_{1}

^{*}

Utilizing Mathematica this report shows how from a practitioner’s point of view useful quantities some known, and some unknown and fresh properties about the Maxwell-Boltzmann distribution are calculated. We shortcut circling the usage of antiquated incomplete tabulated error functions given in the textbooks and professional literature replacing them with efficient upgrades. And, utilizing the animation features of Mathematica displaying the temperature-dependence of the distribution function assists in visualizing the character of the distribution.

Reviewing a few popular undergraduate physics texts [

In this investigating report, we show utilizing a CAS (Computer Algebra System) such as Mathematica [

We begin with MB distribution,

f M B ( m , T , v ) = 4 π ( m 2 π k T ) 3 2 v 2 e − m 2 k T v 2 , (1)

where m, T, and v are the mass, temperature, and the speed of an individual particle (molecule) of the gas, respectively. The k is the Boltzmann constant, k = 1.38 10^{−23} J/mol·K. Utilizing (1) a classic calculation yields the first and the second moments. Noting the latter is conducive to v_{RMS}. Equation (1) if needed may be formatted in terms of the molar mass, m_{mol}, and the universal ideal gas constant, R = 8.31 J/moll·K such that the ratio m/k can be replaced by the m m o l R . Applying (1) the mentioned quantities are,

{ v m p , v ¯ , v R M S } = { 2 , 8 π , 3 } k T m , (2)

The common quantity listed in (2) is k T / m . It is useful tabulating this for different values of m and temperature T. C.f.

T ( K ) | m (kg) | k T m ( m s ) | v m p ( m s ) | v ¯ ( m s ) | v R M S ( m s ) |
---|---|---|---|---|---|

200 | 3.34 × 10^{−27} | 909.036 | 1285.57 | 1450.61 | 1574.5 |

200 | 1.534 × 10^{−26} | 424.172 | 599.87 | 676.88 | 734.687 |

200 | 2.734 × 10^{−26} | 317.728 | 449.335 | 507.02 | 550.321 |

200 | 3.934 × 10^{−26} | 264.873 | 374.587 | 422.676 | 458.773 |

200 | 5.134 × 10^{−26} | 231.86 | 327.9 | 369.996 | 401.594 |

500 | 3.34 × 10^{−27} | 1437.31 | 2032.67 | 2293.62 | 2489.5 |

500 | 1.534 × 10^{−26} | 2489.5 | 948.477 | 1070.24 | 1161.64 |

500 | 2.734 × 10^{−26} | 502.372 | 710.461 | 801.669 | 870.134 |

500 | 3.934 × 10^{−26} | 418.801 | 592.274 | 668.309 | 725.384 |

500 | 5.134 × 10^{−26} | 366.604 | 518.456 | 585.015 | 634.976 |

800 | 3.34 × 10^{−27} | 1818.07 | 2571.14 | 2901.22 | 3148.99 |

800 | 1.534 × 10^{−26} | 848.344 | 1199.74 | 1353.76 | 1469.37 |

800 | 2.734 × 10^{−26} | 635.456 | 898.67 | 1014.04 | 1100.64 |

800 | 3.934 × 10^{−26} | 529.746 | 749.173 | 845.352 | 917.546 |

800 | 5.134 × 10^{−26} | 463.721 | 655.8 | 739.991 | 803.188 |

The usefulness of

Another way to utilize _{2} molecule displays the profile of the distribution at different temperatures. This is shown in

In addition to the observations, one is led to the question about the sensitivity of the ordinate of the maximum values of the distribution at its abscissa, at thev_{mp}. And, ordinarily, when one works with a cusped function, such as Gaussian, the qualitative measure of the sharpness of the function is determined by the width of the function at 1/e-th of its maximum height. As such, noting the MB distribution

is not too far off from Gaussian, its width must be evaluated. This report quantitatively addresses these issues. The steps conducive to our goals are as follows.

As discussed, at v = v m p the distribution given by (1) is at maximum. To form the needed function and to handle the calculation in the upcoming subsections we define an auxiliary function λ : = m K T . In terms of λ and m = m_{0} the mass of an H_{2} molecule, we evaluate the maximum height as a function of temperature. This is shown on the left panel of

To determine the width of the distribution and explore its temperature sensitivity we need to solve an equation that is conducive to speeds associated with the

intersections of a horizontal line 1 e ( f M B ) max = f M B . Where ( f M B ) max and f M B are subject to (2a) and (1), respectively. This algebraic equation is not easily solvable, applying Mathematica yields to a set of complicated roots that are to be sorted. A brief outline is given.

