equilibrium between particles i
TRANSCRIPT
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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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Free Energy and Chemical Potential (1)
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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Free Energy and Chemical Potential (2)
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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Free Energy and Chemical Potential (3)
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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Chemical Potential of an Ideal Gas (2)
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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Chemical Potential of an Ideal Gas (3)
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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Chemical Potential of an Ideal Gas (4)
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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