The Boltzmann constant and its physical meaning. Boltzmann constant

Boltzmann's constant (k (\displaystyle k) or k B (\displaystyle k_(\rm (B)))) is a physical constant that determines the relationship between temperature and energy. Named after the Austrian physicist Ludwig Boltzmann, who made major contributions to statistical physics, in which this constant plays a key role. Its value in the International System of Units SI according to the change in the definitions of the basic SI units (2018) is exactly equal to

k = 1.380 649 × 10 − 23 (\displaystyle k=1(,)380\,649\times 10^(-23)) J / .

Relationship between temperature and energy

In a homogeneous ideal gas at absolute temperature T (\displaystyle T), the energy per translational degree of freedom is, as follows from the Maxwell distribution, kT / 2 (\displaystyle kT/2). At room temperature (300 ), this energy is 2 , 07 × 10 − 21 (\displaystyle 2(,)07\times 10^(-21)) J, or 0.013 eV. In a monatomic ideal gas, each atom has three degrees of freedom corresponding to three spatial axes, which means that each atom has energy in 3 2 k T (\displaystyle (\frac (3)(2))kT).

Knowing the thermal energy, one can calculate the root-mean-square atomic velocity, which is inversely proportional to the square root of the atomic mass. The root mean square velocity at room temperature varies from 1370 m/s for helium to 240 m/s for xenon. In the case of a molecular gas, the situation becomes more complicated, for example, a diatomic gas has 5 degrees of freedom - 3 translational and 2 rotational (at low temperatures, when vibrations of atoms in a molecule are not excited and additional degrees of freedom are not added).

Definition of entropy

The entropy of a thermodynamic system is defined as the natural logarithm of the number of different microstates Z (\displaystyle Z) corresponding to a given macroscopic state (for example, a state with a given total energy).

S = k log ⁡ Z . (\displaystyle S=k\ln Z.)

Proportionality factor k (\displaystyle k) and is the Boltzmann constant. This is an expression that defines the relationship between microscopic ( Z (\displaystyle Z)) and macroscopic states ( S (\displaystyle S)), expresses the central idea of ​​statistical mechanics.

Plan:

Introduction

• 1 Relationship between temperature and energy
• 2 Definition of entropy
Notes

Introduction

Boltzmann constant (k or k B ) is a physical constant that determines the relationship between temperature and energy. It is named after the Austrian physicist Ludwig Boltzmann, who made a great contribution to statistical physics, in which this constant plays a key role. Its experimental value in the SI system is

J/K .

The numbers in parentheses indicate the standard error in the last digits of the value. Boltzmann's constant can be derived from the definition of absolute temperature and other physical constants. However, the calculation of the Boltzmann constant using basic principles is too complicated and impossible with the current level of knowledge. In Planck's natural system of units, the natural unit of temperature is given in such a way that the Boltzmann constant is equal to one.

The universal gas constant is defined as the product of the Boltzmann constant and the Avogadro number, R = kN A. The gas constant is more convenient when the number of particles is given in moles.

1. Relationship between temperature and energy

In a homogeneous ideal gas at absolute temperature T, the energy per translational degree of freedom is, as follows from the Maxwell distribution kT/ 2 . At room temperature (300 K), this energy is J, or 0.013 eV. In a monatomic ideal gas, each atom has three degrees of freedom corresponding to three spatial axes, which means that each atom has energy in .

Knowing the thermal energy, one can calculate the root-mean-square atomic velocity, which is inversely proportional to the square root of the atomic mass. The rms velocity at room temperature varies from 1370 m/s for helium to 240 m/s for xenon. In the case of a molecular gas, the situation becomes more complicated, for example, a diatomic gas already has approximately five degrees of freedom.

2. Definition of entropy

The entropy of a thermodynamic system is defined as the natural logarithm of the number of different microstates Z corresponding to a given macroscopic state (for example, a state with a given total energy).

S = k ln Z.

Proportionality factor k and is the Boltzmann constant. This is an expression that defines the relationship between microscopic ( Z) and macroscopic states ( S), expresses the central idea of ​​statistical mechanics.

