Temperature is a measure of kinetic energy. Absolute temperature

It represents the energy that is determined by the speed of movement of various points belonging to this system. In this case, one should distinguish between the energy that characterizes the translational movement and the rotational movement. At the same time, the average kinetic energy is the average difference between the total energy of the entire system and its rest energy, that is, in essence, its value is the average potential energy.

Its physical value is determined by the formula 3 / 2 kT, in which are indicated: T - temperature, k - Boltzmann constant. This value can serve as a kind of comparison criterion (standard) for the energies contained in various types of thermal motion. For example, the average kinetic energy for gas molecules in the study of translational motion is 17 (- 10) nJ at a gas temperature of 500 C. As a rule, electrons have the highest energy in translational motion, but the energy of neutral atoms and ions is much less.

This value, if we consider any solution, gas or liquid at a given temperature, has a constant value. This statement is also true for colloidal solutions.

The situation is somewhat different for solids. In these substances, the average kinetic energy of any particle is too small to overcome the forces of molecular attraction, and therefore it can only move around a certain point, which conditionally fixes a certain equilibrium position of the particle over a long period of time. This property allows the solid to be sufficiently stable in shape and volume.

If we consider the conditions: translational motion and an ideal gas, then here the average kinetic energy is not a quantity dependent on the molecular weight, and therefore is defined as a value directly proportional to the value of the absolute temperature.

We have given all these judgments in order to show that they are valid for all types of aggregate states of matter - in any of them, temperature acts as the main characteristic that reflects the dynamics and intensity of the thermal motion of elements. And this is the essence of the molecular-kinetic theory and the content of the concept of thermal equilibrium.

As you know, if two physical bodies come into interaction with each other, then a process of heat transfer occurs between them. If the body is a closed system, that is, it does not interact with any bodies, then its heat exchange process will last as long as it takes to equalize the temperatures of this body and the environment. This state is called thermodynamic equilibrium. This conclusion has been repeatedly confirmed by experimental results. To determine the average kinetic energy, one should refer to the characteristics of the temperature of a given body and its heat transfer properties.

It is also important to take into account that microprocesses inside bodies do not end even when the body enters thermodynamic equilibrium. In this state, molecules move inside the bodies, change their velocities, impacts and collisions. Therefore, only one of several of our statements is true - the volume of the body, the pressure (if we are talking about gas), may differ, but the temperature will still remain constant. This once again confirms the assertion that the average kinetic energy of thermal motion in isolated systems is determined solely by the temperature index.

This pattern was established in the course of experiments by J. Charles in 1787. While conducting experiments, he noticed that when bodies (gases) are heated by the same amount, their pressure changes in accordance with a directly proportional law. This observation made it possible to create many useful instruments and things, in particular, a gas thermometer.

This value, if we consider any solution, gas or liquid at a given temperature, has a constant value. This statement is also true for colloidal solutions.

If we consider the conditions: translational motion and then here the average kinetic energy is not a quantity dependent on and therefore is defined as a value directly proportional to the value

In order to compare ideal gas equation of state and the basic equation of molecular kinetic theory, we write them in the most consistent form.

From these ratios it can be seen that:

(1.48)

quantity, which is called permanent Boltzmann- coefficient allowing energy movements molecules(of course average) to express in units temperature, and not only in joules like so far.

As already mentioned, "to explain" in physics means to establish a connection between a new phenomenon, in this case - thermal, with the already studied - mechanical movement. This is the explanation of thermal phenomena. It is with the aim of finding such an explanation that a whole science has now been developed - statisticalphysics. The word "statistical" means that the objects of study are phenomena in which many particles with random (for each particle) properties participate. The study of such objects in human multitudes - peoples, populations - is the subject of statistics.

It is statistical physics that is the basis of chemistry as a science, and not like in a cookbook - “drain this and that, it will turn out what you need!” Why will it work? The answer lies in the properties (statistical properties) of the molecules.

