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84-430: (Redirected from Maxwell-Boltzmann ) Maxwell–Boltzmann may refer to: Maxwell–Boltzmann statistics , statistical distribution of material particles over various energy states in thermal equilibrium Maxwell–Boltzmann distribution , particle speeds in gases See also [ edit ] Maxwell (disambiguation) Boltzmann (disambiguation) Topics referred to by

168-440: A {\displaystyle N_{a}} balls from a total of N {\displaystyle N} balls to place into box a {\displaystyle a} , and continue to select for each box from the remaining balls, ensuring that every ball is placed in one of the boxes. The total number of ways that the balls can be arranged is As every ball has been placed into a box, ( N − N

252-418: A ) − 2 π x a exp ⁡ ( − x 2 2 a 2 ) {\displaystyle \operatorname {erf} \left({\frac {x}{{\sqrt {2}}a}}\right)-{\sqrt {\frac {2}{\pi }}}\,{\frac {x}{a}}\,\exp \left({\frac {-x^{2}}{2a^{2}}}\right)} In physics (in particular in statistical mechanics ),

336-436: A − N b − ⋯ − N k ) ! = 0 ! = 1 {\displaystyle (N-N_{a}-N_{b}-\cdots -N_{k})!=0!=1} , and we simplify the expression as This is just the multinomial coefficient , the number of ways of arranging N items into k boxes, the l -th box holding N l items, ignoring the permutation of items in each box. Now, consider

420-576: A 2 x f ′ ( x ) + ( x 2 − 2 a 2 ) f ( x ) , f ( 1 ) = 1 a 3 2 π exp ⁡ ( − 1 2 a 2 ) . {\displaystyle {\begin{aligned}0&=a^{2}xf'(x)+\left(x^{2}-2a^{2}\right)f(x),\\[4pt]f(1)&={\frac {1}{a^{3}}}{\sqrt {\frac {2}{\pi }}}\exp \left(-{\frac {1}{2a^{2}}}\right).\end{aligned}}} With

504-1020: A = k B T / m . {\textstyle a={\sqrt {k_{\text{B}}T/m}}\,.} The simplest ordinary differential equation satisfied by the distribution is: 0 = k B T v f ′ ( v ) + f ( v ) ( m v 2 − 2 k B T ) , f ( 1 ) = 2 π [ m k B T ] 3 / 2 exp ⁡ ( − m 2 k B T ) ; {\displaystyle {\begin{aligned}0&=k_{\text{B}}Tvf'(v)+f(v)\left(mv^{2}-2k_{\text{B}}T\right),\\[4pt]f(1)&={\sqrt {\frac {2}{\pi }}}\,{\biggl [}{\frac {m}{k_{\text{B}}T}}{\biggr ]}^{3/2}\exp \left(-{\frac {m}{2k_{\text{B}}T}}\right);\end{aligned}}} or in unitless presentation: 0 =

588-443: A certain type (e.g., electrons, protons,photon etc.) as principally indistinguishable. Once this assumption is made, the particle statistics change. The change in entropy in the entropy of mixing example may be viewed as an example of a non-extensive entropy resulting from the distinguishability of the two types of particles being mixed. Quantum particles are either bosons (following Bose–Einstein statistics ) or fermions (subject to

672-608: A corresponding number of microstates available to the reservoir. Call this number Ω R ( s 1 ) {\displaystyle \;\Omega _{R}(s_{1})} . By assumption, the combined system (of the system we are interested in and the reservoir) is isolated, so all microstates are equally probable. Therefore, for instance, if Ω R ( s 1 ) = 2 Ω R ( s 2 ) {\displaystyle \;\Omega _{R}(s_{1})=2\;\Omega _{R}(s_{2})} , we can conclude that our system

756-465: A different energy–momentum relation , such as relativistic particles (resulting in Maxwell–Jüttner distribution ), and to other than three-dimensional spaces. Maxwell–Boltzmann statistics is often described as the statistics of "distinguishable" classical particles. In other words, the configuration of particle A in state 1 and particle B in state 2 is different from the case in which particle B

840-566: A distillation of the underlying technique. The distribution was first derived by Maxwell in 1860 on heuristic grounds. Boltzmann later, in the 1870s, carried out significant investigations into the physical origins of this distribution. The distribution can be derived on the ground that it maximizes the entropy of the system. Maxwell–Boltzmann statistics is used to derive the Maxwell–Boltzmann distribution of an ideal gas. However, it can also be used to extend that distribution to particles with