Notes

1. 1 2 3 http://physics.nist.gov/cuu/Constants/Table/allascii.txt - physics.nist.gov/cuu/Constants/Table/allascii.txt Fundamental Physical Constants - Complete Listing

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Boltzmann's constant ($k$ or $k\_(\backslash rm\; B)$) is a physical constant that determines the relationship between temperature and energy. Named after the Austrian physicist Ludwig Boltzmann, who made major contributions to statistical physics, in which this constant plays a key role. Its experimental value in the International System of Units (SI) is:

$k=1(,)380\backslash ,648\backslash ,52(79)\backslash times\; 10^(-23)$ J / .

The numbers in parentheses indicate the standard error in the last digits of the value. In Planck's natural system of units, the natural unit of temperature is given in such a way that the Boltzmann constant is equal to one.

Relationship between temperature and energy

In a homogeneous ideal gas at absolute temperature $T$, the energy per translational degree of freedom is, as follows from the Maxwell distribution, $kT/2$. At room temperature (300 ), this energy is $2(,)07\backslash times\; 10^(-21)$ J, or 0.013 eV. In a monatomic ideal gas, each atom has three degrees of freedom corresponding to three spatial axes, which means that each atom has energy in $\backslash frac\; 3\; 2\; kT$.

Knowing the thermal energy, one can calculate the root-mean-square atomic velocity, which is inversely proportional to the square root of the atomic mass. The root mean square velocity at room temperature varies from 1370 m/s for helium to 240 m/s for xenon. In the case of a molecular gas, the situation becomes more complicated, for example, a diatomic gas has five degrees of freedom (at low temperatures, when vibrations of atoms in a molecule are not excited).

Definition of entropy

The entropy of a thermodynamic system is defined as the natural logarithm of the number of different microstates $Z$ corresponding to a given macroscopic state (for example, a state with a given total energy).

$S=k\backslash lnZ.$

Proportionality factor $k$ and is the Boltzmann constant. This is an expression that defines the relationship between microscopic ( $Z$) and macroscopic states ( $S$), expresses the central idea of ​​statistical mechanics.

Assumed value fix

XXIV General Conference on Weights and Measures, held on October 17-21, 2011, adopted a resolution in which, in particular, it was proposed to make a future revision of the International System of Units in such a way as to fix the value of the Boltzmann constant, after which it will be considered certain exactly. As a result, it will run exact equality k\u003d 1.380 6X 10 −23 J / K. Such an alleged fixation is associated with the desire to redefine the unit of thermodynamic temperature, the kelvin, by relating its value to the value of the Boltzmann constant.

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An excerpt characterizing the Boltzmann constant

“But what does that mean? Natasha said thoughtfully.
“Ah, I don’t know how extraordinary all this is! Sonya said, clutching her head.
A few minutes later, Prince Andrei called, and Natasha went in to him; and Sonya, experiencing a feeling of excitement and tenderness rarely experienced by her, remained at the window, pondering the whole unusualness of what had happened.
On this day there was an opportunity to send letters to the army, and the countess wrote a letter to her son.
“Sonya,” said the countess, looking up from her letter as her niece passed her. - Sonya, will you write to Nikolenka? said the countess in a low, trembling voice, and in the look of her tired eyes, peering through glasses, Sonya read everything that the countess meant by these words. This look expressed both prayer, and fear of refusal, and shame at what had to be asked, and readiness for irreconcilable hatred in case of refusal.
Sonya went up to the countess and, kneeling down, kissed her hand.
“I will write, maman,” she said.
Sonya was softened, agitated and touched by everything that happened that day, especially by the mysterious performance of divination that she just saw. Now that she knew that on the occasion of the resumption of relations between Natasha and Prince Andrei, Nikolai could not marry Princess Marya, she gladly felt the return of that mood of self-sacrifice in which she loved and used to live. And with tears in her eyes and with joy in the consciousness of committing a generous deed, she, interrupted several times by tears that clouded her velvety black eyes, wrote that touching letter, the receipt of which so struck Nikolai.

