Ludwig Boltzmann: Personal achievements. Boltzmann constant

Plan:

Introduction

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

Introduction

Boltzmann's constant (k or k B) is a physical constant that defines 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 SI system is

J/K .

The numbers in parentheses indicate the standard error in the last digits of the quantity value. Boltzmann's constant can be obtained from the definition of absolute temperature and other physical constants. However, calculating Boltzmann's constant using first principles is too complex and infeasible with the current state of knowledge. In the natural system of Planck units, the natural unit of temperature is given so that Boltzmann's constant is equal to unity.

The universal gas constant is defined as the product of Boltzmann's constant and Avogadro's 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 each translational degree of freedom is equal, 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 an energy of .

Knowing the thermal energy, we can calculate the root mean square velocity of the atoms, 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 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 Boltzmann's 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, 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 molecular kinetic theory. Let us formulate the definition of Boltzmann's constant:

Definition 1

Boltzmann constant is a physical constant that is used to determine the relationship between energy and temperature.

It should not be confused with the Stefan-Boltzmann constant, which is associated with the radiation of energy from a completely solid body.

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

Finding Boltzmann's constant through the ideal gas equation

This constant can be found using the equation describing the state of an ideal gas. It can be determined experimentally that heating any gas from T 0 = 273 K to T 1 = 373 K leads to a change in its pressure from p 0 = 1.013 10 5 P a to p 0 = 1.38 10 5 P a . This is a fairly simple experiment that can be done even just with 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 atm is equal to V = 22.4 liters. Taking into account all these parameters, we can proceed to calculating 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 parameter k:

Finding Boltzmann's constant through the Brownian motion formula

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

Denoting the torsion modulus as L, the moment of inertia of the mirror relative 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 associated with the direction of the moment of elastic forces, which tends to return the mirror to an equilibrium position. Now let's multiply both sides by φ, integrate the result and get:

The following equation is the law of conservation of energy, which will be satisfied for these vibrations (that is, potential energy will transform into kinetic energy and vice versa). We can consider these vibrations 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 already 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 Boltzmann's constant by measuring macroparameters.

Boltzmann constant value

The significance of the coefficient under study is that it can be used to relate the parameters of the microworld with those parameters that describe the macroworld, for example, thermodynamic temperature with the energy of 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 gases, the Boltzmann-Maxwell distribution and many others. Boltzmann's constant is also needed to determine 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 temperature T, knowing that all degrees of freedom are excited in the molecules - rotational, translational, vibrational. All molecules are considered to be volumetric.

Solution

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 represents the sum of translational rotational degrees of freedom. The letter k denotes 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, which means m k o l = 3 N - 6.

i = 6 + 6 N - 12 = 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 equal to p. Determine what the concentration of one gas in the mixture will be, provided that we know the molar masses of both gases μ 1, μ 2.

Solution

First, let's 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 – the mass of a molecule of another, n 2 – the concentration of molecules of one gas, n 2 – the concentration of the second. The density of the mixture is ρ.

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

n 1 = 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 .

Let's 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 we know the molar masses of gases, we can find the masses of the molecules of the first and second gas:

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 a t m, and the temperature is 290 K. This means that we can consider the problem solved.

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According to the Stefan–Boltzmann law, the density of integral hemispherical radiation E 0 depends only on temperature and varies proportionally to the fourth power of absolute temperature T:

The Stefan–Boltzmann constant σ 0 is a physical constant included in the law that determines the volumetric density of the equilibrium thermal radiation of an absolutely black body:

Historically, the Stefan-Boltzmann law was formulated before Planck's radiation law, from which it follows as a consequence. Planck's law establishes the dependence of the spectral flux density of radiation E 0 on wavelength λ and temperature T:

where λ – wavelength, m; With=2.998 10 8 m/s – speed of light in vacuum; T– body temperature, K;
h= 6.625 ×10 -34 J×s – Planck’s constant.

Physical constant k, equal to the ratio of the universal gas constant R=8314J/(kg×K) to Avogadro’s number N.A.=6.022× 10 26 1/(kg×mol):

Number of different system configurations from N particles for a given set of numbers n i(number of particles in i-the state to which the energy e i corresponds) is proportional to the value:

Magnitude W there is a number of ways of distribution N particles by energy levels. If relation (6) is true, then it is considered that the original system obeys Boltzmann statistics. Set of numbers n i, at which the number W maximum, occurs most frequently and corresponds to the most probable distribution.

Physical kinetics– microscopic theory of processes in statistically nonequilibrium systems.

The description of a large number of particles can be successfully carried out using probabilistic methods. For a monatomic gas, the state of a set of molecules is determined by their coordinates and the values ​​of velocity projections on the corresponding coordinate axes. Mathematically, this is described by the distribution function, which characterizes the probability of a particle being in a given state:

is the expected number of molecules in a volume d d whose coordinates are in the range from to +d, and whose velocities are in the range from to +d.

If the time-averaged potential energy of interaction of molecules can be neglected in comparison with their kinetic energy, then the gas is called ideal. An ideal gas is called a Boltzmann gas if the ratio of the path length of the molecules in this gas to the characteristic size of the flow L of course, i.e.

because the path length is inversely proportional nd 2(n is the numerical density 1/m 3, d is the diameter of the molecule, m).

Size

called H-Boltzmann function for a unit volume, which is associated with the probability of detecting a system of gas molecules in a given state. Each state corresponds to certain numbers of filling six-dimensional space-velocity cells into which the phase space of the molecules under consideration can be divided. Let's denote W the probability that there will be N 1 molecules in the first cell of the space under consideration, N 2 in the second, etc.