924-451: A fixed energy ( E = ∑ N i ε i ) {\textstyle \left(E=\sum N_{i}\varepsilon _{i}\right)} in the container. The maxima of W {\displaystyle W} and ln ⁡ ( W ) {\displaystyle \ln(W)} are achieved by the same values of N i {\displaystyle N_{i}} and, since it

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1008-412: A marking on each one, e.g., drawing a different number on each one as is done with lottery balls. The particles are moving inside that container in all directions with great speed. Because the particles are speeding around, they possess some energy. The Maxwell–Boltzmann distribution is a mathematical function that describes about how many particles in the container have a certain energy. More precisely,

1092-972: A molecule with these values of momentum components, so: The normalizing constant can be determined by recognizing that the probability of a molecule having some momentum must be 1. Integrating the exponential in equation 4 over all p x , p y , and p z yields a factor of ∭ − ∞ + ∞ exp ⁡ ( − p x 2 + p y 2 + p z 2 2 m k B T ) d p x d p y d p z = [ π 2 m k B T ] 3 {\displaystyle \iiint _{-\infty }^{+\infty }\exp \left(-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mk_{\text{B}}T}}\right)dp_{x}\,dp_{y}\,dp_{z}={\Bigl [}{\sqrt {\pi }}{\sqrt {2mk_{\text{B}}T}}{\Bigr ]}^{3}} So that

1176-485: A particular state rather than the set of all states with energy ε i {\displaystyle \varepsilon _{i}} , and Z = ∑ i e − ε i / k T {\textstyle Z=\sum _{i}e^{-\varepsilon _{i}/kT}} . Maxwell–Boltzmann statistics grew out of the Maxwell–Boltzmann distribution, most likely as

1260-551: A rectangle. They interact via perfectly elastic collisions . The system is initialized out of equilibrium, but the velocity distribution (in blue) quickly converges to the 2D Maxwell–Boltzmann distribution (in orange). The mean speed ⟨ v ⟩ {\displaystyle \langle v\rangle } , most probable speed ( mode ) v p , and root-mean-square speed ⟨ v 2 ⟩ {\textstyle {\sqrt {\langle v^{2}\rangle }}} can be obtained from properties of

1344-677: A standard spherical coordinate system, where d Ω = sin ⁡ v θ   d v ϕ   d v θ {\displaystyle d\Omega =\sin {v_{\theta }}~dv_{\phi }~dv_{\theta }} is an element of solid angle and v 2 = | v | 2 = v x 2 + v y 2 + v z 2 {\textstyle v^{2}=|\mathbf {v} |^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}} . The Maxwellian distribution function for particles moving in only one direction, if this direction

1428-399: A system. Alternatively, one can make use of the canonical ensemble . In a canonical ensemble, a system is in thermal contact with a reservoir. While energy is free to flow between the system and the reservoir, the reservoir is thought to have infinitely large heat capacity as to maintain constant temperature, T , for the combined system. In the present context, our system is assumed to have

1512-726: A velocity vector v {\displaystyle \mathbf {v} } of magnitude v {\displaystyle v} , is given by f ( v )   d 3 v = [ m 2 π k B T ] 3 / 2 exp ⁡ ( − m v 2 2 k B T )   d 3 v , {\displaystyle f(\mathbf {v} )~d^{3}\mathbf {v} ={\biggl [}{\frac {m}{2\pi k_{\text{B}}T}}{\biggr ]}^{{3}/{2}}\,\exp \left(-{\frac {mv^{2}}{2k_{\text{B}}T}}\right)~d^{3}\mathbf {v} ,} where: One can write

1596-506: Is x , is f ( v x )   d v x = m 2 π k B T exp ⁡ ( − m v x 2 2 k B T )   d v x , {\displaystyle f(v_{x})~dv_{x}={\sqrt {\frac {m}{2\pi k_{\text{B}}T}}}\,\exp \left(-{\frac {mv_{x}^{2}}{2k_{\text{B}}T}}\right)~dv_{x},} which can be obtained by integrating

1680-451: Is a kind of partition function (for the single-particle system, not the usual partition function of the entire system). Because velocity and speed are related to energy, Equation ( 1 ) can be used to derive relationships between temperature and the speeds of gas particles. All that is needed is to discover the density of microstates in energy, which is determined by dividing up momentum space into equal sized regions. The potential energy

