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 isThe 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.
Physical meaning: Gas constant i is numerically equal to the work of expansion of one mole of an ideal gas in an isobaric process with an increase in temperature by 1 K
In the GHS system, the Gas constant is equal to:
The specific gas constant is equal to:
In the formula we used:
Universal gas constant (Mendeleev's constant)
Boltzmann's constant
Avogadro's number
Avogadro's Law - Equal volumes of different gases at constant temperature and pressure contain the same number of molecules.
Two corollaries are derived from Avogadro's Law:
Corollary 1: One mole of any gas under the same conditions occupies the same volume
In particular, under normal conditions (T=0 °C (273K) and p=101.3 kPa), the volume of 1 mole of gas is 22.4 liters. This volume is called the molar volume of the gas Vm. This value can be recalculated to other temperatures and pressures using the Mendeleev-Clapeyron equation
1) Charles's Law:
2) Gay-Lussac's Law:
3) Bohl-Mariotte Law:
Corollary 2: The ratio of the masses of equal volumes of two gases is a constant value for these gases
This constant value is called the relative density of gases and is denoted D. Since the molar volumes of all gases are the same (1st consequence of Avogadro’s law), the ratio of the molar masses of any pair of gases is also equal to this constant:
In the Formula we used:
Relative gas density
Molar masses
Pressure
Molar volume
Universal gas constant
Absolute temperature
Boyle-Mariotte's law - At constant temperature and mass of an ideal gas, the product of its pressure and volume is constant.
This means that as the pressure on the gas increases, its volume decreases, and vice versa. For a constant amount of gas, the Boyle-Mariotte law can also be interpreted as follows: at a constant temperature, the product of pressure and volume is a constant value. The Boyle-Mariotte law is strictly true for an ideal gas and is a consequence of the Mendeleev-Clapeyron equation. For real gases, the Boyle-Mariotte law is satisfied approximately. Almost all gases behave as ideal gases at not too high pressures and not too low temperatures.
To make it easier to understand Boyle Marriott's Law Let's imagine that you are squeezing an inflated balloon. Since there is enough free space between the air molecules, you can easily, by applying some force and doing some work, compress the ball, reducing the volume of gas inside it. This is one of the main differences between gas and liquid. In a bead of liquid water, for example, the molecules are packed tightly together, as if the bead were filled with microscopic pellets. Therefore, unlike air, water does not lend itself to elastic compression.
There is also:
Charles's Law:
Gay Lussac's Law:
In the law we used:
Pressure in 1 vessel
Volume of 1 vessel
Pressure in vessel 2
Volume 2 vessels
Gay Lussac's law - at constant pressure, the volume of a constant mass of gas is proportional to the absolute temperature
The volume V of a given mass of gas at constant gas pressure is directly proportional to the change in temperature
Gay-Lussac's law is valid only for ideal gases; real gases obey it at temperatures and pressures far from critical values. It is a special case of the Clayperon equation.
There is also:
Mendeleev's Clapeyron equation:
Charles's Law:
Boyle Marriott's Law:
In the law we used:
Volume in 1 vessel
Temperature in 1 vessel
Volume in 1 vessel
Temperature in 1 vessel
Initial gas volume
Volume of gas at temperature T
Thermal expansion coefficient of gases
Difference between initial and final temperatures
Henry's law is a law according to which, at a constant temperature, the solubility of a gas in a given liquid is directly proportional to the pressure of this gas above the solution. The law is suitable only for ideal solutions and low pressures.
Henry's law describes the process of dissolving a gas in a liquid. We know what a liquid in which gas is dissolved is from the example of carbonated drinks - non-alcoholic, low-alcohol, and on major holidays - champagne. All of these drinks contain dissolved carbon dioxide (chemical formula CO2), a harmless gas used in the food industry due to its good solubility in water, and all these drinks foam after opening a bottle or can because the dissolved gas begins to be released from the liquid into atmosphere, since after opening a sealed container the pressure inside drops.
Actually, Henry's law states a fairly simple fact: the higher the gas pressure above the surface of the liquid, the more difficult it is for the gas dissolved in it to be released. And this is completely logical from the point of view of molecular kinetic theory, since a gas molecule, in order to break free from the surface of a liquid, needs to overcome the energy of collisions with gas molecules above the surface, and the higher the pressure and, as a consequence, the number of molecules in the boundary region, the higher it is more difficult for a dissolved molecule to overcome this barrier.
In the formula we used:
Gas concentration in solution in fractions of a mole
Henry's coefficient
Partial pressure of gas above solution
Kirchhoff's law of radiation - the ratio of emission and absorption abilities does not depend on the nature of the body, it is the same for all bodies.
By definition, an absolutely black body absorbs all radiation incident on it, that is, for it (Absorptivity of the body). Therefore the function coincides with the emissivity
In the formula we used:
Body emissivity
Body absorption capacity
Kirchhoff function
Stefan-Boltzmann Law - The energetic luminosity of a black body is proportional to the fourth power of absolute temperature.
From the formula it is clear that with increasing temperature, the luminosity of a body does not just increase - it increases to a much greater extent. Double the temperature and the luminosity increases 16 times!
