MaxwellBoltzmann - Maple Help

Maxwell-Boltzmann Distribution

 Overview In the 1800s James Clerk Maxwell and Ludwig Boltzmann independently investigated the distribution of velocities of atoms or molecules in an ideal gas at a given temperature.  The distribution of molecular speeds became known as the Maxwell-Boltzmann distribution.  In this lessons we will explore this distribution as a function of temperature and mass of the gaseous particles.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution is

where v is the speed of the gas particles, M is the molar mass, T is the temperature, and R is the gas constant.

From the distribution we can approximate the probability P(v) of finding a gas particle with a speed v between v-dv/2 and v+dv/2 as

where dv is a small (differential) interval about the velocity v.

If the distribution is correctly defined, the total probability of finding the particle at some speed must be unity, that is

Before we begin, we define the ideal gas constant in SI units

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 ${{R}}_{{\mathrm{SI}}}{≔}{8.3144598}$ (2.1)

We can define the Maxwell-Boltzmann distribution with Maple

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 ${f}{≔}\left({v}{,}{M}{,}{T}{,}{R}\right){↦}{\mathrm{\pi }}{\cdot }\sqrt{{2}}{\cdot }{\left(\frac{{M}}{{\mathrm{\pi }}{\cdot }{R}{\cdot }{T}}\right)}^{{3}}{{2}}}{\cdot }{{v}}^{{2}}{\cdot }{{ⅇ}}^{{-}\frac{{M}{\cdot }{{v}}^{{2}}}{{2}{\cdot }{R}{\cdot }{T}}}$ (2.2)

Let us check that the total probability of the particle having some speed is unity for a gas with a molar mass of 0.1 kg/mol at a temperature of 273 K

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 ${\mathrm{P_total}}{≔}{0.9999999999}$ (2.3)

(a) Does the total probability equal the expected value of unity?

We can explore the Maxwell-Boltzmann distribution as a function of the temperature and molar mass of the gas.

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$\mathbf{T}$

$\mathbf{M}$

Using the slide rules to change temperature T in K and molar mass M in kg/mol (SI units), answer the following questions:

(b) Approximately from the graph, what is the most probably velocity for water H2O at 275 K?

(c) Approximately from the graph, what is the most probably velocity for methane CH4 at 550 K?

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Molecular Speeds

Most Probable Speed

The most probable speed for a gas molecule occurs at the peak (maximum) of the Maxwell-Boltzmann distribution.

We can compute an analytical expression for the most probable speed by solving for the speed at which the derivative of the distribution vanishes

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 ${\mathrm{eq}}{≔}{2}{}{\mathrm{\pi }}{}\sqrt{{2}}{}{\left(\frac{{M}}{{\mathrm{\pi }}{}{R}{}{T}}\right)}^{{3}}{{2}}}{}{v}{}{{ⅇ}}^{{-}\frac{{M}{}{{v}}^{{2}}}{{2}{}{R}{}{T}}}{-}\frac{{\mathrm{\pi }}{}\sqrt{{2}}{}{\left(\frac{{M}}{{\mathrm{\pi }}{}{R}{}{T}}\right)}^{{3}}{{2}}}{}{{v}}^{{3}}{}{M}{}{{ⅇ}}^{{-}\frac{{M}{}{{v}}^{{2}}}{{2}{}{R}{}{T}}}}{{R}{}{T}}{=}{0}$
 ${\mathrm{v_mp}}{≔}{0}{,}\frac{\sqrt{{2}}{}\sqrt{{M}{}{R}{}{T}}}{{M}}{,}{-}\frac{\sqrt{{2}}{}\sqrt{{M}{}{R}{}{T}}}{{M}}$ (3.1.1)

(d) Which of the three answers for the most probable speed is physically realistic?  Explain briefly.

(e) Show the simplification of this result to match the textbook result.

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Average Speed

The average speed for a gas molecule can be derived by integrating the Maxwell-Boltzmann distribution with the speed for all possible speeds

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 ${\mathrm{v_ave}}{≔}\underset{{v}{\to }{\mathrm{\infty }}}{{lim}}{}\left({-}\frac{{T}{}{R}{}\sqrt{{2}}{}{\left(\frac{{M}}{{R}{}{T}}\right)}^{{3}}{{2}}}{}\left({M}{}{{ⅇ}}^{{-}\frac{{M}{}{{v}}^{{2}}}{{2}{}{R}{}{T}}}{}{{v}}^{{2}}{+}{2}{}{{ⅇ}}^{{-}\frac{{M}{}{{v}}^{{2}}}{{2}{}{R}{}{T}}}{}{T}{}{R}{-}{2}{}{R}{}{T}\right)}{\sqrt{{\mathrm{\pi }}}{}{{M}}^{{2}}}\right)$ (3.2.1)

Taking this limit by substitution and simplification, we have

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 ${\mathrm{v_ave}}{≔}\frac{{2}{}{T}{}{R}{}\sqrt{{2}}{}\sqrt{\frac{{M}}{{R}{}{T}}}}{{M}{}\sqrt{{\mathrm{\pi }}}}$ (3.2.2)

(f) Show the simplification of this result to match the textbook result.

(g) Take the ratio of the formulas for vmp and vave.  Which speed is always greater?  Explain briefly.

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