Note that, of course, it is possible to use the found connections between the energy of motion of molecules and the temperature of the gas in another direction to reveal the properties of the motion of molecules, in general, the properties of the gas. For example, it is clear that molecules inside a gas have energy:

(1.50)

This energy is called internal.Internal energy there is always! Even when the body is at rest and does not interact with any other bodies, it has internal energy.

If the molecule is not a “round ball”, but is a “dumbbell” (diatomic molecule), then the kinetic energy is the sum of the energy of translational motion (only translational motion has actually been considered so far) and rotational motion ( rice. 1.18 ).

Rice. 1.18. Molecule rotation

Arbitrary rotation can be imagined as a sequential rotation first around the axis x, and then around the axis z.

The energy reserve of such a movement should not differ in any way from the reserve of movement in a straight line. The molecule "does not know" whether it is flying or spinning. Then in all formulas it is necessary to put the number "five" instead of the number "three".

(1.51)

Gases such as nitrogen, oxygen, air, etc., must be considered precisely according to the last formulas.

In general, if for strict fixation of a molecule in space it is necessary i numbers (say "i degrees of freedom"), then

(1.52)

As they say, "on the floor kT for each degree of freedom.

1.9. Solute as an ideal gas

Ideas about an ideal gas find interesting applications in explaining osmotic pressure that occurs in solution.

Let there be particles of some other solute among the solvent molecules. As is known, particles of a dissolved substance tend to occupy the entire available volume. The solute expands in exactly the same way as it expandsgas,to take up the space given to him.

Just as a gas exerts pressure on the walls of a vessel, the solute exerts pressure on the boundary that separates the solution from the pure solvent. This extra pressure is called osmotic pressure. This pressure can be observed if the solution is separated from the pure solvent semi-tight partition, through which the solvent easily passes, but the solute does not pass ( rice. 1.19 ).

Rice. 1.19. Occurrence of osmotic pressure in the solute compartment

The solute particles tend to move the partition apart, and if the partition is soft, then it bulges. If the partition is rigidly fixed, then the liquid level actually shifts, the level solution in the solute compartment rises (see rice. 1.19 ).

Solution level rise h will continue until the resulting hydrostatic pressure ρ gh(ρ is the density of the solution) will not be equal to the osmotic pressure. There is a complete similarity between gas molecules and solute molecules. Both those and others are far from each other, and they both move chaotically. Of course, there is a solvent between the molecules of the solute, and there is nothing between the molecules of the gas (vacuum), but this is not important. Vacuum was not used in the derivation of laws! Hence it follows that solute particlesin a weak solution behave in the same way as the molecules of an ideal gas. In other words, osmotic pressure exerted by a solute,equal to the pressure that the same substance would produce in a gaseousin the same volume and at the same temperature. Then we get that osmotic pressureπ proportional to the temperature and concentration of the solution(number of particles n per unit volume).

(1.53)

This law is called van't Hoff's law, formula ( 1.53 ) -van't Hoff formula.

The complete similarity of the van't Hoff law with the Clapeyron–Mendeleev equation for an ideal gas is obvious.

The osmotic pressure, of course, does not depend on the type of semi-permeable partition or the type of solvent. Any solutions with the same molar concentration have the same osmotic pressure.

The similarity in the behavior of a solute and an ideal gas is due to the fact that in a dilute solution, the particles of the solute practically do not interact with each other, just as the molecules of an ideal gas do not interact.

The magnitude of the osmotic pressure is often quite significant. For example, if a liter of solution contains 1 mole of a solute, then van't Hoff formula at room temperature, we have π ≈ 24 atm.

If the solute, upon dissolution, decomposes into ions (dissociates), then according to the van't Hoff formula

π V = NkT(1.54)

it is possible to determine the total number N formed particles - ions of both signs and neutral (non-dissociated) particles. And, therefore, one can know degree dissociation substances. Ions can be solvated, but this circumstance does not affect the validity of the van't Hoff formula.

The van't Hoff formula is often used in chemistry to definitions of molecularmass of proteins and polymers. To do this, to the volume solvent V add m grams of the test substance, measure the pressure π. From the formula

(1.55)

find the molecular weight.

So far we have not dealt with temperature; we deliberately avoided talking about this topic. We know that if you compress a gas, the energy of the molecules increases, and we usually say that the gas heats up. Now we need to understand what this has to do with temperature. We know what adiabatic compression is, but how can we set up an experiment so that we can say that it was carried out at a constant temperature? If we take two identical boxes of gas, put them one on top of the other, and hold them like that for a long time, then even if at the beginning these boxes had what we called different temperatures, in the end their temperatures will become the same. What does this mean? Only that the boxes have reached the state they would eventually reach if they were left to themselves for a long time! The state in which the temperatures of two bodies are equal is exactly the final state that is reached after a long contact with each other.

Let's see what happens if the box is divided into two parts by a moving piston and each compartment is filled with a different gas, as shown in Fig. 39.2 (for simplicity, let's assume that there are two monatomic gases, say helium and neon). In compartment 1, mass atoms move at a speed, and in a unit volume there are pieces, in compartment 2, these numbers are respectively equal to , and . Under what conditions is equilibrium achieved?

Fig. 39.2. Atoms of two different monatomic gases separated by a moving piston.

Of course, the bombardment on the left causes the piston to move to the right and compresses the gas in the second compartment, then the same thing happens on the right and the piston goes back and forth until the pressure on both sides is equal, and then the piston stops. We can arrange so that the pressure on both sides is the same, for this it is necessary that the internal energies per unit volume be the same, or that the products of the number of particles per unit volume and the average kinetic energy be the same in both compartments. Now we will try to prove that at equilibrium the individual factors must also be the same. So far, we only know that the products of the numbers of particles in unit volumes and the average kinetic energies are equal

;

this follows from the condition of equality of pressures and from (39.8). We have to establish that as the equilibrium is gradually approached, when the temperatures of the gases are equal, not only this condition is satisfied, but something else also happens.

To be clearer, let's assume that the desired pressure on the left side of the box is achieved by very high density but low velocities. For large and small, you can get the same pressure as for small and large. The atoms, if tightly packed, may move slowly, or there may be very few atoms, but they hit the piston with more force. Will the balance be established forever? At first it seems that the piston will not move anywhere and will always be so, but if you think it over again, it becomes clear that we have missed one very important thing. The fact is that the pressure on the piston is not at all uniform, the piston swings just like the eardrum, which we talked about at the beginning of the chapter, because each new blow is not like the previous one. It turns out not a constant uniform pressure, but rather something like a drum roll - the pressure is constantly changing, and our piston seems to constantly tremble. Let us suppose that the atoms of the right compartment hit the piston more or less evenly, and that there are fewer atoms on the left, and their impacts are rare, but very energetic. Then the piston will continually receive a very strong impulse from the left and move to the right, towards slower atoms, and the speed of these atoms will increase. (When colliding with a piston, each atom gains or loses energy depending on which direction the piston moves at the time of the collision.) After several collisions, the piston will swing, then another, another, and another ..., gas in the right compartment will be from time to time shaken, and this will lead to an increase in the energy of its atoms, and their movement will accelerate. This will continue until the piston swings are balanced. And equilibrium will be established when the speed of the piston becomes such that it will take away energy from atoms as quickly as it gives it away. So, the piston moves at some average speed, and we have to find it. If we succeed in this, we will come closer to solving the problem, because the atoms must adjust their velocities so that each gas receives exactly as much energy through the piston as it loses.

It is very difficult to calculate the movement of the piston in all details; although all this is very easy to understand, it turns out that it is somewhat more difficult to analyze. Before embarking on such an analysis, let's solve another problem: let the box be filled with molecules of two kinds with masses and , velocities, etc.; now the molecules can get to know each other better. If at first all No. 2 molecules are at rest, then this cannot continue for a long time, because No. 1 molecules will hit them and impart some speed to them. If No. 2 molecules can move much faster than No. 1 molecules, then sooner or later they will have to give up part of their energy to slower molecules. Thus, if the box is filled with a mixture of two gases, then the problem is to determine the relative velocity of the molecules of both kinds.

