Graph of potential energy versus distance. Dependence of the potential energy of intermolecular interaction on the distance between molecules. Dependence of potential energy on the distance between molecules

Allows you to analyze the general patterns of motion, if the dependence of the potential energy on the coordinates is known. Consider, for example, the one-dimensional motion 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 the positions of stable and unstable equilibrium

Since the potential energy in a uniform gravity field is proportional to the height of the body, we can imagine an ice hill (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 the kinetic energy K = E - P is always non-negative, it follows that the particle can only be located in regions where E > P. In the figure, a particle with total energy E can only move in areas

In the first region, its motion will be limited (finitely): with a given total energy, the particle cannot overcome the "hills" on its way (they are called potential barriers) and doomed to remain forever in the "valley" between them. Forever - 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 get out of captivity in a potential well - the region

In the second region, the motion of the particle is not limited (infinitely), it can move infinitely far from the origin to the right, but on the left its motion is still limited by the potential barrier:

Video 4.6. Demonstration of finite and infinite movements.

At the extremum points of the potential energy x MIN And x MAX the force acting on the particle is zero, because the derivative of the potential energy is zero:

If a particle at rest is placed at these points, then it would remain there ... again, forever, if not for fluctuations in its position. In this world, there is nothing strictly at rest; a particle can experience small deviations (fluctuations) from the equilibrium position. This naturally creates forces. If they return the particle to an equilibrium position, then such an equilibrium is called sustainable. If, when the particle deviates, the forces that arise take it even further away from the equilibrium position, then we are dealing with unstable equilibrium, and the particle in this position usually does not stay long. By analogy with an ice slide, one can guess that the position will be stable at the minimum of potential energy, and unstable at the maximum.

We will prove that this is indeed the case. For a particle at an extremum point x M (x MIN or x MAX) the 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 dashed line denotes the derivative with respect to the coordinate x). Given that F x \u003d -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 strength is positive. In both cases, the force prevents a change in the particle's coordinate, and the equilibrium position at the potential energy minimum is stable.

Conversely, 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 also contributes to the further deflection of the particle. This state of equilibrium is unstable.

Thus, the position of stable equilibrium can be found by jointly solving the equation and the inequality

Video 4.7. Potential wells, potential barriers and equilibrium: stable and unstable.

Example. The potential energy of a diatomic molecule (for example, H 2 or About 2) is described by an expression of the form

Where r is the distance between atoms, and A, B are positive constants. Determine the equilibrium distance r M between the atoms of the molecule. Is a diatomic molecule stable?

Solution. The first term describes the repulsion of atoms at short distances (the molecule resists compression), the second - the attraction at large distances (the molecule resists rupture). In accordance with what has been said, the equilibrium distance is found by solving the equation

Differentiating the potential energy, we obtain

We now find the second derivative of the potential energy

and substitute there the value of the equilibrium distance rM :

The equilibrium position is stable.

On fig. 4.13 presents the experience of studying the potential curves and equilibrium conditions of the ball. If, on the model of the potential curve, the 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 transition of the ball 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 action 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 "Kvant" - 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 - the initial provisions of statics.

A chemical bond is formed only if, when atoms (two or more) approach each other, the total energy of the system (the sum of kinetic and potential energy) decreases.

The most important information about the structure of molecules is provided by studying the dependence of the potential energy of a system on the distance between its constituent atoms. For the first time this dependence was studied in 1927 by the German scientists W. Heitler and F. London, investigating the causes of the occurrence of a chemical bond in the hydrogen molecule. Using the Schrödinger equation, they concluded that the energy of a system consisting of two nuclei and two electrons in a hydrogen molecule can be expressed as follows:

E = ~ K ± O,

Where TO is the Coulomb integral including all electrostatic interactions, i.e. repulsion between electrons, repulsion between nuclei, as well as the attraction of electrons to the nuclei of atoms. ABOUT- the exchange integral, it characterizes the occurrence of an electron pair and is due to the movement of electrons around both hydrogen nuclei. This integral has a very large negative value. Thus, according to calculations, the energy of this system can take two values:

E \u003d ~ K + O And E \u003d ~ K - O

Consequently, there are such states of electrons, during the interaction of which the energy of the system can change within the limits 0 < E < 0 .

The first equation corresponds to a decrease in the energy of the system E< 0 .

