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Maxwell's Distributions law

Derivation of The Fundamental Equation of The Kinetic Theory

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Kinetic-Molecular Theory (model & fundamental laws)

N number of molecules of ideal gas
m mass of each molecules present in a cube has length l cm
Each molecule move in deferent speed u₁, u₂, u₃, u_N (cm s^-1)
Each molecule move in three dimensions (u₁_x, u₁_y, u₁_z) and each molecule make one collision when it cut distance l cm

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Derivation of kinetic energy based on temperature only

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Derivation of ideal gas speed from Boltzmann Distribution and Maxwell-Boltzmann Distribution

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Distribution of Molecular Speeds in an Ideal Gas

Root mean square speed is assumed that all molecules move at the same speed.

The motions of gas molecules should have distribution of molecular speeds in equilibrium.

Statistical mechanics help us to understand gas molecules behaviour.

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Maxwell-Boltzmann Distribution

The M-B distribution function can be written in terms of energy as:

The distribution function can be written in terms of velocity:

But the velocity is in three dimensional:

u = u_x, u_y , u_z and du = du_x.du_y .du_z = d³u

The constant A is determined by normalization. We will treat this as a probability distribution function normalized so that:

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So:

From calculus :

Therefore:

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dN_i : number of gas molecules that have speed between u and u+du

N: total number of molecules

k: boltzmann constant which equal to R/N_A

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Let

mN_A= M

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To find the most probable speed u_P :

By differentiating f with respect to u and looking for the value of u at which the derivative is zero

i.e.:

Therefore:

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To find the root mean square (rms) speed u_rms:

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Derivation of The collision frequency and the mean free path

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The collision frequency and the mean free path

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To know how many collision that happened by molecules let suppose a vessel its volume is V and contains gas molecules A. The diameter of A molecule is d and move in average speed u in a cylindrical path its diameter is twice of A (i.e. 2d). Also supposed all other molecules which collide with A molecule is solid.

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Therefore the number of collision that happed by one A molecules with B molecules:

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