| Values of k | Units |
|---|---|
| 1. 380 6504(24)×10−23 | J·K-1 |
| 8. The joule (written in lower case ˈdʒuːl or /ˈdʒaʊl/ (symbol J) is the SI unit of Energy measuring heat, Electricity The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic 617 343(15)×10−5 | eV·K-1 |
| 1. 3807×10−16 | erg·K-1 |
| For details, see Value in different units below. An erg is the unit of Energy and Mechanical work in the centimetre-gram-second (CGS system of units symbol "erg" Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics | |
The Boltzmann constant (k or kB) is the physical constant relating energy at the particle level with temperature observed at the bulk level. A physical Constant is a Physical quantity that is generally believed to be both universal in nature and constant in time In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature It is the gas constant divided by the Avogadro constant:
It has the same units as entropy. Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working The Avogadro constant (symbols L, N A also called Avogadro's number, is the number of "elementary entities" (usually Atoms In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy It is named after the Austrian physicist Ludwig Boltzmann. Austria (Österreich ( officially the Republic of Austria (Republik Österreich Ludwig Eduard Boltzmann ( February 20, 1844 &ndash September 5, 1906) was an Austrian Physicist famous for his founding
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Bridge from macroscopic to microscopic physics
Boltzmann's constant k is a bridge between macroscopic and microscopic physics. Macroscopic is commonly used to describe physical objects that are measurable and observable by the Naked eye. Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure p and volume V is proportional to the product of amount of substance n (in number of moles) and absolute temperature T. The ideal gas law is the Equation of state of a hypothetical Ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834 These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically The amount of substance, n, of a sample or system is a Physical quantity which is proportional to the number of elementary entities present The mole (symbol mol) is a unit of Amount of substance: it is an SI base unit, and almost the only unit to be used to measure this Thermodynamic temperature is the absolute measure of Temperature and is one of the principal parameters of Thermodynamics.
where
is called the gas constant [8. Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working 314 472 m3·Pa·K−1·mol−1],
Introducing Boltzmann's constant transforms this into an equation about the microscopic properties of molecules,
where N is the number of molecules of gas, and k is Boltzmann's constant.
Role in the equipartition of energy
Given a thermodynamic system at an absolute temperature T, the thermal energy carried by each microscopic "degree of freedom" in the system is on the order of magnitude of kT/2 (i. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " Absolute zero is the point at which molecules do not move (relative to the rest of the body more than they are required to by a quantum mechanical effect called Zero-point An order of magnitude is the class of scale or magnitude of any amount where each class contains values of a fixed ratio to the class preceding it e. , about 2. 07×10−21 J, or 0. 013 eV at room temperature).
Application to simple gas thermodynamics
In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion Monatomic ideal gases possess 3 degrees of freedom per atom, corresponding to the three spatial directions, which means a thermal energy of 1. 5kT per atom. As indicated in the article on heat capacity, this corresponds very well with experimental data. Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the Temperature of a unit quantity The thermal energy can be used to calculate the root mean square speed of the atoms, which is inversely proportional to the square root of the atomic mass. In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the The atomic mass (ma is the Mass of an atom most often expressed in unified atomic mass units The atomic mass may be considered to be the total mass The root mean square speeds found at room temperature accurately reflect this, ranging from 1370 m/s for helium, down to 240 m/s for xenon. Helium ( He) is a colorless odorless tasteless non-toxic Inert Monatomic Chemical Xenon (ˈzɛnɒn or) is a Chemical element represented by the symbol Xe.
