under the influence of a dipole of another molecule is called an induced dipole moment, and the phenomenon itself is called induction. It is known that H 2, O 2, N 2 and noble gases are liquefied. To explain this fact, the concept of dispersion forces of intermolecular interaction was introduced. These forces act between any atoms and molecules, regardless of their structure. These forces and the van der Waals forces are very weak.
A special type of chemical bond is a hydrogen bond.. Hydrogen is a chemical bond between positively polarized hydrogen atoms of one molecule (or part of it) and negatively polarized atoms of strongly electronegative elements having lone pairs of electrons. The mechanism of hydrogen bond formation is partly electrostatic, partly donor-acceptor in nature (page 147 figure); in the presence of such a bond, even low-molecular substances can be liquids under normal conditions. In proteins, there is a hydrogen bond inside the molecule between the carbonyl oxygen and the amino hydrogen group. In DNA, two chains of nucleotides are linked to each other by hydrogen bonds. Substances with hydrogen bonding have a molecular crystal lattice. The energy of a hydrogen bond (21-29 kJ∕mol) is almost 10 times less than the energy of a conventional chemical bond. But it cross-links all the molecules, and when heated they are the first to break.
Allows you to analyze the general patterns of motion if the dependence of potential energy on coordinates is known. Let us consider, for example, the one-dimensional movement of a material point (particle) along the axis 0x in the potential field shown in Fig. 4.12.
Fig.4.12. Motion of a particle near stable and unstable equilibrium positions
Since in a uniform field of gravity the potential energy is proportional to the height of the body's rise, one can imagine an ice slide (neglecting friction) with a profile corresponding to the function P(x) on the image.
From the law of conservation of energy E = K + P and from the fact that kinetic energy K = E - P is always non-negative, it follows that the particle can only be in regions where E > P. The figure shows a particle with total energy E can only move in areas
In the first region, its movement will be limited (finitely): with a given supply of total energy, the particle cannot overcome the “slides” on its way (they are called potential barriers) and is doomed to remain forever in the “valley” between them. Eternally - from the point of view of classical mechanics, which we are now studying. At the end of the course we will see how quantum mechanics helps a particle escape from imprisonment in a potential well - a region
In the second region, the particle’s movement is not limited (infinitely), it can move infinitely far from the origin to the right, but on the left its movement is still limited by the potential barrier:
Video 4.6. Demonstration of finite and infinite movements.
At potential energy extremum points x MIN And x MAX the force acting on the particle is zero because the derivative of potential energy is zero:
If you place a particle at rest at these points, then it would remain there... again forever, if not for fluctuations in its position. There is nothing strictly at rest in this world; a particle can experience small deviations (fluctuations) from the equilibrium position. In this case, naturally, forces arise. If they return the particle to the equilibrium position, then such equilibrium is called sustainable. If, when the particle deviates, the resulting forces take it even further from its equilibrium position, then we are dealing with unstable equilibrium, and the particle usually does not stay in this position for long. By analogy with an ice slide, one can guess that a stable position will be at a minimum of potential energy, and an unstable one at a maximum.
Let us prove that this is indeed the case. For a particle at the extremum point x M (x MIN or x MAX) force acting on it F x (x M) = 0. Let the particle coordinate change by a small amount due to fluctuation x. With such a change in coordinates, a force will begin to act on the particle
(the prime indicates the derivative with respect to the coordinate x). Considering that F x =-P", we obtain the expression for the force
At the minimum point, the second derivative of the potential energy is positive: U"(x MIN) > 0. Then, for positive deviations from the equilibrium position x > 0 the resulting force is negative, and when x<0 the force is positive. In both cases, the force prevents the particle from changing its coordinates, and the equilibrium position at the minimum potential energy is stable.
On the contrary, at the maximum point the second derivative is negative: U"(x MAX)<0 . Then an increase in the particle coordinate Δx leads to the emergence of a positive force, which further increases the deviation from the equilibrium position. At x<0 the force is negative, that is, in this case it contributes to further deflection of the particle. This equilibrium position is unstable.
Thus, the position of stable equilibrium can be found by solving the equation and inequality together
Video 4.7. Potential holes, potential barriers and equilibrium: stable and unstable.
Example. The potential energy of a diatomic molecule (for example, H 2 or O 2) is described by an expression of the form
Where r is the distance between atoms, and A, B- positive constants. Determine the equilibrium distance r M between the atoms of a molecule. Is a diatomic molecule stable?
Solution. The first term describes the repulsion of atoms at short distances (the molecule resists compression), the second describes the attraction at large distances (the molecule resists breaking). In accordance with what has been said, the equilibrium distance is found by solving the equation
Differentiating the potential energy, we get
We now find the second derivative of the potential energy
and substitute the value of the equilibrium distance there r M :
The equilibrium position is stable.
In Fig. 4.13 presents an experiment in studying potential curves and equilibrium conditions of a ball. If, on the potential curve model, a ball is placed at a height greater than the height of the potential barrier (the energy of the ball is greater than the energy of the barrier), then the ball overcomes the potential barrier. If the initial height of the ball is less than the height of the barrier, then the ball remains within the potential well.
A ball placed at the highest point of the potential barrier is in unstable equilibrium, since any external influence leads to the ball moving to the lowest point of the potential well. At the lower point of the potential well, the ball is in stable equilibrium, since any external influence leads to the return of the ball to the lower point of the potential well.
Rice. 4.13. Experimental study of potential curves
Additional Information
http://vivovoco.rsl.ru/quantum/2001.01/KALEID.PDF – Supplement to the journal “Quantum” - discussions about stable and unstable equilibrium (A. Leonovich);
http://mehanika.3dn.ru/load/24-1-0-3278 – Targ S.M. A short course in theoretical mechanics, Publishing House, Higher School, 1986 – pp. 11–15, §2 – initial provisions of statics.