Up to a constant that determines the origin of the probability, the following relation is valid:

,

Where – H-function of a region of space A occupied by gas. From (9) it is clear that W And H interconnected, i.e. a change in the probability of a state leads to a corresponding evolution of the H function.

Boltzmann's principle establishes the connection between entropy S physical system and thermodynamic probability W her states:

(published according to the publication: Kogan M.N. Dynamics of a rarefied gas. - M.: Nauka, 1967.)

General view of the CUBE:

where is the mass force due to the presence of various fields (gravitational, electric, magnetic) acting on the molecule; J– collision integral. It is this term of the Boltzmann equation that takes into account the collisions of molecules with each other and the corresponding changes in the velocities of interacting particles. The collision integral is a five-dimensional integral and has the following structure:

Equation (12) with integral (13) was obtained for collisions of molecules in which no tangential forces arise, i.e. colliding particles are considered to be perfectly smooth.

During the interaction, the internal energy of the molecules does not change, i.e. these molecules are assumed to be perfectly elastic. We consider two groups of molecules that have velocities and before colliding with each other (collision) (Fig. 1), and after the collision, respectively, velocities and . The difference in speed is called relative speed, i.e. . It is clear that for a smooth elastic collision . Distribution functions f 1 ", f", f 1 , f describe the molecules of the corresponding groups after and before collisions, i.e. ; ; ; .

Rice. 1. Collision of two molecules.

(13) includes two parameters characterizing the location of colliding molecules relative to each other: b and ε; b– aiming distance, i.e. the smallest distance that molecules would approach in the absence of interaction (Fig. 2); ε is called the collision angular parameter (Fig. 3). Integration over b from 0 to ¥ and from 0 to 2p (two external integrals in (12)) covers the entire plane of force interaction perpendicular to the vector

Rice. 2. The trajectory of the molecules.

Rice. 3. Consideration of the interaction of molecules in a cylindrical coordinate system: z, b, ε

The Boltzmann kinetic equation is derived under the following assumptions and assumptions.

1. It is believed that mainly collisions of two molecules occur, i.e. the role of collisions of three or more molecules simultaneously is insignificant. This assumption allows us to use a single-particle distribution function for analysis, which above is simply called the distribution function. Taking into account the collision of three molecules leads to the need to use a two-particle distribution function in the study. Accordingly, the analysis becomes significantly more complicated.

2. Assumption of molecular chaos. It is expressed in the fact that the probabilities of detecting particle 1 at the phase point and particle 2 at the phase point are independent of each other.

3. Collisions of molecules with any impact distance are equally probable, i.e. the distribution function does not change at the interaction diameter. It should be noted that the analyzed element must be small so that f within this element does not change, but at the same time so that the relative fluctuation ~ is not large. The interaction potentials used in calculating the collision integral are spherically symmetric, i.e. .

Maxwell-Boltzmann distribution

The equilibrium state of the gas is described by the absolute Maxwellian distribution, which is an exact solution of the Boltzmann kinetic equation:

where m is the mass of the molecule, kg.

The general local Maxwellian distribution, otherwise called the Maxwell-Boltzmann distribution:

in the case when the gas moves as a whole with speed and the variables n, T depend on the coordinate
and time t.

In the Earth's gravitational field, the exact solution of the Boltzmann equation shows:

Where n 0 = density at the Earth's surface, 1/m3; g– gravity acceleration, m/s 2 ; h– height, m. Formula (16) is an exact solution of the Boltzmann kinetic equation either in unlimited space or in the presence of boundaries that do not violate this distribution, while the temperature must also remain constant.

This page was designed by Puzina Yu.Yu. with the support of the Russian Foundation for Basic Research - project No. 08-08-00638.

Born in 1844 in Vienna. Boltzmann is a pioneer and pioneer in science. His works and research were often incomprehensible and rejected by society. However, with the further development of physics, his works were recognized and subsequently published.

The scientist's scientific interests covered such fundamental areas as physics and mathematics. Since 1867, he worked as a teacher in a number of higher educational institutions. In his research, he established that this is due to the chaotic impacts of molecules on the walls of the vessel in which they are located, while the temperature directly depends on the speed of movement of particles (molecules), in other words, on their Therefore, the higher the speed these particles move, the higher the temperature. Boltzmann's constant is named after the famous Austrian scientist. It was he who made an invaluable contribution to the development of static physics.

Physical meaning of this constant quantity

Boltzmann's constant defines the relationship between temperature and energy. In static mechanics it plays a major key role. Boltzmann's constant is equal to k=1.3806505(24)*10 -23 J/K. The numbers in parentheses indicate the permissible error of the value relative to the last digits. It is worth noting that Boltzmann's constant can also be derived from other physical constants. However, these calculations are quite complex and difficult to perform. They require deep knowledge not only in the field of physics, but also

(k or k B) is a physical constant that defines the relationship between temperature and energy. Named after the Austrian physicist Ludwig Boltzmann, who made major contributions to statistical physics, in which this became a key position. Its experimental value in the SI system is

The numbers in parentheses indicate the standard error in the last digits of the quantity value. In principle, Boltzmann's constant can be obtained from the definition of absolute temperature and other physical constants (to do this, you need to be able to calculate the temperature of the triple point of water from first principles). But determining the Boltzmann constant using first principles is too complex and unrealistic with the current development of knowledge in this field.
Boltzmann's constant is a redundant physical constant if you measure temperature in units of energy, which is very often done in physics. It is, in fact, a connection between a well-defined quantity - energy and degree, the meaning of which has developed historically.
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, states with a given total energy).

Proportionality factor k and is Boltzmann's constant. This expression, which defines the relationship between microscopic (Z) and macroscopic (S) characteristics, expresses the main (central) idea of ​​statistical mechanics.