In the guardhouse, where Pierre was taken, the officer and soldiers who took him treated him with hostility, but at the same time respectfully. There was also a sense of doubt in their attitude towards him about who he was (isn't he a very important person), and hostility due to their still fresh personal struggle with him.
But when, on the morning of the next day, the shift came, Pierre felt that for the new guard - for officers and soldiers - he no longer had the meaning that he had for those who took him. And indeed, in this big, fat man in a peasant's caftan, the guards of the other day no longer saw that living person who fought so desperately with the marauder and the escort soldiers and uttered a solemn phrase about saving the child, but they saw only the seventeenth of those held for some reason, according to the order of the higher authorities, taken by the Russians. If there was anything special in Pierre, it was only his timid, concentrated, thoughtful look and the French language, in which, surprisingly for the French, he spoke well. Despite the fact that on the same day Pierre was connected with other suspects taken, since the officer needed a separate room that he occupied.
All the Russians kept with Pierre were people of the lowest rank. And all of them, recognizing the gentleman in Pierre, shunned him, especially since he spoke French. Pierre sadly heard ridicule over himself.
The next day, in the evening, Pierre learned that all these detainees (and, probably, including himself) were to be tried for arson. On the third day, Pierre was taken with others to a house where a French general with a white mustache, two colonels and other Frenchmen with scarves on their hands were sitting. Pierre, along with others, was asked questions about who he is with that allegedly exceeding human weaknesses, accuracy and determination with which defendants are usually treated. where was he? for what purpose? etc.
These questions, leaving aside the essence of life's work and excluding the possibility of disclosing this essence, like all questions asked at the courts, aimed only at substituting the groove along which the judges wanted the defendant's answers to flow and lead him to the desired goal, that is, to the accusation. As soon as he began to say something that did not satisfy the purpose of the accusation, they accepted the groove, and the water could flow wherever it wanted. In addition, Pierre experienced the same thing that the defendant experiences in all courts: bewilderment, why did they ask him all these questions. He felt that it was only out of condescension or, as it were, courtesy that this trick of the substituted groove was used. He knew that he was in the power of these people, that only power had brought him here, that only power gave them the right to demand answers to questions, that the only purpose of this meeting was to accuse him. And therefore, since there was power and there was a desire to accuse, there was no need for the trick of questions and trial. It was obvious that all answers had to lead to guilt. When asked what he was doing when they took him, Pierre answered with some tragedy that he was carrying a child to his parents, qu "il avait sauve des flammes [whom he saved from the flame]. - Why did he fight with a marauder? Pierre answered, that he defended a woman, that the protection of an offended woman is the duty of every man, that... He was stopped: it did not go to the point. Why was he in the yard of the house on fire, where witnesses saw him? He answered that he was going to see what was being done in Moscow. They stopped him again: they did not ask him where he was going, but why he was near the fire? Who is he? They repeated the first question to which he said that he did not want to answer. Again he answered that he could not say this .

Boltzmann Ludwig (1844-1906)- the great Austrian physicist, one of the founders of the molecular kinetic theory. In the works of Boltzmann, the molecular-kinetic theory first appeared as a logically coherent, consistent physical theory. Boltzmann gave a statistical interpretation of the second law of thermodynamics. He has done a lot for the development and popularization of Maxwell's theory of the electromagnetic field. A fighter by nature, Boltzmann passionately defended the need for a molecular interpretation of thermal phenomena and took upon himself the brunt of the fight against scientists who denied the existence of molecules.

Equation (4.5.3) includes the ratio of the universal gas constant R to the Avogadro constant N A . This ratio is the same for all substances. It is called the Boltzmann constant, in honor of L. Boltzmann, one of the founders of the molecular kinetic theory.

Boltzmann's constant is:

Equation (4.5.3), taking into account the Boltzmann constant, is written as follows:

The physical meaning of the Boltzmann constant

Historically, temperature was first introduced as a thermodynamic quantity, and a unit of measurement was established for it - a degree (see § 3.2). After establishing the relationship between temperature and the average kinetic energy of molecules, it became obvious that temperature can be defined as the average kinetic energy of molecules and expressed in joules or ergs, i.e., instead of the quantity T enter value T* so that

The temperature thus determined is related to the temperature expressed in degrees as follows:

Therefore, the Boltzmann constant can be considered as a quantity that relates the temperature, expressed in energy units, with the temperature, expressed in degrees.

The dependence of gas pressure on the concentration of its molecules and temperature

Expressing E from relation (4.5.5) and substituting into formula (4.4.10), we obtain an expression showing the dependence of gas pressure on the concentration of molecules and temperature:

From formula (4.5.6) it follows that at the same pressures and temperatures, the concentration of molecules in all gases is the same.

