An Analogy for Chemical Equilibrium,DG, and DG°

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We will consider the apparatus below as a chemical reactor in which we can carry out the reaction:

H₂O(left)  -> H₂O(right)
 

We will work with 90 cm³of water, which is approximately 5.0 moles.

Note that the cylinder on the right has a larger diameter than the cylinder on the left.
We will take the area of the left cylinder as 10 cm², and that of the right cylinder as 20cm².

We can start with all of the water on either the left or the right.
 

When the stopcock is opened,the outcome will be the same:

This is the nature of Chemical Equilibrium -
whether you start with only reactants or only products,
the system comes to equilibrium to satisfy some inherent property.

The equilibrium condition could have been determined by the relative volumes,
relative masses, relative potential energy, or some other property of the system.

In this case, the equilibrium is determined by the relative height of the liquid
in the two cylinders.

(More precisely, the equilibrium is determined by equal pressures on both sides of the stopcock.)

Irrespective of the starting condition, water will move from the cylinder
with the higher level to the cylinder with the lower level.
 

In a chemical system at fixed temperature and pressure,
the reaction proceeds so as to lower the Gibbs Free Energy (G) of the system.

The equilibrium condition is determined by the minimum free energy of the system.
However, our description of the equilibrium condition involves
an equilibrium constant (K) which is related to
the relative amounts of the reactants and products.

The situation is further complicated by the fact that the value of the equilibrium constant
is determined by the Gibbs free energy required (DG^o)
to convert a specific amount of reactants to a specific amount of products
(specified by the balanced chemical reaction).

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Let's try to apply the ideas of chemical equilibrium to this very simple system.

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In this very simple case of a "reaction",

H₂O(left)  -> H₂O(right)

there is no difference between Gibbs Free Energy (G) and Potential Energy (E),
so we will work with the simpler quantity, E.

The potential energy of an object is the product of its mass (m),
the gravitational constant (g= 9.80 m/sec²), and the height of its center of gravity (h):

E = mgh  .

Using mass values in grams and height values in meters
gives the potential energy in milliJoules (mJ).

For a column of water of height H(meters), volume V (cm³), and Density (g/cm³):

m = DV    ;     h = H/2

E = gDVH/2  .

The energy is different in the left and right columns,

E_L = gDV_LH_L/2   ;   E_R = gDV_RH_R/2

E_total = E_L+ E_R  = (gD/2)(V_LH_L + V_RH_R)

While the transfer is most easily considered in terms of volume,
Chemists prefer to work with the Chemical Amount (moles):

moles = DV/M   (M =Molar Mass in grams) .

We can then describe the energies on the left and right in molar terms:

(E_m)_L =gMH_L/2   ;    (E_m)_R= gMH_R/2

The difference between these two terms is called DE_rx:
DE_rx=(E_m)_R -  (E_m)_L = (gM/2)(H_R- H_L)

Now we consider the changes in these quantities as we allow the liquid
to flow from left to right until it reaches equilibrium,
then we will force all of the liquid to the right.

The equilibrium condition of equal heights of the two columns
now has two additional conditions:

1.  The total energy hasa minimum value at this point.
2. The energy/mole has the same value in both columns.

The quantity DE_rx is related to the driving force of the process
(the difference in heights of the two columns):
When this quantity is negative, the "reaction" proceeds as it is written

H₂O(left)  -> H₂O(right)
and if this quantity is positive the reverse "reaction" is favored.

While chemical reactions are much more complex than this simple transfer,
the equilibrium condition is very similar:

1. The total Gibbs Free Energy of the system has a minimum value at equilibrium.
2. The sums of the Partial Molar Gibbs Free Energies (Chemical Potentials) are
the same for the Reactants and the Products (each multiplied by its
stoichiometric constant).

A quantity DG_rx  is defined as the difference between the
molar Gibbs free energies of the products and the
molar Gibbs free energies of the reactants.

When this quantity is negative, the reaction proceeds as it is written,
and if this quantity is positive, the reverse reaction is favored.

One other point may be made about this system:
The standard energy change (DE°) for the reaction

H₂O(left)  -> H₂O(right)
is given by the difference in the molar energy when all of the water
is on the right (3.97 mJ/mole) and when all of the water is on the left
(7.94 mJ/mole):

DE°= - 3.97 mJ/mole

This represents the energy (mechanical work) per mole of water
required to convert  the initial state (all on the left) to
the final state (all on the right).

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For Chemical Reactions, there is a DG°, which represents the
Chemical Work required to convert the reactants
completely to the products at constant T & P.

This quantity is not to be confused with DG_rx, which represents the
Chemical work required to make this conversion
while the reaction is in progress.

 If DG°is negative, the equilibrium condition will favor
products over reactants
and the opposite is true if DG° is positive.
 

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