The second equation corresponds to an increase in the energy of the system E > 0.

The condition for reducing the energy of the system is satisfied by “y”- a function that determines the state of interacting electrons with oppositely directed (antiparallel) spins. This “y”- the function is called symmetrical “y”- function.

This leads to the conclusion that a chemical bond between atoms should arise only if the electrons belonging to different atoms have oppositely directed spins. Only under this condition will the energy of the molecular system be less than the energy of atomic systems, i.e. a stable molecule is formed. Consequently, the antiparallelism of the electron spins of interacting atoms is a necessary condition for the formation of a covalent bond.

Rice. 8. Change in potential energy in a system of two hydrogen atoms depending on the distance between the nuclei

When two atoms approach each other, if the electron spins are parallel, then their total energy increases, a repulsive force arises and increases between the atoms (Fig. 8).

With oppositely directed spins, the approach of atoms to a certain distance r0 accompanied by a decrease in the energy of the system.

At r = r0 the system has the lowest energy, i.e. is in the most stable state, characterized by the formation of hydrogen molecules H 2. With further approach of the atoms, the energy increases sharply.

The emergence of the molecule H 2 of atoms can be explained by the overlapping of atomic electron clouds to form a molecular cloud that surrounds two positively charged nuclei.

Rice. 9. Overlapping electron clouds

in the formation of a hydrogen molecule

In the place where the electron clouds overlap (ie, in the space between the nuclei), the electron density of the binding cloud is maximum (Fig. 9). In other words, the probability of electrons being in the space between the nuclei is greater than in other places. Due to this, attractive forces arise between the positive charge of the nucleus and the negative charges of the electrons and the nuclei approach each other - the distance between the hydrogen nuclei in the molecule H 2 noticeably less (0.74Å) the sum of the radii of two free hydrogen atoms (1.06Å)

The bond formed as a result of the generalization of the electron densities of interacting atoms is called covalent.

According to quantum mechanical concepts, the interaction of atoms can lead to the formation of a molecule only under the condition that the electron spins of the approaching atoms have oppositely directed spins. When electrons with parallel spins approach each other, only repulsive forces act.

H ­ + H ¯ → H ­¯ H → H 2

+1/2 -1/2

Since the exact solution of the Schrödinger equation for atomic-molecular systems is impossible, various approximate methods have arisen for calculating the wave function, and hence the distribution of electron density in a molecule. Two methods are most widely used: the method of valence bonds (Sun) and the molecular orbital method (MO). In the development of the first method, special merit belongs to Heitler and London, Slater and Pauling. The development of the second method is associated mainly with the names of Mulliken and Hund.

The main provisions of the method sun. 1) A covalent chemical bond is formed by two electrons with oppositely directed spins, and this electron pair belongs to two atoms.

2) When a covalent bond is formed, the electron clouds of interacting atoms overlap, the electron density increases in the internuclear space, which leads to a decrease in the energy of the system.

3) The stronger the covalent bond, the more the interacting electron clouds overlap. Therefore, the covalent bond is formed in the direction in which this overlap is maximum.

This method justifies the designation using a dash of a chemical bond in the structural formulas of compounds.

So in the method view sun a chemical bond is localized between two atoms, i.e. it is two-center and two-electron.

Even in ancient times, the golden rule of mechanics was discovered: winning in strength, you lose in distance. Indeed, if, for example, a load is lifted along an inclined plane, then one has to do work against gravity (we will assume that work against friction forces can be neglected). If the inclined plane is gentle, then the path is long, but less force can be applied to the load. On a steep plane it is harder to lift the load, but the path is shorter. The work that must be done in order to lift a load of mass m to a height is always the same and equal to .

This is the most important property of gravity forces: the work does not depend on the shape of the path, but is determined only by the initial and final positions of the body. On fig. 1 shows three possible movements of the body from point M to point N. The acceleration of the gravitational field is indicated by an arrow. It is easy to prove that, moving the bodies along the segment MN and along the broken line MON, one will have to do the same work, since the work on the MO segment is zero. By dividing the curved path into many straight segments, we can make sure that in this case the work is the same.

Forces with this property are called potential or conservative. For them, you can determine the potential energy. It is enough to choose the origin - to assume that in some position (for example, on the surface of the Earth) the potential energy is equal to zero, and then at any other point it will be equal to the work of moving the body from the initial position to this point.

Potential energy together with kinetic energy is the total mechanical energy of the body. If the body is only in the field of potential forces, then the total energy is conserved (the law of conservation of mechanical energy). To launch a rocket capable of leaving the solar system, it is necessary to tell it a huge speed (about 11 km / s). The stock of kinetic energy compensates for the increase in potential energy as the rocket moves away from the Earth.

Not only the forces of gravity are potential, but also the forces of electrostatic interaction. After all, Coulomb's law is very similar to Newton's law of universal gravitation. Even the formulas for potential energy are almost the same: in both cases, the energy is inversely proportional to the distance between the interacting bodies.

At the same time, the work of the friction forces depends on the shape of the path (for example, with dry friction, the shortest path is the best), and such forces are not potential.

With the help of potential energy it is convenient to describe the interaction of particles in the microcosm, for example, two atoms. Forces of attraction act at large distances between atoms. Although each atom is neutral, under the influence of the electric field of another atom, it turns into a small dipole, and these dipoles are attracted to each other (Fig. 2). Therefore, when the atoms approach each other, they must be restrained and negative work must be done against these forces. At small distances between atoms, on the contrary, repulsive forces act, which are mainly due to the Coulomb interaction of approaching nuclei. In this case, positive work must be done to bring the atoms closer together.

A graph of the potential energy of atoms depending on the distance between them is shown in Fig. 3. Potential energy has a minimum, and this position of atoms corresponds to a stable formation - a molecule. In this case, the atoms are said to be in a potential well.

Similarly, in a crystal, atoms are arranged in space in such a way that it has a minimum potential energy. As a result, a periodic structure is formed - a crystal lattice (see Crystal Physics).

The stable position of the system always corresponds to the minimum potential energy. On fig. 4 shows the relief of the surface on which the ball is located. There are three equilibrium positions, but only one, corresponding to the minimum potential energy, is stable (in this case, the ball is literally in the hole).

Interestingly, if only the forces of electrostatic interaction act between the particles (a system of fixed charges), then they cannot be in a state of stable equilibrium at all. Potential energy does not have a minimum, and the system will definitely fall apart (the charges will scatter). This theorem of Earnshaw served as the most important proof of the failure of the static model of the atom.

If the mass of the body molecule is denoted, and the speed of its translational motion, then the kinetic energy of the translational motion of the molecule will be equal to

The molecules of the body can have different speeds and values, therefore, to characterize the state of the body, the average energy of translational motion is used

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 motion of molecules:

Since there are interaction forces between the 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 due to the repulsive forces will be positive, and the attractive forces will be negative (Fig. 2.1, b), since when the molecules approach each other, a certain work must be done to overcome the repulsive forces, and the attractive forces, on the contrary, do the work themselves . On fig. 2.1, b shows a graph of the change in the potential energy of the interaction 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, it is necessary to do work to overcome the forces of interaction of molecules,

equal in magnitude (in other words, the molecules must overcome a potential barrier with a height

Since in reality the molecules always have kinetic energy, the distance between them is constantly changing and can be either greater or less. If the kinetic energy of the molecule B is less, for example, in Fig. then the molecule will move within the potential well. Overcoming the opposition of forces of attraction (or repulsion), the molecule B can move away from A (or approach) to distances at which all of its kinetic energy is converted into the 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). The forces of attraction (or repulsion) then push the molecule B away from these extreme positions. Thus, the interaction forces keep the molecules near each other at a certain average distance.

If the kinetic energy of the molecule B is greater than Ymiv (Epost" in Fig. 2.1, b), then it will overcome the potential barrier and the distance between the molecules can increase indefinitely.

When a molecule moves within a potential well, then the greater its kinetic energy (in Fig. 2.1, b), i.e., the higher the body temperature, the greater the average distance between 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 rises much steeper to the left than to the right. This asymmetry of the graph is obtained due to the fact that the repulsive forces decrease with increasing much faster than the attractive forces (Fig. 2.1, a).

Dependence of intermolecular interaction forces on the distance between molecules

Between the molecules of matter simultaneously operate attractive forces And repulsive forces. On distance r = r0 force F= 0, i.e., the forces of attraction and repulsion balance each other (see Fig. 1). So the distance r0 corresponds to the equilibrium state between molecules, in which they would be in the absence of thermal motion. At r< r 0 repulsive forces prevail (Fo > 0), at r > r 0- forces of attraction (Fn< 0). At distances r > 10 -9 m, intermolecular forces of interaction are practically absent (F → 0).

Dependence of the potential energy of intermolecular interaction on the distance between molecules

elementary work δA strength F with an increase in the distance between molecules by dr, it occurs due to a decrease in the mutual potential energy of the molecules, i.e. δ A= Fdr= - dP. As per picture b, if the molecules are at a distance from each other at which intermolecular forces of interaction do not act (r→∞), then П = 0. With the gradual approach of the molecules, attractive forces appear between them (F< 0) that do positive work (δA= F dr > 0). Then the potential energy of interaction decreases, reaching a minimum at r = r 0 . At r< r 0 with decreasing r repulsive force (F > 0) increase sharply and the work done against them is negative ( δA = Fdr< 0). Potential energy also begins to increase sharply and becomes positive. It follows from this potential curve that the system of two interacting molecules is in a state of stable equilibrium ( r = r0) has the minimum potential energy.

Figure 1 - Dependence of forces and potential energy of intermolecular interaction on the distance between molecules

F o- repulsion force; F u- force of gravity; F- their resultant

the ideal gas equation of state is converted to van der Waals equation:

. (1.6)

for one mole of gas

Isotherms

Let us analyze the isotherms of the van der Waals equation, the dependences R from V for real gas at constant temperature. Multiplying the van der Waals equation by V 2 and expanding the brackets, we get

PV 3 – (RT + bP) vV 2 + av 2 V - abv 3= 0.

Since this equation has a third degree with respect to V, and the coefficients at V are real, then it has either one or three real roots, i.e. isobar R= const intersects the curve P = P(V) at one or three points, as shown in figure 7.4. Moreover, with an increase in temperature, we will move from the nonmonotonic dependence P = P(V) to a monotone single-valued function. Isotherm at T cr, which separates the nonmonotonic T< T кр and monotonous T > T cr isotherm, corresponds to the isotherm at the critical temperature. At temperatures above the critical dependence P = P(V) is a single-valued monotonic volume function. This means that at T > T cr the substance is in only one state, the gaseous state, as was the case with an ideal gas. At a gas temperature below the critical one, this unambiguity disappears, which means that the transition of a substance from gaseous to liquid and vice versa is possible. Location on DIA isotherms T 1 pressure increases with volume ( dP/dV) > 0. This state is unstable, since the slightest density fluctuations should be amplified here. Therefore, the area BCA cannot exist sustainably. In the regions DLB And AGE pressure drops as volume increases (dP/dV) T< 0 is a necessary but not sufficient condition for stable equilibrium. The experiment shows that the system moves from the region of stable states G.E.(gas) to the region of stable states LD(liquid) through a two-phase state (gas - liquid) GL along the horizontal isotherm GCL.

Under quasi-static compression, starting from the point G, the system breaks up into 2 phases - liquid and gas, and the densities of the liquid and gas remain unchanged under compression and are equal to their values ​​at the points L And G respectively. During compression, the amount of substance in the gaseous phase decreases continuously, and in the liquid phase it increases until the point L, in which all the substance will go into a liquid state.

Rice. 7.4

The presence of a critical point on the van der Waals isotherm means that for each liquid there is a temperature above which the substance can exist only in the gaseous state. D.I. also came to this conclusion. Mendeleev in 1861. He noticed that at a certain temperature, the rise of liquid in the capillaries ceased, i.e. surface tension vanishes. At the same temperature, the latent heat of vaporization vanishes. Mendeleev called this temperature the temperature of absolute boiling. Above this temperature, according to Mendeleev, the gas cannot be condensed into a liquid by any increase in pressure.

We defined the critical point K as the inflection point of the critical isotherm, at which the tangent to the isotherm is horizontal (Fig. 7.5). It can also be defined as the point at which the horizontal sections of the isotherms pass in the limit when the temperature rises to the critical one. This is the basis of the method for determining critical parameters P k, V k , T k owned by Andrews. A system of isotherms is constructed at various temperatures. The limiting isotherm, in which the horizontal section LG(Fig. 7.4) goes to a point, will be a critical isotherm, and the indicated point will be a critical point (Fig. 7.5).

Rice. 7.5

The disadvantage of the Andrews method lies in its bulkiness.