Maxwell Boltzmann Statistic

Ideal Gas Heat Type MonatomicPUTU YULIA ANGGA DEWI1213021016Suppose a monatomic gas consisting of N molecules, each of mass m, in an enclosed space whose volume V. Each molecule is characterized by the position coordinates x, y, z, and coordinate velocity vx, vy, and vz.

If among the molecules no style that works, then there is no potential energy between molecules together. In this discussion of the effect of gravitational forces of the earth is not taken into account. Thus the potential energy can be considered constant and taken nol.Kenyataan showed that the walls of the vessel impenetrable molecules, thus the potential energy can be considered infinite for position coordinates x, y, z which is located outside the vessel.If the molecules can be considered a point object, then the kinetic energy of the molecules only translational kinetic energy. For a cell to i, which coordinates the velocity is vx, vy, and vz, kinetic energy is

Then for any cell in the space occupied by the gas:wi =

and for cell out the space is : wi =

Partition function is given by:Z = S exp(-

/2kT)To calculate the partition function is taken the following steps.Divide the phase space into cells with the same volume of H = dxdydzdvxdvydvz.Change sigma in equation (1) with integral, differential multiplied by six and shared with H.

Equation (5) states the number of molecules per unit volume of a regular space is constant, irrespective of the position and the same as the total number of molecules N divided by the total volume V. In other words, molecules are uniformly distributed in the space occupied by the gas ,

To get distribution in velocity space we integration equation (4) for all x, y, and z, because

Equation of state can be obtained from the Helmholtz function, F = -NkTlnZ (k is the Boltzmann constant) and of the relationship with the style of the thermo-dynamic pressure, namely: p = -

ideal gas equation can be determined as follows.PV = NkT = nRT Furthermore, entropy (S) can be expressed by V and T as follows. S =Nk ln Z + U/TWith U can be determined:

s = R lnV + 3/2 R lnT + RAU system's internal energy of the gas is given by:U = NkTandlnZ =lnV -lnH + 3/2ln(2pk/m)+ 3/2ln(T)The molar specific heat at constant volume are:

But the result is matched with a price above and the prices found in the kinetic theory of gases. Maxwell Boltzmann Statistic For Barometric Equation AND Principle of Equipartition of EnergyBAROMETRIC EQUATION

By subtituting then it is obtained that

And integrating over z yields

Untuk dua lapisan yang tebalnya sama pada ketinggian z1 dan z2 , jumlah relatif molekul-molekulnya adalah : = ex[-mg(z2-z1)/kT] By integrating the previous equation with respect to vx vy and vz yields

with

Recalling that

By stating that po expressing the pressure at z =0 (earth surface), thus:

And finally the equation for P is given byPRICIPLE OF EQUIPARTITION OF ENERGYThe energy (w) of a molecule is a function of all cell coordinate in phase space where it exists, from the equation

Which can be written asdNz = A exp (-wz/kT) dz

The total number of molecules is given byAnd the total energy Uz associated to z coordinate is given by

With average of energyIf the energy wz is a quadratic function of z, in form wz = az2 Therefore

Based on classical theory of specific heat of diatomic gases, the molecules described as the dumbbell is composed of two atoms (treated as a point mass) have a certain separation distanceSpecific Heat of Diatomic GasesMolal specific internal energy (u) is given by:u = U/n = 3/2 RT + RT + RT

Molal specific heat thus obtained:cv = =(3/2)R +R + R =(7/2)Rcv = (cv) trans + (cv) rot + (cv) vib

rotational energy vibrational energy

Partition function for rotational and vibrational: Zvib=exp(-wvib/kT)= exp-[(n+1/2)hf/T) = exp-[(n+1/2)vib/T)wvib = (n+1/2)hfwot = n(n+1)

Total energy at any temperature is given by:U = wi N I

Can be calculated based on the equation:

It is obtained:

The average energy to an oscillator (single oscillator) is:

At extremely high temperatures, precious exponential rate is very small:

Thus:

molar vibration energy that is:

molar specific heat for vibration are:

the temperature characteristics of the rotation is:

Theory of diamagnetism and paramagnetism begins by assuming that the molecule consists of one or more electrons that spin in a circular orbit and spin on each axis, thus the molecule is a system with magnetic moment and angular momentum.Theory of Paramagnetismmagnetic moment per unit volume is:

In a strong field and low temperature with x = (B/kT):In weak fields and high temperatures x very low

for conformity with the law of Curie, the constant C is equal to:

If the Curie law is fulfilled:

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