Basic equation of the MKT. Temperature as a measure of average kinetic energy random movement of molecules.
Why does gas exert pressure? Gas molecules continuously randomly move, collide with the walls of the vessel and transfer their momentum p=m to them v Pressure is the total impulse transmitted by molecules of 1 sq. m wall for 1s.
Thermal equilibrium
- this is such a state of a system of bodies in thermal contact, in which there is no heat transfer from one body to another, and all the macroscopic parameters of the bodies remain unchanged. Temperature is a physical parameter same for all bodies in thermal equilibrium. The possibility of introducing the concept of temperature follows from experience and is called zero law of thermodynamics.
In a system of bodies in a state of thermodynamic equilibrium, the volumes and pressures can be different, but the temperatures are necessarily the same. Thus, temperature characterizes the state of thermodynamic equilibrium of an isolated system of bodies.
Temperature T, pressure R and volumeV
– macroscopic quantities characterizing the state of a huge number of molecules, i.e. gas condition in general Gas thermometers. To calibrate a constant volume gas thermometer, you can measure the pressure at two temperatures (for example, 0 °C and 100 °C), plot the points p 0 and p 100 on a graph, and then draw a straight line between them. Using the calibration curve thus obtained, temperatures corresponding to other pressures can be determined.
By extrapolating the graph to the region of low pressures, it is possible to determine some "hypothetical" temperature, at which the pressure of the gas would be zero. Experience shows that this temperature is -273.15 ° C and does not depend on the properties of the gas. The English physicist W. Kelvin (Thomson) in 1848 suggested using the point of zero gas pressure to build a new temperature scale (the Kelvin scale). In this scale, the temperature unit is the same as in the Celsius scale, but the zero point is shifted:T= t +273.15. An ideal gas is a gas consisting of vanishingly small spherical molecules that interact with each other and with the walls only during elastic collisions. Ideal gas (model) 1. Set a large number molecules of mass m0, the dimensions of the molecules are neglected (they take the molecules as material points) 2. Molecules are at large distances from each other and move randomly. 3. Molecules interact according to the laws of elastic collisions, the forces of attraction between molecules are neglected. 4. The velocities of the molecules are varied, but at a certain temperature the average speed of the molecules remains constant. Real gas 1. Real gas molecules are not point formations, the diameters of molecules are only tens of times less distances between molecules. 2. Molecules do not interact according to the laws of elastic collisions
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 thermal movement. For example, the average kinetic energy for gas molecules in the study forward movement, is equal to 17 (- 10) nJ at a gas temperature of 500 C. As a rule, the most energy in translational motion, electrons have, 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 solid matter 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 for the purpose of showing that they are valid for all types. aggregate states substances - in any of them, temperature acts as the main characteristic, reflecting the dynamics and intensity of the thermal movement 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 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 a constant value. 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.
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 constant Boltzmann- coefficient allowing energy movements molecules(of course average) to express v units temperature, and not only in joules like so far.
As already mentioned, "to explain" in physics means to establish the connection of a new phenomenon, in this case- thermal, with 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.
With a decrease in the absolute temperature of an ideal gas by 1.5 times, the average kinetic energy of the thermal motion of molecules
1) will increase by 1.5 times
2) will decrease by 1.5 times
3) will decrease by 2.25 times
4) will not change
Solution.
With a decrease in absolute temperature by 1.5 times, the average kinetic energy will also decrease by 1.5 times.
Correct answer: 2.
Answer: 2
With a decrease in the absolute temperature of an ideal gas by a factor of 4, the root-mean-square velocity of the thermal motion of its molecules
1) decrease by 16 times
2) will decrease by 2 times
3) will decrease by 4 times
4) will not change
Solution.
The absolute temperature of an ideal gas is proportional to the square of the root-mean-square velocity: Thus, with a decrease in the absolute temperature by 4 times, the root-mean-square velocity of its molecules will decrease by 2 times.
Correct answer: 2.
Vladimir Pokidov (Moscow) 21.05.2013 16:37
We were sent such a wonderful formula as E \u003d 3 / 2kT, The average kinetic energy of the thermal motion of molecules of an ideal gas is directly proportional to its temperature, as the temperature changes, so does the average kinetic energy of the thermal motion of molecules
Alexei
Good day!