Heated bodies emit energy in the form of electromagnetic waves of various lengths. When we say that a body is “red hot,” this means that its temperature is high enough for thermal radiation to occur in the visible, light part of the spectrum. At the atomic level, radiation results from the emission of photons by excited atoms.
To understand how this law works, imagine an atom emitting light in the depths of the Sun. Light is immediately absorbed by another atom, re-emitted by it - and thus transmitted along a chain from atom to atom, due to which the entire system is in a state energy balance. In an equilibrium state, light of a strictly defined frequency is absorbed by one atom in one place simultaneously with the emission of light of the same frequency by another atom in another place. As a result, the light intensity of each wavelength of the spectrum remains unchanged.
The temperature inside the Sun drops as it moves away from its center. Therefore, as you move towards the surface, the spectrum of light radiation appears to correspond to higher temperatures than the ambient temperature. As a result, upon re-radiation, according to Stefan-Boltzmann law, it will occur at lower energies and frequencies, but at the same time, due to the law of conservation of energy, a larger number of photons will be emitted. Thus, by the time it reaches the surface, the spectral distribution will correspond to the temperature of the surface of the Sun (about 5,800 K) and not the temperature at the center of the Sun (about 15,000,000 K).
Energy arriving at the surface of the Sun (or the surface of any hot object) leaves it in the form of radiation. The Stefan-Boltzmann law tells us exactly what is the energy emitted.
In the above formulation Stefan-Boltzmann law extends only to a completely black body, which absorbs all radiation falling on its surface. Real physical bodies absorb only part of the radiation energy, and the remaining part is reflected by them, however, the pattern according to which the specific radiation power from their surface is proportional to T in 4, as a rule, remains the same in this case, however, in this case the Boltzmann constant must be replaced by another coefficient that will reflect the properties of a real physical body. Such constants are usually determined experimentally.
In the formula we used:
Energy luminosity of the body
Stefan-Boltzmann constant
Absolute temperature
Charles's law - the pressure of a given mass of ideal gas at constant volume is directly proportional to the absolute temperature
To make it easier to understand Charles's law, imagine the air inside a balloon. At a constant temperature, the air in the balloon will expand or contract until the pressure produced by its molecules reaches 101,325 pascals and equals atmospheric pressure. In other words, until for every blow of an air molecule from the outside, directed into the ball, there will be a similar blow of an air molecule, directed from the inside of the ball outward.
If you lower the temperature of the air in the ball (for example, by placing it in a large refrigerator), the molecules inside the ball will begin to move more slowly, hitting the walls of the ball less energetically from the inside. The molecules of the outside air will then put more pressure on the ball, compressing it, as a result, the volume of gas inside the ball will decrease. This will happen until the increase in gas density compensates for the decreased temperature, and then equilibrium will be established again.
There is also:
Mendeleev's Clapeyron equation:
Gay Lussac's Law:
Boyle Marriott's Law:
In the law we used:
Pressure in 1 vessel
Temperature in 1 vessel
Pressure in vessel 2
Temperature in vessel 2
First law of thermodynamics - The change in internal energy ΔU of a non-isolated thermodynamic system is equal to the difference between the amount of heat Q transferred to the system and the work A of external forces
Instead of the work A performed by external forces on a thermodynamic system, it is often more convenient to consider the work A’ performed by the thermodynamic system on external bodies. Since these works are equal in absolute value, but opposite in sign:
Then after such a transformation first law of thermodynamics will look like:
First law of thermodynamics - In a non-isolated thermodynamic system, the change in internal energy is equal to the difference between the amount of heat Q received and the work A’ performed by this system
In simple terms first law of thermodynamics speaks of energy that cannot be created on its own and disappear into nowhere; it is transferred from one system to another and turns from one form to another (mechanical to thermal).
An important consequence first law of thermodynamics is that it is impossible to create a machine (engine) that is capable of performing useful work without consuming external energy. Such a hypothetical machine was called a perpetual motion machine of the first kind.
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 2 3 http://physics.nist.gov/cuu/Constants/Table/allascii.txt - physics.nist.gov/cuu/Constants/Table/allascii.txt Fundamental Physical Constants - Complete Listing
This abstract is based on an article from Russian Wikipedia. Synchronization completed 07/10/11 01:04:29
Similar abstracts:
For a constant related to the energy of blackbody radiation, see Stefan-Boltzmann Constant
|
Constant value k |
Dimension |
|
1,380 6504(24) 10 −23 |
|
|
8,617 343(15) 10 −5 |
|
|
1,3807 10 −16 |
|
|
See also Values in various units below. |
|
Boltzmann's constant (k or k B) is a physical constant that determines the relationship between the temperature of a substance and the energy of thermal motion of particles of this substance. 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
In the table, the last numbers in parentheses indicate the standard error of the constant value. In principle, Boltzmann's constant can be obtained from the definition of absolute temperature and other physical constants. However, accurately calculating Boltzmann's constant using first principles is too complex and infeasible with the current state of knowledge.
Boltzmann's constant can be determined experimentally using Planck's law of thermal radiation, which describes the energy distribution in the spectrum of equilibrium radiation at a certain temperature of the emitting body, as well as other methods.