This is also a very difficult task, but we will still solve it. First we have to solve the "subproblem" (again, this is one of those cases where, no matter how the problem is solved, the final result is easy to remember, and the conclusion requires great art). Suppose we have two colliding molecules with different masses; in order to avoid complications, we observe the collision from the system of their center of mass (c.m.), from where it is easier to follow the impact of molecules. According to the laws of collisions, derived from the laws of conservation of momentum and energy, after a collision, molecules can only move in such a way that each retains the value of its original speed, and they can only change the direction of movement. A typical collision looks like it is depicted in Fig. 39.3. Suppose for a moment that we observe collisions whose center-of-mass systems are at rest. In addition, it must be assumed that all molecules move horizontally. Of course, after the first collision, some of the molecules will move at some angle to the original direction. In other words, if at the beginning all the molecules moved horizontally, then after some time we will find molecules already moving vertically. After a series of other collisions, they will change direction again and turn another angle. Thus, even if someone manages to put the molecules in order at first, they will still very soon disperse in different directions and each time will be more and more dispersed. Where will this eventually lead? Answer: Any pair of molecules will move in an arbitrarily chosen direction as readily as in any other. After that, further collisions can no longer change the distribution of molecules.

Fig. 39. 3. Collision of two unequal molecules, as viewed from the center of mass system.

What is meant when one speaks of equiprobable motion in any direction? Of course, one cannot talk about the probability of movement along a given straight line - the straight line is too thin for probability to be attributed to it, but one should take the unit of "something". The idea is that as many molecules pass through a given section of the sphere centered at the collision point as through any other section of the sphere. As a result of collisions, the molecules are distributed in directions so that any two segments of the sphere equal in area will have equal probabilities (i.e., the same number of molecules that have passed through these segments).

By the way, if we compare the original direction and the direction that forms some angle with it, then it is interesting that the elementary area on a sphere of unit radius is equal to the product of , or, what is the same, the differential . This means that the cosine of the angle between two directions is equally likely to take on any value between and .

Now we need to remember what is actually there; because we do not have collisions in the center of mass system, but two atoms collide with arbitrary vector velocities and . What happens to them? We will do this: we will again go to the center of mass system, only now it moves with a “mass-averaged” speed . If you follow the collision from the center of mass system, then it will look like it is shown in Fig. 39.3, only one has to think about the relative speed of the collision. The relative speed is . The situation, therefore, is as follows: the center-of-mass system moves, and in the center-of-mass system the molecules approach each other with a relative velocity ; colliding, they move in new directions. While all this is happening, the center of mass is moving at the same speed all the time without change.

Well, what happens in the end? From the previous reasoning, we draw the following conclusion: at equilibrium, all directions are equally probable relative to the direction of motion of the center of mass. This means that in the end there will be no correlation between the direction of the relative velocity and the movement of the center of mass. Even if such a correlation existed at the beginning, collisions would destroy it and it would eventually disappear completely. Therefore, the average value of the cosine of the angle between and is zero. It means that

The scalar product is easy to express in terms of and :

Let's do it first; what is the average? In other words, what is the average of the projection of the velocity of one molecule onto the direction of the velocity of another molecule? It is clear that the probabilities of a molecule moving both in one direction and in the opposite direction are the same. The average speed in any direction is zero. Therefore, the mean value in the direction is also zero. So the mean is zero! Therefore, we came to the conclusion that the mean should be equal to . This means that the average kinetic energies of both molecules must be equal:

If a gas consists of two kinds of atoms, then it can be shown (and we even believe that we have succeeded in doing so) that the average kinetic energies of atoms of each kind are equal when the gas is in equilibrium. This means that heavy atoms move more slowly than light ones; it is easy to verify this by setting up an experiment with "atoms" of various masses in an air trough.