That's right, in fact, the temperature and the average energy of thermal motion are one and the same. But in this problem we are asked about speed, not about energy.
With an increase in the absolute temperature of an ideal gas by a factor of 2, the average kinetic energy of the thermal motion of molecules
1) will not change
2) will increase by 4 times
3) will decrease by 2 times
4) will increase by 2 times
Solution.
The average kinetic energy of the thermal motion of molecules of an ideal gas is directly proportional to the absolute temperature, for example, for a monatomic gas:
When the absolute temperature doubles, the average kinetic energy also doubles.
Correct answer: 4.
Answer: 4
With a decrease in the absolute temperature of an ideal gas by a factor of 2, the average kinetic energy of the thermal motion of molecules
1) will not change
2) will decrease by 4 times
3) will decrease by 2 times
4) will increase by 2 times
Solution.
The average kinetic energy of the thermal motion of ideal gas molecules is directly proportional to the absolute temperature:
When the absolute temperature decreases by a factor of 2, the average kinetic energy will also decrease by a factor of 2.
Correct answer: 3.
Answer: 3
With an increase in the root-mean-square velocity of the thermal motion of molecules by a factor of 2, the average kinetic energy of the thermal motion of molecules
1) will not change
2) will increase by 4 times
3) will decrease by 4 times
4) will increase by 2 times
Solution.
Therefore, an increase in the root mean square velocity of thermal motion by a factor of 2 will lead to an increase in the average kinetic energy by a factor of 4.
Correct answer: 2.
Answer: 2
Alexey (St. Petersburg)
Good day!
Both formulas are valid. The formula used in the solution (the first equality) is simply mathematical notation definitions of the average kinetic energy: that you need to take all the molecules, calculate their kinetic energies, and then take the arithmetic mean. The second (identical) equality in this formula is just the definition of what root mean square speed is.
Your formula is actually much more serious, it shows that the average energy of thermal motion can be used as a measure of temperature.
With a 2-fold decrease in the mean square velocity of the thermal motion of molecules, the average kinetic energy of the thermal motion of molecules
1) will not change
2) will increase by 4 times
3) will decrease by 4 times
4) will increase by 2 times
Solution.
The average kinetic energy of the thermal motion of molecules is proportional to the square of the root mean square velocity of the thermal motion of molecules:
Therefore, a 2-fold decrease in the root-mean-square velocity of thermal motion will lead to a 4-fold decrease in the average kinetic energy.
Correct answer: 3.
Answer: 3
With an increase in the average kinetic energy of the thermal motion of molecules by a factor of 4, their root-mean-square velocity
1) will decrease by 4 times
2) will increase by 4 times
3) will decrease by 2 times
4) will increase by 2 times
Solution.
Consequently, with an increase in the average kinetic energy of the thermal motion of molecules by a factor of 4, their root-mean-square velocity will increase by a factor of 2.
Correct answer: 4.
Answer: 4
Alexey (St. Petersburg)
Good day!
A sign is an identical equality, that is, an equality that always holds, in fact, when there is such a sign, it means that the values \u200b\u200bare equal by definition.
Yana Firsova (Gelendzhik) 25.05.2012 23:33
Yuri Shoitov (Kursk) 10.10.2012 10:00
Hello Alexey!
There is an error in your solution that does not affect the answer. Why did you need to talk about the square of the average value of the speed modulus in your decision? There is no such term in the assignment. Moreover, it is not at all equal to the mean square value, but only proportional. Therefore your identity is false.
Yuri Shoitov (Kursk) 10.10.2012 22:00
Good evening, Alexey!
If so, what is the joke that you designate the same value in different ways in the same formula?! Is that to give more science. Believe in our method of teaching physics and without you this "good" is enough.
Alexey (St. Petersburg)
I can't figure out what's bothering you. I have written that the square of rms speed is, by definition, the average of the square of speed. The dash is just part of the rms speed designation, and the b is the averaging procedure.
With a decrease in the average kinetic energy of the thermal motion of molecules by 4 times, their root-mean-square velocity
1) will decrease by 4 times
2) will increase by 4 times
3) will decrease by 2 times
4) will increase by 2 times
Solution.