1764-417: Is a total of k {\displaystyle k} boxes labelled a , b , … , k {\displaystyle a,b,\ldots ,k} . With the concept of combination , we could calculate how many ways there are to arrange N {\displaystyle N} into the set of boxes, where the order of balls within each box isn’t tracked. First, we select N

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1848-517: Is considered to be a continuous variable, the Thomas–Fermi approximation yields a continuous degeneracy g proportional to ε {\displaystyle {\sqrt {\varepsilon }}} so that: which is just the Maxwell–Boltzmann distribution for the energy. In the above discussion, the Boltzmann distribution function was obtained via directly analysing the multiplicities of

1932-423: Is different from Wikidata All article disambiguation pages All disambiguation pages Maxwell%E2%80%93Boltzmann statistics The expected number of particles with energy ε i {\displaystyle \varepsilon _{i}} for Maxwell–Boltzmann statistics is where: Equivalently, the number of particles is sometimes expressed as where the index i now specifies

2016-600: Is directly related to the speed of sound c in the gas, by c = γ 3   v r m s = f + 2 3 f   v r m s = f + 2 2 f   v p , {\displaystyle c={\sqrt {\frac {\gamma }{3}}}\ v_{\mathrm {rms} }={\sqrt {\frac {f+2}{3f}}}\ v_{\mathrm {rms} }={\sqrt {\frac {f+2}{2f}}}\ v_{\text{p}},} where γ = 1 + 2 f {\textstyle \gamma =1+{\frac {2}{f}}}

2100-683: Is easier to accomplish mathematically, we will maximize the latter function instead. We constrain our solution using Lagrange multipliers forming the function: Finally In order to maximize the expression above we apply Fermat's theorem (stationary points) , according to which local extrema, if exist, must be at critical points (partial derivatives vanish): By solving the equations above ( i = 1 … n {\displaystyle i=1\ldots n} ) we arrive to an expression for N i {\displaystyle N_{i}} : Substituting this expression for N i {\displaystyle N_{i}} into

2184-517: Is essentially a division by N ! of Boltzmann's original expression for W , and this correction is referred to as correct Boltzmann counting . We wish to find the N i {\displaystyle N_{i}} for which the function W {\displaystyle W} is maximized, while considering the constraint that there is a fixed number of particles ( N = ∑ N i ) {\textstyle \left(N=\sum N_{i}\right)} and

2268-557: Is found imposing where d 3 p {\displaystyle d^{3}\mathbf {p} } is the infinitesimal phase-space volume of momenta corresponding to the energy interval dE . Making use of the spherical symmetry of the energy-momentum dispersion relation E = | p | 2 2 m , {\displaystyle E={\tfrac {|\mathbf {p} |^{2}}{2m}},} this can be expressed in terms of dE as Using then ( 8 ) in ( 7 ), and expressing everything in terms of

2352-514: Is in state 1 and particle A is in state 2. This assumption leads to the proper (Boltzmann) statistics of particles in the energy states, but yields non-physical results for the entropy, as embodied in the Gibbs paradox . At the same time, there are no real particles that have the characteristics required by Maxwell–Boltzmann statistics. Indeed, the Gibbs paradox is resolved if we treat all particles of

2436-423: Is invariant.) where the index s runs through all microstates of the system. Z is sometimes called the Boltzmann sum over states (or "Zustandssumme" in the original German). If we index the summation via the energy eigenvalues instead of all possible states, degeneracy must be taken into account. The probability of our system having energy ε i {\displaystyle \varepsilon _{i}}

2520-406: Is seen to be the product of three independent normally distributed variables p x {\displaystyle p_{x}} , p y {\displaystyle p_{y}} , and p z {\displaystyle p_{z}} , with variance m k B T {\displaystyle mk_{\text{B}}T} . Additionally, it can be seen that

2604-531: Is simply the sum of the probabilities of all corresponding microstates: where, with obvious modification, Maxwell%E2%80%93Boltzmann distribution 2 π x 2 a 3 exp ⁡ ( − x 2 2 a 2 ) {\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {x^{2}}{a^{3}}}\,\exp \left({\frac {-x^{2}}{2a^{2}}}\right)} erf ⁡ ( x 2

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2688-445: Is taken to be zero, so that all energy is in the form of kinetic energy. The relationship between kinetic energy and momentum for massive non- relativistic particles is where p is the square of the momentum vector p = [ p x , p y , p z ] . We may therefore rewrite Equation ( 1 ) as: where: This distribution of N i  : N is proportional to the probability density function f p for finding