There is a relationship between the universal gas constant and Avogadro's number, from which the value of Boltzmann's constant follows:
The dimension of Boltzmann's constant is the same as that of entropy.
|
Story
In 1877, Boltzmann was the first to connect entropy and probability, but a fairly accurate value of the constant k as a coupling coefficient in the formula for entropy appeared only in the works of M. Planck. When deriving the law of black body radiation, Planck in 1900–1901. for the Boltzmann constant, he found a value of 1.346 10 −23 J/K, almost 2.5% less than the currently accepted value.
Before 1900, the relations that are now written with the Boltzmann constant were written using the gas constant R, and instead of the average energy per molecule, the total energy of the substance was used. Laconic formula of the form S = k log W on the bust of Boltzmann became such thanks to Planck. In his Nobel lecture in 1920, Planck wrote:
This constant is often called Boltzmann's constant, although, as far as I know, Boltzmann himself never introduced it - a strange state of affairs, despite the fact that Boltzmann's statements did not talk about the exact measurement of this constant.
This situation can be explained by the ongoing scientific debate at that time to clarify the essence of the atomic structure of matter. In the second half of the 19th century, there was considerable disagreement as to whether atoms and molecules were real or just a convenient way of describing phenomena. There was also no consensus as to whether the "chemical molecules" distinguished by their atomic mass were the same molecules as in the kinetic theory. Further in Planck's Nobel lecture one can find the following:
“Nothing can better demonstrate the positive and accelerating rate of progress than the art of experiment during the last twenty years, when many methods have been discovered at once for measuring the mass of molecules with almost the same accuracy as measuring the mass of a planet.”
Ideal gas equation of state
For an ideal gas, the unified gas law relating pressure is valid P, volume V, amount of substance n in moles, gas constant R and absolute temperature T:
In this equality, you can make a substitution. Then the gas law will be expressed in terms of the Boltzmann constant and the number of molecules N in gas volume V:
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.
Gas thermodynamics relations
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 3 kT/ 2 . This agrees well with experimental data. 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.
Kinetic theory gives a formula for average pressure P ideal gas:
Considering that the average kinetic energy of rectilinear motion is equal to:
we find the equation of state of an ideal gas:
This relationship holds well for molecular gases; however, the dependence of the heat capacity changes, since the molecules can have additional internal degrees of freedom in relation to those degrees of freedom that are associated with the movement of molecules in space. For example, a diatomic gas already has approximately five degrees of freedom.
Boltzmann multiplier
In general, the system is in equilibrium with a thermal reservoir at a temperature T has a probability p occupy a state of energy E, which can be written using the corresponding exponential Boltzmann multiplier:
This expression involves the quantity kT with the dimension of energy.
Probability calculation is used not only for calculations in the kinetic theory of ideal gases, but also in other areas, for example in chemical kinetics in the Arrhenius equation.
Role in the statistical determination of entropy
Main article: Thermodynamic entropy
Entropy S of an isolated thermodynamic system in thermodynamic equilibrium is determined through the natural logarithm of the number of different microstates W, corresponding to a given macroscopic state (for example, a state with a given total energy E):
Proportionality factor k is Boltzmann's constant. This is an expression that defines the relationship between microscopic and macroscopic states (via W and entropy S accordingly), expresses the central idea of statistical mechanics and is the main discovery of Boltzmann.
Classical thermodynamics uses the Clausius expression for entropy:
Thus, the appearance of the Boltzmann constant k can be seen as a consequence of the connection between thermodynamic and statistical definitions of entropy.
Entropy can be expressed in units k, which gives the following:
In such units, entropy exactly corresponds to information entropy.
Characteristic energy kT equal to the amount of heat required to increase entropy S"for one nat.
Role in semiconductor physics: thermal stress
Unlike other substances, in semiconductors there is a strong dependence of electrical conductivity on temperature:
where the factor σ 0 depends rather weakly on temperature compared to the exponential, E A– conduction activation energy. The density of conduction electrons also depends exponentially on temperature. For the current through a semiconductor p-n junction, instead of the activation energy, consider the characteristic energy of a given p-n junction at temperature T as the characteristic energy of an electron in an electric field:
Where q- , A V T there is thermal stress depending on temperature.
This relationship is the basis for expressing the Boltzmann constant in units of eV∙K −1. At room temperature (≈ 300 K) the thermal voltage value is about 25.85 millivolts ≈ 26 mV.
In classical theory, a formula is often used, according to which the effective speed of charge carriers in a substance is equal to the product of the carrier mobility μ and the electric field strength. Another formula relates the carrier flux density to the diffusion coefficient D and with a carrier concentration gradient n :
According to the Einstein-Smoluchowski relation, the diffusion coefficient is related to mobility:
Boltzmann's constant k is also included in the Wiedemann-Franz law, according to which the ratio of the thermal conductivity coefficient to the electrical conductivity coefficient in metals is proportional to the temperature and the square of the ratio of the Boltzmann constant to the electric charge.
Applications in other areas
To delimit temperature regions in which the behavior of matter is described by quantum or classical methods, the Debye temperature is used: