The order of molecules in a solid. Abstract model ideas about the structure of liquids, gases and crystals
Kinetic energy of a molecule
In a gas, molecules move freely (isolated from other molecules), only occasionally colliding with each other or with the walls of the container. As long as a molecule moves freely, it only has kinetic energy. During a collision, the molecules also gain potential energy. Thus, the total energy of a gas is the sum of the kinetic and potential energies of its molecules. The more rarefied the gas, the more molecules at each moment of time are in a state of free movement, having only kinetic energy. Consequently, when the gas is rarefied, the proportion of potential energy decreases in comparison with kinetic energy.
The average kinetic energy of a molecule at equilibrium of an ideal gas has one very important feature: in a mixture of different gases, the average kinetic energy of a molecule for different components of the mixture is the same.
For example, air is a mixture of gases. The average energy of an air molecule for all its components under normal conditions, when air can still be considered an ideal gas, is the same. This property of ideal gases can be proven on the basis of general statistical considerations. An important corollary follows from this: if two different gases (in different vessels) are in thermal equilibrium with each other, then the average kinetic energies of their molecules are the same.
In gases, the distance between molecules and atoms is usually much greater than the size of the molecules themselves; the interaction forces between molecules are not great. As a result, the gas does not have its own shape and constant volume. Gas is easily compressed and can expand without limit. Gas molecules move freely (translationally, they can rotate), only sometimes colliding with other molecules and the walls of the vessel in which the gas is located, and they move at very high speeds.
Movement of particles in solids
The structure of solids is fundamentally different from the structure of gases. In them, the intermolecular distances are small and the potential energy of the molecules is comparable to the kinetic energy. Atoms (or ions, or whole molecules) cannot be called motionless; they perform random oscillatory motion around average positions. The higher the temperature, the greater the oscillation energy, and therefore the average amplitude of oscillations. Thermal vibrations of atoms also explain the heat capacity of solids. Let us consider in more detail the movements of particles in crystalline solids. The entire crystal as a whole is a very complex coupled oscillatory system. The deviations of atoms from their average positions are small, and therefore we can assume that the atoms are subject to the action of quasi-elastic forces that obey Hooke’s linear law. Such oscillatory systems are called linear.
There is a developed mathematical theory of systems subject to linear oscillations. It proves a very important theorem, the essence of which is as follows. If the system performs small (linear) interconnected oscillations, then by transforming the coordinates it can be formally reduced to a system of independent oscillators (whose oscillation equations do not depend on each other). A system of independent oscillators behaves like an ideal gas in the sense that the atoms of the latter can also be considered as independent.
It is by using the idea of independence of gas atoms that we arrive at Boltzmann's law. This very important conclusion provides a simple and reliable basis for the entire theory of solids.
Boltzmann's law
The number of oscillators with given parameters (coordinates and velocities) is determined in the same way as the number of gas molecules in a given state, according to the formula:
Oscillator energy.
Boltzmann's law (1) in the theory of solid bodies has no restrictions, but formula (2) for the oscillator energy is taken from classical mechanics. When theoretically considering solids, one must rely on quantum mechanics, which is characterized by discrete changes in the energy of the oscillator. The discreteness of the oscillator energy becomes insignificant only at sufficiently high values of its energy. This means that (2) can only be used at sufficiently high temperatures. At high temperatures of a solid, close to the melting point, the law of uniform distribution of energy over the degrees of freedom follows from Boltzmann’s law. If in gases for each degree of freedom there is on average an amount of energy equal to (1/2) kT, then the oscillator has one degree of freedom, in addition to the kinetic one, with potential energy. Therefore, per one degree of freedom in a solid at a sufficiently high temperature there is an energy equal to kT. Based on this law, it is not difficult to calculate the total internal energy of a solid body, and after it its heat capacity. A mole of a solid contains NA atoms, and each atom has three degrees of freedom. Therefore, the mole contains 3 NA oscillators. Energy of a mole of a solid
and the molar heat capacity of a solid at sufficiently high temperatures is
Experience confirms this law.