2772-659: Is the adiabatic index , f is the number of degrees of freedom of the individual gas molecule. For the example above, diatomic nitrogen (approximating air ) at 300 K , f = 5 {\displaystyle f=5} and c = 7 15 v r m s ≈ 68 %   v r m s ≈ 84 %   v p ≈ 353   m / s , {\displaystyle c={\sqrt {\frac {7}{15}}}v_{\mathrm {rms} }\approx 68\%\ v_{\mathrm {rms} }\approx 84\%\ v_{\text{p}}\approx 353\ \mathrm {m/s} ,}

2856-447: Is the occupation number of the energy level i . {\displaystyle i.} If we know all the occupation numbers { N i ∣ i = 1 , 2 , 3 , … } , {\displaystyle \{N_{i}\mid i=1,2,3,\ldots \},} then we know the total energy of the system. However, because we can distinguish between which particles are occupying each energy level,

2940-763: Is the temperature , P is pressure, V is volume , and μ is the chemical potential . Boltzmann's equation S = k ln ⁡ W {\displaystyle S=k\ln W} is the realization that the entropy is proportional to ln ⁡ W {\displaystyle \ln W} with the constant of proportionality being the Boltzmann constant . Using the ideal gas equation of state ( PV  =  NkT ), It follows immediately that β = 1 / k T {\displaystyle \beta =1/kT} and α = − μ / k T {\displaystyle \alpha =-\mu /kT} so that

3024-413: Is twice as likely to be in state s 1 {\displaystyle \;s_{1}} than s 2 {\displaystyle \;s_{2}} . In general, if P ( s i ) {\displaystyle \;P(s_{i})} is the probability that our system is in state s i {\displaystyle \;s_{i}} , Since the entropy of

3108-475: Is used for describing systems in equilibrium. However, most systems do not start out in their equilibrium state. The evolution of a system towards its equilibrium state is governed by the Boltzmann equation . The equation predicts that for short range interactions, the equilibrium velocity distribution will follow a Maxwell–Boltzmann distribution. To the right is a molecular dynamics (MD) simulation in which 900  hard sphere particles are constrained to move in

3192-442: Is zero. Similarly, d V R = 0. {\displaystyle dV_{R}=0.} This gives where U R ( s i ) {\displaystyle U_{R}(s_{i})} and E ( s i ) {\displaystyle E(s_{i})} denote the energies of the reservoir and the system at s i {\displaystyle s_{i}} , respectively. For

3276-568: The g i {\displaystyle g_{i}} boxes, the second object can also go into any of the g i {\displaystyle g_{i}} boxes, and so on). Thus the number of ways W {\displaystyle W} that a total of N {\displaystyle N} particles can be classified into energy levels according to their energies, while each level i {\displaystyle i} having g i {\displaystyle g_{i}} distinct states such that

3360-682: The Darwin–Fowler method of mean values, the Maxwell–Boltzmann distribution is obtained as an exact result. For particles confined to move in a plane, the speed distribution is given by P ( s < | v | < s + d s ) = m s k B T exp ⁡ ( − m s 2 2 k B T ) d s {\displaystyle P(s<|\mathbf {v} |<s+ds)={\frac {ms}{k_{\text{B}}T}}\exp \left(-{\frac {ms^{2}}{2k_{\text{B}}T}}\right)ds} This distribution

3444-585: The Maxwell–Boltzmann distribution , or Maxwell(ian) distribution , is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann . It was first defined and used for describing particle speeds in idealized gases , where the particles move freely inside a stationary container without interacting with one another, except for very brief collisions in which they exchange energy and momentum with each other or with their thermal environment. The term "particle" in this context refers to gaseous particles only ( atoms or molecules ), and

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3528-490: The Pauli exclusion principle , following instead Fermi–Dirac statistics ). Both of these quantum statistics approach the Maxwell–Boltzmann statistics in the limit of high temperature and low particle density. Maxwell–Boltzmann statistics can be derived in various statistical mechanical thermodynamic ensembles: In each case it is necessary to assume that the particles are non-interacting, and that multiple particles can occupy

3612-980: The equipartition theorem , given that the energy is evenly distributed among all three degrees of freedom in equilibrium, we can also split f E ( E ) d E {\displaystyle f_{E}(E)dE} into a set of chi-squared distributions , where the energy per degree of freedom, ε is distributed as a chi-squared distribution with one degree of freedom, f ε ( ε ) d ε = 1 π ε k B T   exp ⁡ ( − ε k B T ) d ε {\displaystyle f_{\varepsilon }(\varepsilon )\,d\varepsilon ={\sqrt {\frac {1}{\pi \varepsilon k_{\text{B}}T}}}~\exp \left(-{\frac {\varepsilon }{k_{\text{B}}T}}\right)\,d\varepsilon } At equilibrium, this distribution will hold true for any number of degrees of freedom. For example, if

3696-403: The i -th level accommodates N i {\displaystyle N_{i}} particles is: This is the form for W first derived by Boltzmann . Boltzmann's fundamental equation S = k ln ⁡ W {\displaystyle S=k\,\ln W} relates the thermodynamic entropy S to the number of microstates W , where k is the Boltzmann constant . It

3780-630: The velocity vector in Euclidean space ), with a scale parameter measuring speeds in units proportional to the square root of T / m {\displaystyle T/m} (the ratio of temperature and particle mass). The Maxwell–Boltzmann distribution is a result of the kinetic theory of gases , which provides a simplified explanation of many fundamental gaseous properties, including pressure and diffusion . The Maxwell–Boltzmann distribution applies fundamentally to particle velocities in three dimensions, but turns out to depend only on

3864-471: The Maxwell distribution. This works well for nearly ideal , monatomic gases like helium , but also for molecular gases like diatomic oxygen . This is because despite the larger heat capacity (larger internal energy at the same temperature) due to their larger number of degrees of freedom , their translational kinetic energy (and thus their speed) is unchanged. For diatomic nitrogen ( N 2 ,

3948-528: The Maxwell–Boltzmann distribution gives the non-normalized probability (this means that the probabilities do not add up to 1) that the state corresponding to a particular energy is occupied. In general, there may be many particles with the same amount of energy ε {\displaystyle \varepsilon } . Let the number of particles with the same energy ε 1 {\displaystyle \varepsilon _{1}} be N 1 {\displaystyle N_{1}} ,

4032-446: The Maxwell–Boltzmann form. However, rarefied gases at ordinary temperatures behave very nearly like an ideal gas and the Maxwell speed distribution is an excellent approximation for such gases. This is also true for ideal plasmas , which are ionized gases of sufficiently low density. The distribution was first derived by Maxwell in 1860 on heuristic grounds. Boltzmann later, in the 1870s, carried out significant investigations into

4116-547: The case where there is more than one way to put N i {\displaystyle N_{i}} particles in the box i {\displaystyle i} (i.e. taking the degeneracy problem into consideration). If the i {\displaystyle i} -th box has a "degeneracy" of g i {\displaystyle g_{i}} , that is, it has g i {\displaystyle g_{i}} "sub-boxes" ( g i {\displaystyle g_{i}} boxes with

4200-413: The distribution again under the framework of statistical thermodynamics . The derivations in this section are along the lines of Boltzmann's 1877 derivation, starting with result known as Maxwell–Boltzmann statistics (from statistical thermodynamics). Maxwell–Boltzmann statistics gives the average number of particles found in a given single-particle microstate . Under certain assumptions, the logarithm of

4284-444: The element of velocity space as d 3 v = d v x d v y d v z {\displaystyle d^{3}\mathbf {v} =dv_{x}\,dv_{y}\,dv_{z}} , for velocities in a standard Cartesian coordinate system, or as d 3 v = v 2 d v d Ω {\displaystyle d^{3}\mathbf {v} =v^{2}\,dv\,d\Omega } in

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4368-1393: The energy E , we get f E ( E ) d E = [ 1 2 π m k B T ] 3 / 2 exp ⁡ ( − E k B T ) 4 π m 2 m E   d E = 2 E π [ 1 k B T ] 3 / 2 exp ⁡ ( − E k B T ) d E {\displaystyle {\begin{aligned}f_{E}(E)dE&=\left[{\frac {1}{2\pi mk_{\text{B}}T}}\right]^{3/2}\exp \left(-{\frac {E}{k_{\text{B}}T}}\right)4\pi m{\sqrt {2mE}}\ dE\\[1ex]&=2{\sqrt {\frac {E}{\pi }}}\,\left[{\frac {1}{k_{\text{B}}T}}\right]^{3/2}\exp \left(-{\frac {E}{k_{\text{B}}T}}\right)\,dE\end{aligned}}} and finally f E ( E ) = 2 E π [ 1 k B T ] 3 / 2 exp ⁡ ( − E k B T ) {\displaystyle f_{E}(E)=2{\sqrt {\frac {E}{\pi }}}\,\left[{\frac {1}{k_{\text{B}}T}}\right]^{3/2}\exp \left(-{\frac {E}{k_{\text{B}}T}}\right)}    ( 9 ) Since

4452-489: The energy is proportional to the sum of the squares of the three normally distributed momentum components, this energy distribution can be written equivalently as a gamma distribution , using a shape parameter, k shape = 3 / 2 {\displaystyle k_{\text{shape}}=3/2} and a scale parameter, θ scale = k B T . {\displaystyle \theta _{\text{scale}}=k_{\text{B}}T.} Using

4536-461: The energy levels ε i {\displaystyle \varepsilon _{i}} with degeneracies g i {\displaystyle g_{i}} . As before, we would like to calculate the probability that our system has energy ε i {\displaystyle \varepsilon _{i}} . If our system is in state s 1 {\displaystyle \;s_{1}} , then there would be

4620-563: The equation for ln ⁡ W {\displaystyle \ln W} and assuming that N ≫ 1 {\displaystyle N\gg 1} yields: or, rearranging: Boltzmann realized that this is just an expression of the Euler-integrated fundamental equation of thermodynamics . Identifying E as the internal energy, the Euler-integrated fundamental equation states that : where T

4704-429: The factorial: to write: Using the fact that ( 1 + N i / g i ) g i ≈ e N i {\displaystyle (1+N_{i}/g_{i})^{g_{i}}\approx e^{N_{i}}} for g i ≫ N i {\displaystyle g_{i}\gg N_{i}} we can again use Stirling's approximation to write: This

4788-634: The fraction of particles in a given microstate is linear in the ratio of the energy of that state to the temperature of the system: there are constants k {\displaystyle k} and C {\displaystyle C} such that, for all i {\displaystyle i} , − log ⁡ ( N i N ) = 1 k ⋅ E i T + C . {\displaystyle -\log \left({\frac {N_{i}}{N}}\right)={\frac {1}{k}}\cdot {\frac {E_{i}}{T}}+C.} The assumptions of this equation are that

4872-537: The magnitude of momentum will be distributed as a Maxwell–Boltzmann distribution, with a = m k B T {\textstyle a={\sqrt {mk_{\text{B}}T}}} . The Maxwell–Boltzmann distribution for the momentum (or equally for the velocities) can be obtained more fundamentally using the H-theorem at equilibrium within the Kinetic theory of gases framework. The energy distribution

4956-682: The normalized distribution function is: f p ( p x , p y , p z ) = [ 1 2 π m k B T ] 3 / 2 exp ⁡ ( − p x 2 + p y 2 + p z 2 2 m k B T ) {\displaystyle f_{\mathbf {p} }(p_{x},p_{y},p_{z})=\left[{\frac {1}{2\pi mk_{\text{B}}T}}\right]^{3/2}\exp \left(-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mk_{\text{B}}T}}\right)}    ( 6 ) The distribution

5040-524: The number of particles possessing another energy ε 2 {\displaystyle \varepsilon _{2}} be N 2 {\displaystyle N_{2}} , and so forth for all the possible energies { ε i ∣ i = 1 , 2 , 3 , … } . {\displaystyle \{\varepsilon _{i}\mid i=1,2,3,\ldots \}.} To describe this situation, we say that N i {\displaystyle N_{i}}

5124-444: The number of possible states of the system, we must count each and every microstate, and not just the possible sets of occupation numbers. To begin with, assume that there is only one state at each energy level i {\displaystyle i} (there is no degeneracy). What follows next is a bit of combinatorial thinking which has little to do in accurately describing the reservoir of particles. For instance, let's say there

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5208-520: The number of ways of distributing the N i {\displaystyle N_{i}} objects in the g i {\displaystyle g_{i}} "sub-boxes". The number of ways of placing N i {\displaystyle N_{i}} distinguishable objects in g i {\displaystyle g_{i}} "sub-boxes" is g i N i {\displaystyle g_{i}^{N_{i}}} (the first object can go into any of

5292-431: The particles are rigid mass dipoles of fixed dipole moment, they will have three translational degrees of freedom and two additional rotational degrees of freedom. The energy in each degree of freedom will be described according to the above chi-squared distribution with one degree of freedom, and the total energy will be distributed according to a chi-squared distribution with five degrees of freedom. This has implications in

5376-518: The particles do not interact, and that they are classical; this means that each particle's state can be considered independently from the other particles' states. Additionally, the particles are assumed to be in thermal equilibrium. This relation can be written as an equation by introducing a normalizing factor: where: The denominator in equation 1 is a normalizing factor so that the ratios N i : N {\displaystyle N_{i}:N} add up to unity — in other words it

