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Hi folks!

i'm a biologist trying to understand some basics of statistical mechanics.

unfortunately, i cannot get over the following problem(s).

A)

in the boltzmann distribution the fraction of particles with energy Ei is given by:

[tex]\frac{Ni}{N} = \frac{exp(-\beta Ei)}{\sum exp(-\beta Ej)} \:\:\: (1)[/tex]

The most likely state should therefore be Ei = 0 with probability 1/Z.

However, when one derives the distribution of energies via the Maxwell-boltzmann speed distribution one obtains:

[tex]f_E\,dE = 2\sqrt{\frac{E}{\pi(kT)^3}}~\exp\left[\frac{-E}{kT}\right]\,dE[/tex] [tex]\:\:\: (2)[/tex]

.

f(E) goes to zero for E = 0. How is this possible if particles with Ei = 0 are the most frequent species?

B)

In reaction kinetics the reaction constant for one direction is given by:

[tex]k = A exp(-\beta Ea) \:\:\:(3)[/tex]

where A is the Arrhenius constant and Ea is the hight of the energy barrier for the reaction.

The term [tex]exp(-\beta Ea)[/tex] is supposed to correspond to the fraction of particles that are fast enough to get over the energy barrier.

Coming back to equation (2), shouldn't this fraction correspond to [tex]\int f(E)dE[/tex] from Ea to infinity? I don't see how one could get this from integrating (2) ?

Respectively, shouldn't taking the sum in equation (1) over all particles with Ei > Ea give this value as well?

What am i missing here?

thanks in advance!!!

Tim

i'm a biologist trying to understand some basics of statistical mechanics.

unfortunately, i cannot get over the following problem(s).

A)

in the boltzmann distribution the fraction of particles with energy Ei is given by:

[tex]\frac{Ni}{N} = \frac{exp(-\beta Ei)}{\sum exp(-\beta Ej)} \:\:\: (1)[/tex]

The most likely state should therefore be Ei = 0 with probability 1/Z.

However, when one derives the distribution of energies via the Maxwell-boltzmann speed distribution one obtains:

[tex]f_E\,dE = 2\sqrt{\frac{E}{\pi(kT)^3}}~\exp\left[\frac{-E}{kT}\right]\,dE[/tex] [tex]\:\:\: (2)[/tex]

.

f(E) goes to zero for E = 0. How is this possible if particles with Ei = 0 are the most frequent species?

B)

In reaction kinetics the reaction constant for one direction is given by:

[tex]k = A exp(-\beta Ea) \:\:\:(3)[/tex]

where A is the Arrhenius constant and Ea is the hight of the energy barrier for the reaction.

The term [tex]exp(-\beta Ea)[/tex] is supposed to correspond to the fraction of particles that are fast enough to get over the energy barrier.

Coming back to equation (2), shouldn't this fraction correspond to [tex]\int f(E)dE[/tex] from Ea to infinity? I don't see how one could get this from integrating (2) ?

Respectively, shouldn't taking the sum in equation (1) over all particles with Ei > Ea give this value as well?

What am i missing here?

thanks in advance!!!

Tim

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