Liquids occupy an intermediate position between gases and solids. Liquid molecules do not disperse over long distances, and liquid under normal conditions retains its volume. But unlike solids, molecules not only vibrate, but also jump from place to place, that is, they perform free movements. As the temperature increases, liquids boil (there is a so-called boiling point) and turn into gas. As the temperature decreases, liquids crystallize and become solids. There is a point in the temperature field at which the boundary between gas (saturated vapor) and liquid disappears (critical point). The pattern of thermal motion of molecules in liquids near the solidification temperature is very similar to the behavior of molecules in solids. For example, the heat capacity coefficients are exactly the same. Since the heat capacity of a substance changes slightly during melting, we can conclude that the nature of the movement of particles in a liquid is close to the movement in a solid (at the melting temperature). When heated, the properties of the liquid gradually change, and it becomes more like a gas. In liquids, the average kinetic energy of particles is less than the potential energy of their intermolecular interaction. The energy of intermolecular interaction in liquids and solids differs insignificantly. If we compare the heat of fusion and the heat of evaporation, we will see that during the transition from one state of aggregation to another, the heat of fusion is significantly lower than the heat of vaporization. An adequate mathematical description of the structure of a liquid can only be given with the help of statistical physics. For example, if a liquid consists of identical spherical molecules, then its structure can be described by the radial distribution function g(r), which gives the probability of detecting any molecule at a distance r from the given one chosen as a reference point. This function can be found experimentally by studying the diffraction of x-rays or neutrons, or a computer simulation of this function can be carried out using Newtonian mechanics.
The kinetic theory of liquid was developed by Ya.I. Frenkel. In this theory, a liquid is considered, as in the case of a solid, as a dynamic system of harmonious oscillators. But unlike a solid body, the equilibrium position of molecules in a liquid is temporary. After oscillating around one position, the liquid molecule jumps to a new position located nearby. Such a jump occurs with the expenditure of energy. The average “settled life” time of a liquid molecule can be calculated as:
\[\left\langle t\right\rangle =t_0e^(\frac(W)(kT))\left(5\right),\]
where $t_0\ $ is the period of oscillations around one equilibrium position. The energy that a molecule must receive in order to move from one position to another is called the activation energy W, and the time the molecule is in the equilibrium position is called the “settled life” time t.
For a water molecule, for example, at room temperature, one molecule undergoes about 100 vibrations and jumps to a new position. The forces of attraction between the molecules of a liquid are strong so that the volume is maintained, but the limited sedentary life of the molecules leads to the emergence of such a phenomenon as fluidity. During particle oscillations near the equilibrium position, they continuously collide with each other, so even a small compression of the liquid leads to a sharp “hardening” of particle collisions. This means a sharp increase in the pressure of the liquid on the walls of the vessel in which it is compressed.
Example 1
Task: Determine the specific heat capacity of copper. Assume that the temperature of copper is close to the melting point. (Molar mass of copper $\mu =63\cdot 10^(-3)\frac(kg)(mol))$
According to Dulong and Petit's law, a mole of chemically simple substances at temperatures close to the melting point has a heat capacity:
Specific heat capacity of copper:
\[С=\frac(с)(\mu )\to С=\frac(3R)(\mu )\left(1.2\right),\] \[С=\frac(3\cdot 8.31) (63\cdot 10^(-3))=0.39\ \cdot 10^3(\frac(J)(kgK))\]
Answer: Specific heat capacity of copper $0.39\ \cdot 10^3\left(\frac(J)(kgK)\right).$
Assignment: Explain in a simplified way from a physics point of view the process of dissolving salt (NaCl) in water.
The basis of the modern theory of solutions was created by D.I. Mendeleev. He established that during dissolution, two processes occur simultaneously: physical - the uniform distribution of particles of the solute throughout the entire volume of the solution, and chemical - the interaction of the solvent with the solute. We are interested in the physical process. Salt molecules do not destroy water molecules. In this case, it would be impossible to evaporate the water. If salt molecules joined water molecules, we would get some new substance. And salt molecules cannot penetrate inside the molecules.
An ion-dipole bond occurs between the Na+ and Cl- ions of chlorine and polar water molecules. It turns out to be stronger than the ionic bonds in the molecules of table salt. As a result of this process, the bond between the ions located on the surface of NaCl crystals is weakened, sodium and chlorine ions are detached from the crystal, and water molecules form so-called hydration shells around them. The separated hydrated ions, under the influence of thermal motion, are evenly distributed between the solvent molecules.
Molecular physics made easy!
Molecular interaction forces
All molecules of a substance interact with each other through forces of attraction and repulsion.
Evidence of the interaction of molecules: the phenomenon of wetting, resistance to compression and tension, low compressibility of solids and gases, etc.
The reason for the interaction of molecules is the electromagnetic interactions of charged particles in a substance.
How to explain this?
An atom consists of a positively charged nucleus and a negatively charged electron shell. The charge of the nucleus is equal to the total charge of all the electrons, so the atom as a whole is electrically neutral.
A molecule consisting of one or more atoms is also electrically neutral.
Let's consider the interaction between molecules using the example of two stationary molecules.
Gravitational and electromagnetic forces can exist between bodies in nature.