5460-423: The physical origins of this distribution. The distribution can be derived on the ground that it maximizes the entropy of the system. A list of derivations are: For a system containing a large number of identical non-interacting, non-relativistic classical particles in thermodynamic equilibrium, the fraction of the particles within an infinitesimal element of the three-dimensional velocity space d v , centered on

5544-716: The population [B,A] while for indistinguishable particles, they are not. If we carry out the argument for indistinguishable particles, we are led to the Bose–Einstein expression for W : The Maxwell–Boltzmann distribution follows from this Bose–Einstein distribution for temperatures well above absolute zero, implying that g i ≫ 1 {\displaystyle g_{i}\gg 1} . The Maxwell–Boltzmann distribution also requires low density, implying that g i ≫ N i {\displaystyle g_{i}\gg N_{i}} . Under these conditions, we may use Stirling's approximation for

5628-417: The populations may now be written: Note that the above formula is sometimes written: where z = exp ⁡ ( μ / k T ) {\displaystyle z=\exp(\mu /kT)} is the absolute activity . Alternatively, we may use the fact that to obtain the population numbers as where Z is the partition function defined by: In an approximation where ε i

5712-661: The primary component of air ) at room temperature ( 300 K ), this gives v p ≈ 2 ⋅ 8.31   J ⋅ mol − 1 K − 1   300   K 0.028   kg ⋅ mol − 1 ≈ 422   m/s . {\displaystyle v_{\text{p}}\approx {\sqrt {\frac {2\cdot 8.31\ {\text{J}}\cdot {\text{mol}}^{-1}{\text{K}}^{-1}\ 300\ {\text{K}}}{0.028\ {\text{kg}}\cdot {\text{mol}}^{-1}}}}\approx 422\ {\text{m/s}}.} In summary,

5796-445: The probability, per unit speed, of finding the particle with a speed near v . This equation is simply the Maxwell–Boltzmann distribution (given in the infobox) with distribution parameter a = k B T / m . {\textstyle a={\sqrt {k_{\text{B}}T/m}}\,.} The Maxwell–Boltzmann distribution is equivalent to the chi distribution with three degrees of freedom and scale parameter

5880-400: The reservoir S R = k ln ⁡ Ω R {\displaystyle \;S_{R}=k\ln \Omega _{R}} , the above becomes Next we recall the thermodynamic identity (from the first law of thermodynamics ): In a canonical ensemble, there is no exchange of particles, so the d N R {\displaystyle dN_{R}} term

5964-410: The same energy ε i {\displaystyle \varepsilon _{i}} . These states/boxes with the same energy are called degenerate states.), such that any way of filling the i {\displaystyle i} -th box where the number in the sub-boxes is changed is a distinct way of filling the box, then the number of ways of filling the i -th box must be increased by

6048-413: The same state and do so independently. Suppose we have a container with a huge number of very small particles all with identical physical characteristics (such as mass, charge, etc.). Let's refer to this as the system . Assume that though the particles have identical properties, they are distinguishable. For example, we might identify each particle by continually observing their trajectories, or by placing

6132-430: The same term [REDACTED] This disambiguation page lists articles associated with the title Maxwell–Boltzmann . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Maxwell–Boltzmann&oldid=515199088 " Category : Disambiguation pages Hidden categories: Short description

6216-417: The second equality we have used the conservation of energy. Substituting into the first equation relating P ( s 1 ) , P ( s 2 ) {\displaystyle P(s_{1}),\;P(s_{2})} : which implies, for any state s of the system where Z is an appropriately chosen "constant" to make total probability 1. ( Z is constant provided that the temperature T

6300-400: The set of occupation numbers { N i ∣ i = 1 , 2 , 3 , … } {\displaystyle \{N_{i}\mid i=1,2,3,\ldots \}} does not completely describe the state of the system. To completely describe the state of the system, or the microstate , we must specify exactly which particles are in each energy level. Thus when we count

6384-616: The speed (the magnitude of the velocity) of the particles. A particle speed probability distribution indicates which speeds are more likely: a randomly chosen particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The kinetic theory of gases applies to the classical ideal gas , which is an idealization of real gases. In real gases, there are various effects (e.g., van der Waals interactions , vortical flow, relativistic speed limits, and quantum exchange interactions ) that can make their speed distribution different from