The average kinetic energy of the thermal motion of molecules is proportional to the square of the root-mean-square velocity:
Consequently, with a decrease in the average kinetic energy of the thermal motion of molecules by 4 times, their root-mean-square velocity will decrease by 2 times.
Correct answer: 3.
Answer: 3
With an increase in the absolute temperature of a monatomic ideal gas by a factor of 2, the root-mean-square velocity of the thermal motion of molecules
1) decrease by a factor
2) will increase in times
3) will decrease by 2 times
4) will increase by 2 times
Solution.
The absolute temperature of an ideal monatomic gas is proportional to the square of the root-mean-square velocity of the thermal motion of molecules. Really:
Consequently, with an increase in the absolute temperature of an ideal gas by a factor of 2, the root-mean-square velocity of the thermal motion of molecules will increase by a factor of .
Correct answer: 2.
Answer: 2
With a decrease in the absolute temperature of an ideal gas by a factor of 2, the root-mean-square velocity of the thermal motion of molecules
1) decrease by a factor
2) will increase in times
3) will decrease by 2 times
4) will increase by 2 times
Solution.
The absolute temperature of an ideal gas is proportional to the square of the root-mean-square velocity of the thermal motion of molecules. Really:
Consequently, when the absolute temperature of an ideal gas is reduced by a factor of 2, the root-mean-square velocity of the thermal motion of molecules will decrease by a factor of .
Correct answer: 1.
Answer: 1
Alexey (St. Petersburg)
Good day!
Don't confuse average value of the square of the speed is not equal to the square of the average speed, but to the square of the mean square speed. The average velocity for a gas molecule is generally zero.
Yuri Shoitov (Kursk) 11.10.2012 10:07
You confuse all the same and not the guest.
In all school physics the letter v without an arrow denotes the velocity modulus. If there is a line above this letter, then this indicates the average value of the velocity modulus, which is calculated from the Maxwell distribution, and it is equal to 8RT / pi * mu. The square root of the root mean square speed is 3RT/pi*mu. As you can see, there is no equality in your identity.
Alexey (St. Petersburg)
Good day!
I don’t even know what to object, this is probably a question of designations. In Myakishev's textbook, the root-mean-square speed is denoted in this way, Sivukhin uses the notation. How do you use this value?
Igor (Who needs to know) 01.02.2013 16:15
Why did you calculate the temperature of an ideal gas using the kinetic energy formula? After all, the root mean square speed is found by the formula: http://reshuege.ru/formula/d5/d5e3acf50adcde572c26975a0d743de1.png = Root of (3kT/m0)
Alexey (St. Petersburg)
Good day!
If you look closely, you will see that your definition of root mean square speed is the same as the one used in the solution.
By definition, the square of the mean square velocity is equal to the mean square of the velocity, and it is through the latter that the temperature of the gas is determined.
With a decrease in the average kinetic energy of the thermal motion of molecules by a factor of 2, the absolute temperature
1) will not change
2) will increase by 4 times
3) will decrease by 2 times
4) will increase by 2 times
Solution.
The average kinetic energy of the thermal motion of ideal gas molecules is directly proportional to the absolute temperature:
Consequently, with a decrease in the average kinetic energy of thermal motion by a factor of 2, the absolute temperature of the gas will also decrease by a factor of 2.
Correct answer: 3.
Answer: 3
As a result of neon heating, the temperature of this gas increased by 4 times. The average kinetic energy of the thermal motion of its molecules in this case
1) increased by 4 times
2) increased by 2 times
3) decreased by 4 times
4) has not changed
Thus, as a result of heating neon by a factor of 4, the average kinetic energy of the thermal motion of its molecules increases by a factor of 4.
Correct answer: 1.
« 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 least possible value temperature T is the value of T = 0 if the pressure p or the volume V are equal to 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 most low 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 average speed 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 the atoms can only vibrate near the equilibrium positions at the nodes of the crystal lattice.
As the temperature approaches absolute zero, the energy of the thermal motion of molecules approaches zero, i.e., the translational thermal motion molecules.
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 temperature and pressure contain the same number molecules.