Now we take the next step and show that if there are two gases in a box separated by a partition, then as equilibrium is reached, the average kinetic energies of atoms of different gases will be the same, although the atoms are enclosed in different boxes. Reasoning can be structured in different ways. For example, one can imagine that a small hole has been made in the partition (Fig. 39.4), so that the molecules of one gas pass through it, while the molecules of the second are too large and do not fit through. When equilibrium is established, then in the compartment where the mixture of gases is located, the average kinetic energies of the molecules of each type will become equal. But after all, among the molecules that have penetrated through the hole, there are those that have not lost energy, so the average kinetic energy of pure gas molecules must be equal to the average kinetic energy of the molecules of the mixture. This is not a very satisfactory proof, because there might not have been such a hole through which molecules of one gas would pass and molecules of another could not pass.

Fig. 39.4. Two gases in a box separated by a semi-permeable partition.

Let's get back to the piston problem. It can be shown that the kinetic energy of the piston must also be equal to . In fact, the kinetic energy of the piston is associated only with its horizontal movement. Neglecting the possible movement of the piston up and down, we find that the horizontal movement corresponds to the kinetic energy. But in the same way, based on the equilibrium on the other side, it can be shown that the kinetic energy of the piston must be equal to . Although we repeat the previous discussion, some additional difficulties arise due to the fact that, as a result of collisions, the average kinetic energies of the piston and gas molecules are equal, because the piston is not inside the gas, but is displaced to one side.

If you are not satisfied with this proof, then you can think of an artificial example when the balance is provided by a device on which the molecules of each gas hit from both sides. Suppose that a short rod passes through the piston, at the ends of which a ball is planted. The rod can move through the piston without friction. Molecules of the same kind are hitting each of the balls from all sides. Let the mass of our device be , and the masses of the gas molecules, as before, be equal to and . As a result of collisions with molecules of the first kind, the kinetic energy of a body of mass is equal to the average value (we have already proved this). Similarly, collisions with second-class molecules cause the body to have a kinetic energy equal to the average value . If the gases are in thermal equilibrium, then the kinetic energies of both balls must be equal. Thus, the result proved for the case of a mixture of gases can be immediately generalized to the case of two different gases at the same temperature.

So, if two gases have the same temperature, then the average kinetic energies of the molecules of these gases in the center of mass system are equal.

The average kinetic energy of molecules is a property of "temperature" only. And being a property of "temperature" rather than a gas, it can serve as a definition of temperature. The average kinetic energy of a molecule is thus some function of temperature. But who will tell us on what scale to count the temperature? We can define the temperature scale ourselves so that the average energy is proportional to the temperature. The best way to do this is to call the average energy itself “temperature”. This would be the simplest function, but, unfortunately, this scale has already been chosen differently and instead of calling the energy of the molecule simply "temperature", a constant factor is used that relates the average energy of the molecule and the degree of absolute temperature, or degree Kelvin. This multiplier is joules per degree Kelvin. Thus, if the absolute temperature of the gas is equal to , then the average kinetic energy of the molecule is equal (the factor is introduced only for convenience, due to which the factors in other formulas will disappear).

Note that the kinetic energy associated with the component of motion in any direction is only . Three independent directions of movement bring it to .

« Physics - Grade 10 "

absolute temperature.

Instead of temperature Θ, expressed in energy units, we introduce temperature, expressed in degrees familiar to us.

Θ = kТ, (9.12)

where k is the coefficient of proportionality.

>The temperature defined by equation (9.12) is called absolute.

Such a name, as we shall now see, has sufficient grounds. Taking into account definition (9.12), we obtain

This formula introduces a temperature scale (in degrees) that does not depend on the substance used to measure the temperature.

The temperature defined by formula (9.13) obviously cannot be negative, since all the quantities on the left side of this formula are obviously positive. Therefore, the smallest possible value of temperature T is T = 0 if pressure p or volume V are zero.

The limiting temperature at which the pressure of an ideal gas vanishes at a fixed volume, or at which the volume of an ideal gas tends to zero at a constant pressure, is called absolute zero temperature.