Since the masses of molecules are extremely small, negligible forces of gravitational interaction between molecules can be ignored.
At very large distances there is also no electromagnetic interaction between molecules.
But, as the distance between molecules decreases, the molecules begin to orient themselves in such a way that their sides facing each other will have charges of different signs (in general, the molecules remain neutral), and attractive forces arise between the molecules.
With an even greater decrease in the distance between molecules, repulsive forces arise as a result of the interaction of negatively charged electron shells of the atoms of the molecules.
As a result, the molecule is acted upon by the sum of the forces of attraction and repulsion. At large distances, the force of attraction predominates (at a distance of 2-3 diameters of the molecule, the attraction is maximum), at short distances the force of repulsion prevails.
There is a distance between molecules at which the attractive forces become equal to the repulsive forces. This position of the molecules is called the position of stable equilibrium.
Molecules located at a distance from each other and connected by electromagnetic forces have potential energy.
In a stable equilibrium position, the potential energy of the molecules is minimal.
In a substance, each molecule interacts simultaneously with many neighboring molecules, which also affects the value of the minimum potential energy of the molecules.
In addition, all molecules of a substance are in continuous motion, i.e. have kinetic energy.
Thus, the structure of a substance and its properties (solid, liquid and gaseous bodies) are determined by the relationship between the minimum potential energy of interaction of molecules and the reserve of kinetic energy of thermal motion of molecules.
Structure and properties of solid, liquid and gaseous bodies
The structure of bodies is explained by the interaction of particles of the body and the nature of their thermal movement.
Solid
Solids have a constant shape and volume and are practically incompressible.
The minimum potential energy of interaction of molecules is greater than the kinetic energy of molecules.
Strong particle interaction.
The thermal motion of molecules in a solid is expressed only by vibrations of particles (atoms, molecules) around a stable equilibrium position.
Due to the large forces of attraction, molecules practically cannot change their position in matter, this explains the invariability of the volume and shape of solids.
Most solids have a spatially ordered arrangement of particles that form a regular crystal lattice.
Particles of matter (atoms, molecules, ions) are located at the vertices - nodes of the crystal lattice. The nodes of the crystal lattice coincide with the position of stable equilibrium of the particles.
Such solids are called crystalline.
Liquid
Liquids have a certain volume, but do not have their own shape; they take the shape of the vessel in which they are located.
Weak particle interaction.
The thermal motion of molecules in a liquid is expressed by vibrations around a stable equilibrium position within the volume provided to the molecule by its neighbors
Molecules cannot move freely throughout the entire volume of a substance, but transitions of molecules to neighboring places are possible. This explains the fluidity of the liquid and the ability to change its shape.
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In liquids, molecules are quite firmly bound to each other by forces of attraction, which explains the invariance of the volume of the liquid.
In a liquid, the distance between molecules is approximately equal to the diameter of the molecule. When the distance between molecules decreases (compression of the liquid), the repulsive forces increase sharply, so liquids are incompressible.
In terms of their structure and the nature of thermal movement, liquids occupy an intermediate position between solids and gases.
Although the difference between a liquid and a gas is much greater than between a liquid and a solid. For example, during melting or crystallization, the volume of a body changes many times less than during evaporation or condensation.
Gases do not have a constant volume and occupy the entire volume of the vessel in which they are located.
The minimum potential energy of interaction between molecules is less than the kinetic energy of molecules.
Particles of matter practically do not interact.
Gases are characterized by complete disorder in the arrangement and movement of molecules.
In gases, the distance between molecules and atoms is usually much larger than the size of the molecules, but very small. Therefore, gases do not have their own shape and constant volume. Gases are easily compressed because repulsive forces over large distances are also small. Gases have the property of expanding indefinitely, filling the entire volume provided to them. Gas molecules move at very high speeds, collide with each other, and bounce off each other in different directions. Numerous impacts of molecules on the walls of the vessel create gas pressure.
Movement of molecules in liquids
In liquids, molecules not only oscillate around an equilibrium position, but also make jumps from one equilibrium position to the next. These jumps occur periodically. The time interval between such jumps is called average time of settled life(or average relaxation time) and is denoted by the letter τ. In other words, relaxation time is the time of oscillations around one specific equilibrium position. At room temperature this time averages 10 -11 s. The time of one oscillation is 10 -12 ... 10 -13 s.
The time of sedentary life decreases with increasing temperature. The distance between the molecules of a liquid is smaller than the size of the molecules, the particles are located close to each other, and is large. However, the arrangement of liquid molecules is not strictly ordered throughout the volume.
Liquids, like solids, retain their volume, but do not have their own shape. Therefore, they take the shape of the vessel in which they are located. The liquid has the following properties: fluidity. Thanks to this property, the liquid does not resist changing shape, is slightly compressed, and its physical properties are the same in all directions inside the liquid (isotropy of liquids). The nature of molecular motion in liquids was first established by the Soviet physicist Yakov Ilyich Frenkel (1894 - 1952).
Movement of molecules in solids
The molecules and atoms of a solid are arranged in a certain order and form crystal lattice. Such solids are called crystalline. Atoms perform vibrational movements around the equilibrium position, and the attraction between them is very strong. Therefore, solids under normal conditions retain their volume and have their own shape.
Arrangement of molecules in solids. In solids, the distances between molecules are equal to the sizes of the molecules, so solids retain their shape. Molecules are arranged in a certain order, called a crystal lattice, so under normal conditions solids retain their volume.
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Kinetic energy of a molecule
In a gas, molecules move freely (isolated from other molecules), only occasionally colliding with each other or with the walls of the container. As long as a molecule moves freely, it only has kinetic energy. During a collision, the molecules also gain potential energy. Thus, the total energy of a gas is the sum of the kinetic and potential energies of its molecules. The more rarefied the gas, the more molecules at each moment of time are in a state of free movement, having only kinetic energy. Consequently, when the gas is rarefied, the proportion of potential energy decreases in comparison with kinetic energy.
The average kinetic energy of a molecule at equilibrium of an ideal gas has one very important feature: in a mixture of different gases, the average kinetic energy of a molecule for different components of the mixture is the same.
For example, air is a mixture of gases. The average energy of an air molecule for all its components under normal conditions, when air can still be considered an ideal gas, is the same. This property of ideal gases can be proven on the basis of general statistical considerations. An important corollary follows from this: if two different gases (in different vessels) are in thermal equilibrium with each other, then the average kinetic energies of their molecules are the same.
In gases, the distance between molecules and atoms is usually much greater than the size of the molecules themselves; the interaction forces between molecules are not great. As a result, the gas does not have its own shape and constant volume. Gas is easily compressed and can expand without limit. Gas molecules move freely (translationally, they can rotate), only sometimes colliding with other molecules and the walls of the vessel in which the gas is located, and they move at very high speeds.
Movement of particles in solids
The structure of solids is fundamentally different from the structure of gases. In them, the intermolecular distances are small and the potential energy of the molecules is comparable to the kinetic energy. Atoms (or ions, or whole molecules) cannot be called motionless; they perform random oscillatory motion around average positions. The higher the temperature, the greater the oscillation energy, and therefore the average amplitude of oscillations. Thermal vibrations of atoms also explain the heat capacity of solids. Let us consider in more detail the movements of particles in crystalline solids. The entire crystal as a whole is a very complex coupled oscillatory system. The deviations of atoms from their average positions are small, and therefore we can assume that the atoms are subject to the action of quasi-elastic forces that obey Hooke’s linear law. Such oscillatory systems are called linear.
There is a developed mathematical theory of systems subject to linear oscillations. It proves a very important theorem, the essence of which is as follows. If the system performs small (linear) interconnected oscillations, then by transforming the coordinates it can be formally reduced to a system of independent oscillators (whose oscillation equations do not depend on each other). A system of independent oscillators behaves like an ideal gas in the sense that the atoms of the latter can also be considered as independent.
It is by using the idea of independence of gas atoms that we arrive at Boltzmann's law. This very important conclusion provides a simple and reliable basis for the entire theory of solids.
Boltzmann's law
The number of oscillators with given parameters (coordinates and velocities) is determined in the same way as the number of gas molecules in a given state, according to the formula:
Oscillator energy.
Boltzmann's law (1) in the theory of solid bodies has no restrictions, but formula (2) for the oscillator energy is taken from classical mechanics. When theoretically considering solids, one must rely on quantum mechanics, which is characterized by discrete changes in the energy of the oscillator. The discreteness of the oscillator energy becomes insignificant only at sufficiently high values of its energy. This means that (2) can only be used at sufficiently high temperatures. At high temperatures of a solid, close to the melting point, the law of uniform distribution of energy over the degrees of freedom follows from Boltzmann’s law. If in gases for each degree of freedom there is on average an amount of energy equal to (1/2) kT, then the oscillator has one degree of freedom, in addition to the kinetic one, with potential energy. Therefore, per one degree of freedom in a solid at a sufficiently high temperature there is an energy equal to kT. Based on this law, it is not difficult to calculate the total internal energy of a solid body, and after it its heat capacity. A mole of a solid contains NA atoms, and each atom has three degrees of freedom. Therefore, the mole contains 3 NA oscillators. Energy of a mole of a solid
and the molar heat capacity of a solid at sufficiently high temperatures is
Experience confirms this law.
