Equilibrium Between Particles i

8/3/2019 Equilibrium Between Particles i

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By:

Hunh Ngc TnMai Trung Hiu

EQUILIBRIUM BETWEEN

PARTICLES I


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OUTLINE:A. Free Energy and Chemical Potential

B. Absolute Entropy of an Ideal Gas

C. Chemical Potential of an Ideal GasD. Law of Atmospheres

E. Physical Interpretations of Chemical Potential


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How many states of

Physical Equilibriumin our life?


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States of equilibrium There are three states of physical equilibrium:

Stable Equilibrium

Unstable Equilibrium

Neutral Equilibrium


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A. Free Energy and Chemical Potential

We consider the equilibrium between systems that canexchange particles. A wide variety of important

problems involve particle exchange between systems

at temperature T , for example, ionization of atoms,

chemical reactions , dissociation of molecules...


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Free Energy and Chemical Potential (0)

An isolated system is in equilibrium when its total entropy is a maximum. In adddition, if the isolated system

consists of a small system in thermal equilibrium with a

reservoir at temperature T, then equilibrium is determined

by a minimum in the free energy F of the small system, sowe have the following equations:


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Equilibrium corresponds to maximum Stot = Sreservoir + Ssmall systemFree energy Fsys = UsysTreservoirSsys

This is the maximum available work we can get from a system

that is connected to a reservoir (environment) at temperature

Treservoir.


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If two systems in equilibrium with a reservoir at temperature Tare allowed to exchange particles, so we have the equilibrium

condition is:

We saw that minimizing F is equivalent

to maximizing Stot, but with the

advantage that we dont have to deal

explicitly with Sreservoir .


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Whereas, we have:

For two subsystems exchanging particles (one for one). Then

we consider that the condition for chemical equilibrium is:

Finally, we have a conclusion that is so important:

The chemical potential of a system equals the change in free

energy when one particle is added to the system at constant

volume.


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B. Absolute Entropy of an Ideal Gas

The entropy S of a monoatomic ideal gas can be expressedin a famous equation called the Sackur-Tetrode equation.

where_m : mass of monoatomis gas_ k : Boltzmann's constant_ T : Kenvin degree

_ h : Plancks constant_ p : pressure


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Absolute Entropy of an Ideal Gas (1)

The above equation is known as the quantumdensity, which we identify as the density of

quantum cells - the number of cells per unit

volume.


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C. Chemical Potential of an Ideal Gas

We consider that will apply the Principle of MinimumFree Energy to a variety of practical problems involving

the equilibrium between two or more subsystems. In each

case, at least one of the subsystems is an ideal monoatomic

gas, whose entropy is:


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Chemical Potential of an Ideal Gas (1)

With U= (3/2)NkT, the free energy, F= U-TS, of themonoatomic gas is

which simplifies to the compact form:


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Following to the above formula, we obtained the formula ofthe chemical potential of an ideal monoatomic gas by

taking derivative of with respect

to N

where we have substituted n=N/V as the density of particles


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Example:

For one mole of Ar gas at p= 1atm and T =300K, we

have kT=0.026 eV, nQ =1030 x (40)3/2 m-3 and n=2.45x1025 m-

3,yielding a chemical potential,

= (0.026eV)ln(9.8 x 10-8) = -0.42 eV

The following graph of (T):


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If two subsystems with the same exchange a

particle, F remains unchanged (a minimum),

implying that the two subsystems are inequilibrium.


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D. Law of Atmospheres

The molecules in the upper box each have apotential energy of mgh; therfore,

Where as, we have

Setting 1= 2 yields,


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E. Physical Interpretations of Chemical Potential

As the piston is allowed to move

isothermally from volume V1 to V2 ,the

work done by each particles may be simply

viewed as the change in chemical potential,

= = ln2

1

= ln(

2

1)


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Equilibrium Between Particles i