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Maxwell-Boltzmann Distribution
Maxwell Boltzmann
• The Maxwell-Boltzmann distribution describes the
distribution of particle energies; this distribution can be
used to gain a qualitative estimate of the fraction of
collisions with sufficient energy to lead to a reaction, and
also how that fraction depends on temperature.
• The Maxwell-Boltzmann equation, which is based on the
Kinetic Theory of Gases, gives the distribution of speeds for
a gas at a certain temperature. From this distribution
function, the most probable speed, the average speed, and
the root-mean-square speed can be derived.
• The Maxwell-Boltzmann distribution shows that the
distribution of kinetic energies becomes greater (more
disperse) as temperature increases.
Maxwell-Boltzmann Distribution Graph 1
This graph shows the Maxwell-Boltzmann Distribution of speeds for a
certain gas at a certain temperature, such as nitrogen at 298 K. The speed at
the top of the curve is called the most probable speed because the largest
number of molecules have that speed.
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Maxwell-Boltzmann Distribution Graph 2
This graph shows how the Maxwell-Boltzmann Distribution is affected by
temperature. At lower temperatures, the molecules have less energy. Therefore,
the speeds of the molecules are lower and the distribution has a smaller range.
As the temperature of the molecules increases, the distribution flattens out.
Because the molecules have greater energy at higher temperature, the
molecules are moving faster.
Maxwell-Boltzmann Distribution Graph 3
Graph 3 shows the dependence of the Maxwell-Boltzmann
Distribution on molecule size. On average, heavier molecules
move more slowly than lighter molecules. Therefore, heavier
molecules will have a smaller speed distribution, while lighter
molecules will have a speed distribution that is more spread
out.
Maxwell-Boltzmann Distribution
• Three speed expressions can be derived: the most
probable speed, the average speed, and the root-meansquare speed. The most probable speed is the maximum
value on the distribution plot. The average speed is the
sum of the speeds of all the molecules divided by the
number of molecules. The root-mean-square speed is
square root of the average speed-squared.
C mp = 2RT
C avg = 8RT C rms = 3RT
M
πM
M
• where R is the gas constant, T is the temperature and M is
the molar mass of the gas.
• It always follows that C mp <C avg <C rms
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