Kinetic theory gives the average pressure p for an ideal gas as
Substituting that the average translational kinetic energy is
gives
so the ideal gas equation is regained. Kinetic theory (or kinetic theory of gases) attempts to explain Macroscopic properties of Gases such as pressure temperature or volume by considering
The ideal gas equation is also followed quite well for molecular gases; but the form for the heat capacity is more complicated, because the molecules possess new internal degrees of freedom, as well as the three degrees of freedom for movement of the molecule as a whole. Diatomic gases, for example, possess in total approximately 5 degrees of freedom per molecule. (Furthermore, these additional degrees of freedom may be further complicated by quantum mechanics -- see the article on heat capacity for details. Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the Temperature of a unit quantity )
Role in Boltzmann factors
More generally, systems in equilibrium with a reservoir of heat at temperature T have probabilities of occupying states with energy E weighted by the corresponding Boltzmann factor:
Again, it is the energy-like quantity kT which takes central importance. In Physics, the Boltzmann factor is a weighting factor that determines the relative probability of a state i in a multi-state system in Thermodynamic equilibrium
Consequences of this include (in addition to the results for ideal gases above), for example the Arrhenius equation of simple chemical kinetics. The Arrhenius equation is a simple but remarkably accurate formula for the temperature dependence of the Rate constant, and therefore rate of a chemical reaction
Role in definition of entropy
In statistical mechanics, the entropy S of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of Ω, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy E):
This equation, which relates the microscopic details of the system (via Ω) to its macroscopic state (via the entropy S), is the central idea of statistical mechanics. In Statistical thermodynamics, Boltzmann's equation is a probability equation relating the Entropy S of an ideal gas to the quantity W, which In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy In Thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium Mechanical equilibrium, and The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational Such is its importance that it is inscribed on Boltzmann's tombstone. Ludwig Eduard Boltzmann ( February 20, 1844 &ndash September 5, 1906) was an Austrian Physicist famous for his founding
The constant of proportionality k appears in order to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius:
One could choose instead a rescaled entropy in microscopic terms such that
This is a rather more natural form; and this rescaled entropy exactly corresponds to Shannon's subsequent information entropy, and could thereby have avoided much unnecessary subsequent confusion between the two.
The characteristic energy kT is thus the heat required to increase the rescaled entropy by one nat. A nat (sometimes also nit or even nepit) is a Logarithmic unit of Information or entropy, based on Natural logarithms and
Role in semiconductor physics: the thermal voltage
In semiconductors, the relationship between the flow of electrical current and the electrostatic potential across a p-n junction depends on a characteristic voltage called the thermal voltage, denoted VT. A semiconductor' is a Solid material that has Electrical conductivity in between a conductor and an insulator; it can vary over that Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field A p-n junction is a junction formed by combining P-type and N-type Semiconductors together in very close contact The thermal voltage depends on absolute temperature T (in kelvins) as
where q is the magnitude of the electrical charge (in coulombs) on the electron (see elementary charge) with a value 1. The elementary charge, usually denoted e, is the Electric charge carried by a single Proton, or equivalently the negative of the electric charge carried 602 176 487 ×10−19 C . Using the unit of electronvolt, the Boltzmann constant relating temperature to energy can be expressed as 8. 617 343(15)×10−5 eV/K, making it easy to calculate that at room temperature (T ≈ 300 K), the value of the thermal voltage is approximately 25. 85 millivolts ≈ 26 mV (Google calculator). See also semiconductor diodes. Dioden2jpg|thumb|right|150px|Figure 2 Various semiconductor diodes
Boltzmann's constant in Planck units
Planck's system of natural units is one system constructed such that the Boltzmann constant is 1. In Physics, natural units are Physical units of Measurement defined in terms of universal Physical constants, such that some chosen physical This gives
as the average kinetic energy of a gas molecule per degree of freedom; and makes the definition of thermodynamic entropy coincide with that of information entropy:
The value chosen for the Planck unit of temperature is that corresponding to the energy of the Planck mass—a staggering 1.41679×1032 K. The Planck temperature, named after German Physicist Max Planck, is the unit of Temperature, denoted by TP in the system of The Planck mass is the unit of Mass, denoted by m P in the system of Natural units known as Planck units. Detailed list of temperatures from 100 K to 1000 K Most ordinary human activity takes place at temperatures of this order of magnitude
Historical note
Although Boltzmann first linked entropy and probability in 1877, it seems the relation was never expressed with a specific constant until Max Planck first introduced k , and gave an accurate value for it, in his derivation of the law of black body radiation in December 1900. Year 1877 ( MDCCCLXXVII) was a Common year starting on Monday (link will display the full calendar of the Gregorian calendar (or a Common For a general introduction see Black body. In Physics, Planck's law describes the spectral radiance of Electromagnetic radiation The iconic terse form of the equation S = k log W on Boltzmann's tombstone is in fact due to Planck, not Boltzmann.