The simplified algebraic equation given to the formal equation in Sect. 2b is,

2 ln v − λ 2 v 2 = ln 2 − 2 − ln λ , (3)

where λ is defined in (2a). Solution of (3) for v yields six roots, some negative and some complex. By inspection, the acceptable positive roots with manipulation give the interested widths. For a chosen mass, m_{0} these widths are temperature dependent. Their variation vs temperature, T is shown on the right panel of

These features shown in

Displayed MB distributions for various scenarios shown in

any speed. I.e., there are always paired speeds associated with the same population. It is noted that for the paired set on the right v ¯ < v R M S and for the left set this is reversed!

As shown in

The goal is for instance to identify the speed that is less than v ¯ (the black marked abscissa on the right) that is conducive to the same population on the left. The procedure to achieving this goal is straight forward although not doable without utilizing a CAS, specifically Mathematica. After inspection only two out of four roots of the solved equation are meaningful. For readers familiar with Mathematica codes are given for reproduction, these are,

solv1=Solve[f[m,T,v]==Evaluate[f[m,T,v]/.v->sqrt(8/π) sqrt(kT/m)],v]//Simplify

{v/.solv1[

{1127.32,1450.61,1450.61}

{v/.solv2[

{1016.96,1574.49,1574.49}

The first code solves the equation that sets the distribution equals the value of distribution at v ¯ . The third code evaluates the speed at v R M S . The output confirms the accuracy of the computations. The output of the second code is a list of the speeds. The last two numeric in the given list are the v ¯ . These two are the same though they have been evaluated differently. The middle one is one of the roots of the equation on hand, the third one is v ¯ using (2). The same explanation holds for the third code and its output. Note, for each output the value of the first element is less than the second element. The point of the set goal is well achieved.

Customarily, in the general practice, the interest is to utilize a distribution e.g., the MB distribution and count the number the particles at a certain speed. In the case of MB distribution substituting the chosen speed in the distribution function yields the answer. However, counting the particles within a speed range naturally requires integration. The integration of the MB distribution with the integration limits specified by speed range is conducive to the error function. This lends itself to relying on the tabulated numeric values of the error function with severe limitations [

Here, by way of example utilizing Mathematica’s symbolic and then numeric capabilities, we bypass the antiquated usage of tabulated information.

We show two detailed cases. The first one does the integration within the range of zero to v ¯ , and the second within/to v R M S . The same approach may be applied to any desired speed range. To pursue these goals for the sake of simplicity (1) and (2) are reformulated in terms of the aforementioned λ.

Symbolic integration of both cases is,

I1 = 2 π λ 3 Integrate [ v 2 e − 1 2 λ v 2 , { v , 0 , 8 π 1 λ } , Assumptions → λ > 0 ] / / Simplify

e − 4 / π λ 3 ( − 4 + e 4 / π π Erf [ 2 π ] ) π λ 3 / 2

I2 = 2 π λ 3 Integrate [ v 2 e − 1 2 λ v 2 , { v , 8 π 1 λ , 3 1 λ } , Assumptions → λ > 0 ] / / Simplify

λ 3 ( − 2 3 e 3 / 2 + 4 e − 4 / π 2 π + 2 π ( Erf [ 3 2 ] − Erf [ 2 π ] ) ) 2 π λ 3 / 2

As noted, the output is given in error functions. Mathematica uses Erf[_{N}_{2} (nitrogen) at T = 200 K.

The numeric values of the above integrations are 0.53305 and 0.0753246. For instance, assuming the number of molecules is 5000 the population of the molecules is 2665, and 377, respectfully. These are meaningful numbers because the speed range of the first integration is much wider than the second accommodating much more molecules.

The main goal of crafting this investigating practical report is to show how by utilizing a Computer Algebra System, such as Mathematica many of the traditional quantities associated with Maxwell-Boltzmann distribution easily are calculated. As shown our approach explores issues affiliated with MB distribution that are fresh and not discussed in the published textbooks nor researched-based literature. Mathematica codes are provided so that the interested reader may practice the issues. The interested reader may find [

The author acknowledges the John T. and Page S. Smith Professorship funds for completing and publishing this work.

The author declares no conflicts of interest regarding the publication of this paper.

Sarafian, H. (2022) Maxwell-Boltzmann Distribution Mathematica Based Practitioner’s Approach. American Journal of Computational Mathematics, 12, 79-86. https://doi.org/10.4236/ajcm.2022.121006