This implies Avogadro's law: equal volumes of gases at the same temperatures and pressures contain the same number of molecules.

The average kinetic energy of the translational motion of molecules is directly proportional to the absolute temperature. Proportionality factor- Boltzmann's constantk \u003d 10 -23 J / K - need to remember.

§ 4.6. Maxwell distribution

In a large number of cases, knowing the average values ​​of physical quantities alone is not enough. For example, knowing the average height of people does not allow planning the production of clothes of various sizes. You need to know the approximate number of people whose height lies in a certain interval. Similarly, it is important to know the numbers of molecules that have velocities other than the average. Maxwell was the first to find how these numbers can be determined.

Probability of a random event

In §4.1 we have already mentioned that J. Maxwell introduced the concept of probability to describe the behavior of a large set of molecules.

As repeatedly emphasized, in principle it is impossible to follow the change in the speed (or momentum) of one molecule over a long time interval. It is also impossible to accurately determine the speed of all gas molecules at a given time. From the macroscopic conditions in which the gas is located (a certain volume and temperature), certain values ​​of the velocities of the molecules do not necessarily follow. The speed of a molecule can be considered as a random variable, which under given macroscopic conditions can take on different values, just as when throwing a dice, any number of points from 1 to 6 (the number of faces of the die is six) can fall out. It is impossible to predict what number of points will fall out in a given throw of the die. But the probability of rolling, say, five points is defensible.

What is the probability of a random event occurring? Let a very large number be produced N tests (N is the number of rolls of the die). At the same time, in N" cases, there was a favorable outcome of the tests (i.e., the loss of five). Then the probability of this event is equal to the ratio of the number of cases with a favorable outcome to the total number of trials, provided that this number is arbitrarily large:

For a symmetric die, the probability of any chosen number of points from 1 to 6 is .

We see that against the background of many random events, a certain quantitative pattern is revealed, a number appears. This number - the probability - allows you to calculate averages. So, if you make 300 throws of a dice, then the average number of throws of a five, as follows from formula (4.6.1), will be equal to: 300 = 50, and it is completely indifferent to throw the same dice 300 times or simultaneously 300 identical dice .

Undoubtedly, the behavior of gas molecules in a vessel is much more complicated than the movement of a thrown dice. But even here one can hope to discover certain quantitative regularities that make it possible to calculate statistical averages, if only the problem is posed in the same way as in game theory, and not as in classical mechanics. We must abandon the unsolvable problem of determining the exact value of the speed of the molecule at a given moment and try to find the probability that the speed has a certain value.

The Boltzmann constant, which is a coefficient equal to k = 1.38 10 - 23 J K, is part of a significant number of formulas in physics. It got its name from the Austrian physicist, one of the founders of the molecular kinetic theory. We formulate the definition of the Boltzmann constant:

Definition 1

Boltzmann constant called a physical constant, which determines the relationship between energy and temperature.

It should not be confused with the Stefan-Boltzmann constant associated with the radiation of the energy of an absolutely rigid body.

There are various methods for calculating this coefficient. In this article, we will look at two of them.

Finding the Boltzmann constant through the ideal gas equation

This constant can be found using an equation describing the state of an ideal gas. It can be experimentally determined that heating any gas from T 0 \u003d 273 K to T 1 \u003d 373 K leads to a change in its pressure from p 0 \u003d 1.013 10 5 Pa to p 0 \u003d 1.38 10 5 Pa . This is a fairly simple experiment that can be done even with just air. To measure temperature, you need to use a thermometer, and pressure - a manometer. It is important to remember that the number of molecules in a mole of any gas is approximately equal to 6 10 23, and the volume at a pressure of 1 atom is V = 22.4 l. Taking into account all the named parameters, we can proceed to the calculation of the Boltzmann constant k:

To do this, we write the equation twice, substituting the state parameters into it.

Knowing the result, we can find the value of the parameter k:

Finding the Boltzmann constant through the Brownian motion formula

For the second calculation method, we also need to conduct an experiment. For him, you need to take a small mirror and hang it in the air with an elastic thread. Let us assume that the mirror-air system is in a stable state (static equilibrium). Air molecules hit the mirror, which essentially behaves like a Brownian particle. However, taking into account its suspended state, we can observe rotational oscillations around a certain axis coinciding with the suspension (vertically directed thread). Now let's direct a beam of light to the surface of the mirror. Even with slight movements and turns of the mirror, the beam reflected in it will noticeably shift. This gives us the ability to measure the rotational vibrations of an object.