As Planck wrote in his 1918 Nobel Prize lecture,
- "This constant is often referred to as Boltzmann's constant, although, to my knowledge, Boltzmann himself never introduced it — a peculiar state of affairs, which can be explained by the fact that Boltzmann, as appears from his occasional utterances, never gave thought to the possibility of carrying out an exact measurement of the constant. The Nobel Prize (Nobelpriset (Nobelprisen is a Swedish prize established in the 1895 will of Swedish chemist Alfred Nobel; it was first awarded in Peace, Literature Nothing can better illustrate the positive and hectic pace of progress which the art of experimenters has made over the past twenty years, than the fact that since that time, not only one, but a great number of methods have been discovered for measuring the mass of a molecule with practically the same accuracy as that attained for a planet. " [1]
Before 1900, equations involving Boltzmann factors were not written using the energies per molecule and Boltzmann's constant, but rather using the gas constant R, and macroscopic energies for macroscopic quantities of the substance; as for convenience is still generally the case in chemistry to this day. Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working
Value in different units
| Values of k | Units | Comments |
|---|---|---|
| 1. 380 6504(24)×10−23 | J/K | SI units, 2002 CODATA value |
| 8. The joule (written in lower case ˈdʒuːl or /ˈdʒaʊl/ (symbol J) is the SI unit of Energy measuring heat, Electricity The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic CODATA ( Committee on Data for Science and Technology) was established in 1966 as an interdisciplinary committee of the International Council of Science (ICSU formerly 617 343(15)×10−5 | eV/K | 1 electronvolt = 1.602 176 53(14)×10−19 J |
| 6. This list compares various energies in Joules (J organized by Order of magnitude. 336 281(73)×10−6 | Ryd/K | 1 Rydberg = 13. 6 eV |
| 1. 3807×10−16 | erg/K |
The digits in parentheses are the standard measurement uncertainty in the last two digits of the measured value. An erg is the unit of Energy and Mechanical work in the centimetre-gram-second (CGS system of units symbol "erg" In Metrology, measurement uncertainty describes a region about an observed value of a Physical quantity which is likely to enclose the true value of that quantity
The constant can also be expressed in terms of energy per mole instead of energy per entity (such as 1. The mole (symbol mol) is a unit of Amount of substance: it is an SI base unit, and almost the only unit to be used to measure this 99 calories/mol-K); for historical reasons it is then called the gas constant and given the symbol R. Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working
Since k is a constant of proportionality of temperature and energy, the numerical value of k depends on the choice of units for energy and temperature. A physical Constant is a Physical quantity that is generally believed to be both universal in nature and constant in time The Kelvin temperature scale was chosen to conveniently divide up the liquid range of water into one hundred intervals. The very small numerical value of k merely reflects the small energy in joules required to increase a particle's energy through 1oK. The physically fundamental idea is the characteristic energy kT of a particular temperature.
The numerical value of k provides a mapping from this characteristic microscopic energy E to the macroscopically-derived temperature scale T = E/k. On the other hand, the Plank units of temperature and energy are defined in such a way that k=1. If we choose to measure temperature in units of energy then Boltzmann's constant would not be needed at all. [1]
References
- Boltzmann's constant CODATA value at NIST
- Peter J. Mohr, and Barry N. Taylor, "CODATA recommended values of the fundamental physical constants: 1998", Rev. Mod. Phys. , Vol 72, No. 2, April 2000
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