Liquids occupy an intermediate position between gases and solids. Liquid molecules do not disperse over long distances, and liquid under normal conditions retains its volume. But unlike solids, molecules not only vibrate, but also jump from place to place, that is, they perform free movements. As the temperature increases, liquids boil (there is a so-called boiling point) and turn into gas. As the temperature decreases, liquids crystallize and become solids. There is a point in the temperature field at which the boundary between gas (saturated vapor) and liquid disappears (critical point). The pattern of thermal motion of molecules in liquids near the solidification temperature is very similar to the behavior of molecules in solids. For example, the heat capacity coefficients are exactly the same. Since the heat capacity of a substance changes slightly during melting, we can conclude that the nature of the movement of particles in a liquid is close to the movement in a solid (at the melting temperature). When heated, the properties of the liquid gradually change, and it becomes more like a gas. In liquids, the average kinetic energy of particles is less than the potential energy of their intermolecular interaction. The energy of intermolecular interaction in liquids and solids differs insignificantly. If we compare the heat of fusion and the heat of evaporation, we will see that during the transition from one state of aggregation to another, the heat of fusion is significantly lower than the heat of vaporization. An adequate mathematical description of the structure of a liquid can only be given with the help of statistical physics. For example, if a liquid consists of identical spherical molecules, then its structure can be described by the radial distribution function g(r), which gives the probability of detecting any molecule at a distance r from the given one chosen as a reference point. This function can be found experimentally by studying the diffraction of x-rays or neutrons, or a computer simulation of this function can be carried out using Newtonian mechanics.
The kinetic theory of liquid was developed by Ya.I. Frenkel. In this theory, a liquid is considered, as in the case of a solid, as a dynamic system of harmonious oscillators. But unlike a solid body, the equilibrium position of molecules in a liquid is temporary. After oscillating around one position, the liquid molecule jumps to a new position located nearby. Such a jump occurs with the expenditure of energy. The average “settled life” time of a liquid molecule can be calculated as:
\[\left\langle t\right\rangle =t_0e^(\frac(W)(kT))\left(5\right),\]
where $t_0\ $ is the period of oscillations around one equilibrium position. The energy that a molecule must receive in order to move from one position to another is called the activation energy W, and the time the molecule is in the equilibrium position is called the “settled life” time t.
For a water molecule, for example, at room temperature, one molecule undergoes about 100 vibrations and jumps to a new position. The forces of attraction between the molecules of a liquid are strong so that the volume is maintained, but the limited sedentary life of the molecules leads to the emergence of such a phenomenon as fluidity. During particle oscillations near the equilibrium position, they continuously collide with each other, so even a small compression of the liquid leads to a sharp “hardening” of particle collisions. This means a sharp increase in the pressure of the liquid on the walls of the vessel in which it is compressed.
Example 1
Task: Determine the specific heat capacity of copper. Assume that the temperature of copper is close to the melting point. (Molar mass of copper $\mu =63\cdot 10^(-3)\frac(kg)(mol))$
According to Dulong and Petit's law, a mole of chemically simple substances at temperatures close to the melting point has a heat capacity:
Specific heat capacity of copper:
\[С=\frac(с)(\mu )\to С=\frac(3R)(\mu )\left(1.2\right),\] \[С=\frac(3\cdot 8.31) (63\cdot 10^(-3))=0.39\ \cdot 10^3(\frac(J)(kgK))\]
Answer: Specific heat capacity of copper $0.39\ \cdot 10^3\left(\frac(J)(kgK)\right).$
Assignment: Explain in a simplified way from a physics point of view the process of dissolving salt (NaCl) in water.
The basis of the modern theory of solutions was created by D.I. Mendeleev. He established that during dissolution, two processes occur simultaneously: physical - the uniform distribution of particles of the solute throughout the entire volume of the solution, and chemical - the interaction of the solvent with the solute. We are interested in the physical process. Salt molecules do not destroy water molecules. In this case, it would be impossible to evaporate the water. If salt molecules joined water molecules, we would get some new substance. And salt molecules cannot penetrate inside the molecules.
An ion-dipole bond occurs between the Na+ and Cl- ions of chlorine and polar water molecules. It turns out to be stronger than the ionic bonds in the molecules of table salt. As a result of this process, the bond between the ions located on the surface of NaCl crystals is weakened, sodium and chlorine ions are detached from the crystal, and water molecules form so-called hydration shells around them. The separated hydrated ions, under the influence of thermal motion, are evenly distributed between the solvent molecules.