This is the lowest temperature in nature, that “greatest or last degree of cold”, the existence of which Lomonosov predicted.

The English scientist W. Thomson (Lord Kelvin) (1824-1907) introduced the absolute temperature scale. Zero temperature on an absolute scale (also called Kelvin scale) corresponds to absolute zero, and each unit of temperature on this scale is equal to a degree Celsius.

The SI unit of absolute temperature is called kelvin(denoted by the letter K).

Boltzmann's constant.

We define the coefficient k in formula (9.13) so that a change in temperature by one kelvin (1 K) is equal to a change in temperature by one degree Celsius (1 °C).

We know the values of Θ at 0 °С and 100 °С (see formulas (9.9) and (9.11)). Let us denote the absolute temperature at 0 °C through T 1, and at 100 °C through T 2. Then according to formula (9.12)

Θ 100 - Θ 0 \u003d k (T 2 -T 1),

Θ 100 - Θ 0 \u003d k 100 K \u003d (5.14 - 3.76) 10 -21 J.

Coefficient

k = 1.38 10 -23 J/K (9.14)

called Boltzmann constant in honor of L. Boltzmann, one of the founders of the molecular-kinetic theory of gases.

Boltzmann's constant relates the temperature Θ in energy units to the temperature T in kelvins.

This is one of the most important constants in molecular kinetic theory.

Knowing the Boltzmann constant, you can find the value of absolute zero on the Celsius scale. To do this, we first find the value of the absolute temperature corresponding to 0 °C. Since at 0 ° C kT 1 \u003d 3.76 10 -21 J, then

One kelvin and one degree Celsius are the same. Therefore, any value of the absolute temperature T will be 273 degrees higher than the corresponding temperature t in Celsius:

T (K) = (f + 273) (°C). (9.15)

The change in absolute temperature ΔТ is equal to the change in temperature on the Celsius scale Δt: ΔТ(К) = Δt (°С).

Figure 9.5 shows the absolute scale and the Celsius scale for comparison. Absolute zero corresponds to the temperature t = -273 °C.

The US uses the Fahrenheit scale. The freezing point of water on this scale is 32 °F, and the boiling point is 212 °E. The temperature is converted from Fahrenheit to Celsius using the formula t(°C) = 5/9 (t(°F) - 32).

Note the most important fact: absolute zero temperature is unattainable!

Temperature is a measure of the average kinetic energy of molecules.

From the basic equation of the molecular-kinetic theory (9.8) and the definition of temperature (9.13), the most important consequence follows:
absolute temperature is a measure of the average kinetic energy of the movement of molecules.

Let's prove it.

Equations (9.7) and (9.13) imply that This implies the relationship between the average kinetic energy of the translational motion of the molecule and temperature:

The average kinetic energy of the chaotic translational motion of gas molecules is proportional to the absolute temperature.

The higher the temperature, the faster the molecules move. Thus, the previously put forward conjecture about the relationship between temperature and the average velocity of molecules has received a reliable justification. The relation (9.16) between the temperature and the average kinetic energy of the translational motion of molecules has been established for ideal gases.

However, it turns out to be true for any substances in which the motion of atoms or molecules obeys the laws of Newtonian mechanics. It is true for liquids as well as for solids, where atoms can only vibrate around the equilibrium positions at the nodes of the crystal lattice.

When the temperature approaches absolute zero, the energy of the thermal motion of molecules approaches zero, i.e., the translational thermal motion of the molecules stops.

The dependence of gas pressure on the concentration of its molecules and temperature. Considering that from formula (9.13) we obtain an expression showing the dependence of gas pressure on the concentration of molecules and temperature:

From formula (9.17) it follows that at the same pressures and temperatures, the concentration of molecules in all gases is the same.

From here follows Avogadro's law, known to you from the course of chemistry.

Avogadro's law:

Equal volumes of gases at the same temperatures and pressures contain the same number of molecules.