Denoting the modulus of torsion as L, the moment of inertia of the mirror with respect to the axis of rotation as J, and the angle of rotation of the mirror as φ, we can write the oscillation equation of the following form:

The minus in the equation is related to the direction of the moment of elastic forces, which tends to return the mirror to its equilibrium position. Now let's multiply both parts by φ, integrate the result and get:

The following equation is the law of conservation of energy that will be true for these oscillations (that is, potential energy will be converted into kinetic energy and vice versa). We can consider these oscillations to be harmonic, therefore:

When deriving one of the formulas earlier, we used the law of uniform distribution of energy over degrees of freedom. So we can write it like this:

As we have said, the angle of rotation can be measured. So, if the temperature is approximately 290 K, and the torsion modulus L ≈ 10 - 15 N m; φ ≈ 4 10 - 6, then we can calculate the value of the coefficient we need as follows:

Therefore, knowing the basics of Brownian motion, we can find the Boltzmann constant by measuring macro parameters.

The value of the Boltzmann constant

The value of the coefficient under study lies in the fact that it can be used to connect the parameters of the microcosm with those parameters that describe the macrocosm, for example, thermodynamic temperature with the energy of the translational motion of molecules:

This coefficient is included in the equations of the average energy of a molecule, the state of an ideal gas, the kinetic theory of gas, the Boltzmann-Maxwell distribution, and many others. Also, the Boltzmann constant is needed in order to determine the entropy. It plays an important role in the study of semiconductors, for example, in the equation describing the dependence of electrical conductivity on temperature.

Example 1

Condition: calculate the average energy of a gas molecule consisting of N-atomic molecules at a temperature T, knowing that all degrees of freedom are excited in the molecules - rotational, translational, vibrational. All molecules are considered bulk.

Decision

The energy is evenly distributed over the degrees of freedom for each of its degrees, which means that these degrees will have the same kinetic energy. It will be equal to ε i = 1 2 k T . Then to calculate the average energy we can use the formula:

ε = i 2 k T , where i = m p o s t + m υ r + 2 m k o l is the sum of translational rotational degrees of freedom. The letter k stands for Boltzmann's constant.

Let's move on to determining the number of degrees of freedom of the molecule:

m p o s t = 3 , m υ r = 3 , hence m k o l = 3 N - 6 .

i \u003d 6 + 6 N - 12 \u003d 6 N - 6; ε = 6 N - 6 2 k T = 3 N - 3 k T .

Answer: under these conditions, the average energy of the molecule will be equal to ε = 3 N - 3 k T .

Example 2

Condition: is a mixture of two ideal gases whose density under normal conditions is p. Determine what will be the concentration of one gas in the mixture, provided that we know the molar masses of both gases μ 1, μ 2.

Decision

First, calculate the total mass of the mixture.

m = ρ V = N 1 m 01 + N 2 m 02 = n 1 V m 01 + n 2 V m 02 → ρ = n 1 m 01 + n 2 m 02 .

The parameter m 01 denotes the mass of a molecule of one gas, m 02 is the mass of a molecule of another, n 2 is the concentration of molecules of one gas, n 2 is the concentration of the second. The density of the mixture is equal to ρ.

Now, from this equation, we express the concentration of the first gas:

n 1 \u003d ρ - n 2 m 02 m 01; n 2 = n - n 1 → n 1 = ρ - (n - n 1) m 02 m 01 → n 1 = ρ - n m 02 + n 1 m 02 m 01 → n 1 m 01 - n 1 m 02 = ρ - n m 02 → n 1 (m 01 - m 02) = ρ - n m 02 .

p = n k T → n = p k T .

Substitute the resulting equal value:

n 1 (m 01 - m 02) = ρ - p k T m 02 → n 1 = ρ - p k T m 02 (m 01 - m 02) .

Since the molar masses of gases are known to us, we can find the masses of the molecules of the first and second gases:

m 01 = μ 1 N A , m 02 = μ 2 N A .

We also know that the mixture of gases is under normal conditions, i.e. the pressure is 1 atm, and the temperature is 290 K. So, we can consider the problem solved.

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