6468-404: The system of particles is assumed to have reached thermodynamic equilibrium . The energies of such particles follow what is known as Maxwell–Boltzmann statistics , and the statistical distribution of speeds is derived by equating particle energies with kinetic energy . Mathematically, the Maxwell–Boltzmann distribution is the chi distribution with three degrees of freedom (the components of

6552-781: The three-dimensional form given above over v y and v z . Recognizing the symmetry of f ( v ) {\displaystyle f(v)} , one can integrate over solid angle and write a probability distribution of speeds as the function f ( v ) = [ m 2 π k B T ] 3 / 2 4 π v 2 exp ⁡ ( − m v 2 2 k B T ) . {\displaystyle f(v)={\biggl [}{\frac {m}{2\pi k_{\text{B}}T}}{\biggr ]}^{{3}/{2}}\,4\pi v^{2}\exp \left(-{\frac {mv^{2}}{2k_{\text{B}}T}}\right).} This probability density function gives

6636-993: The three-dimensional velocity distribution is f ( v ) ≡ [ 2 π k B T m ] − 3 / 2 exp ⁡ ( − 1 2 m v 2 k B T ) . {\displaystyle f(\mathbf {v} )\equiv \left[{\frac {2\pi k_{\text{B}}T}{m}}\right]^{-3/2}\exp \left(-{\frac {1}{2}}{\frac {m\mathbf {v} ^{2}}{k_{\text{B}}T}}\right).} The integral can easily be done by changing to coordinates u = v 1 − v 2 {\displaystyle \mathbf {u} =\mathbf {v} _{1}-\mathbf {v} _{2}} and U = v 1 + v 2 2 . {\displaystyle \mathbf {U} ={\tfrac {\mathbf {v} _{1}\,+\,\mathbf {v} _{2}}{2}}.} The Maxwell–Boltzmann distribution assumes that

6720-1188: The true value for air can be approximated by using the average molar weight of air ( 29 g/mol ), yielding 347 m/s at 300 K (corrections for variable humidity are of the order of 0.1% to 0.6%). The average relative velocity v rel ≡ ⟨ | v 1 − v 2 | ⟩ = ∫ d 3 v 1 d 3 v 2 | v 1 − v 2 | f ( v 1 ) f ( v 2 ) = 4 π k B T m = 2 ⟨ v ⟩ {\displaystyle {\begin{aligned}v_{\text{rel}}\equiv \langle |\mathbf {v} _{1}-\mathbf {v} _{2}|\rangle &=\int \!d^{3}\mathbf {v} _{1}\,d^{3}\mathbf {v} _{2}\left|\mathbf {v} _{1}-\mathbf {v} _{2}\right|f(\mathbf {v} _{1})f(\mathbf {v} _{2})\\[2pt]&={\frac {4}{\sqrt {\pi }}}{\sqrt {\frac {k_{\text{B}}T}{m}}}={\sqrt {2}}\langle v\rangle \end{aligned}}} where

6804-486: The typical speeds are related as follows: v p ≈ 88.6 %   ⟨ v ⟩ < ⟨ v ⟩ < 108.5 %   ⟨ v ⟩ ≈ v r m s . {\displaystyle v_{\text{p}}\approx 88.6\%\ \langle v\rangle <\langle v\rangle <108.5\%\ \langle v\rangle \approx v_{\mathrm {rms} }.} The root mean square speed

6888-514: The velocities of individual particles are much less than the speed of light, i.e. that T ≪ m c 2 k B {\displaystyle T\ll {\frac {mc^{2}}{k_{\text{B}}}}} . For electrons, the temperature of electrons must be T e ≪ 5.93 × 10 9   K {\displaystyle T_{e}\ll 5.93\times 10^{9}~\mathrm {K} } . The original derivation in 1860 by James Clerk Maxwell

6972-481: Was an argument based on molecular collisions of the Kinetic theory of gases as well as certain symmetries in the speed distribution function; Maxwell also gave an early argument that these molecular collisions entail a tendency towards equilibrium. After Maxwell, Ludwig Boltzmann in 1872 also derived the distribution on mechanical grounds and argued that gases should over time tend toward this distribution, due to collisions (see H-theorem ). He later (1877) derived

7056-411: Was pointed out by Gibbs however, that the above expression for W does not yield an extensive entropy, and is therefore faulty. This problem is known as the Gibbs paradox . The problem is that the particles considered by the above equation are not indistinguishable . In other words, for two particles ( A and B ) in two energy sublevels the population represented by [A,B] is considered distinct from

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