Principle of Microscopic Reversibility - Chemical
equilibrium
One can determine the expression for K from the balanced chemical
equation:
aA + bB --> cC + dD
K = ([C]c[D]d)/([A]a[B]b
Unlike kinetics, the coefficients found in the balanced equation are
automatically the exponents even for reactions which involve more than
one step (non-elementary reactions).
For example a two-step mechanism for the reaction 2A + Z --> C:
(1) A + A--> B
(2) B + Z--> C
The reverse reaction will look like:
(1) C --> B + Z
(2) B --> A + A
This is the principle of microscopic reversibility: the forward and reverse of a
particular pathway are related as mirror images. Thus, each mechanistic step
must be at equilibrium when the whole system is at equilibrium:
(1) A + A<===> B
(2) B + Z <====> C
k1 [A]2 = k-1[B]
and
k2 [B][Z] = k-2 [C]
The equilibrium expressions for the two steps are as follows:
k1 / k-1 = [B]/[A]2
k2 / k-2 = [C]/([B][Z])
Multiplying the two equations yields:
(k1k2 / k-1k-2) = [C]/([A]2[Z])
or
K = [C]/([A]2[Z])
the equation one gets directly from the stoichiometry of the reaction: 2A + Z --> C
It should be clear how an understanding of kinetics helps in understanding
chemical equilibrium. Now, let us move a little bit off-tangent and begin asking
what a catalyst does to a system in chemical equilibrium. From the previous
topic, we know that catalysts provide an alternate mechanism or route with lower
activation energy. Using the Arrhenius equation, the rate constant for the forward
reaction can be expressed as follows for the catalyzed reaction:
kf = Af exp(-(Eaf - C)/RT)
where C is the amount the energy of activation for the forward reaction (Eaf) is
lowered. The rate constant of the reverse is likewise affected and the Energy of
activation (Ear) is lowered by the same amount, C. Thus, the rate constant for the
reverse reaction is given by:
kr = Ar exp(-(Ear - C)/RT)
The uncatalyzed K is given by:
kf / kr = (Af exp (-Eaf/RT) / Ar exp (-Ear/RT))
kf / kr = (Af/Ar) exp (-(Eaf - Ear)/RT)
For the catalyzed reaction, K is given by:
kf / kr = (Af exp (-(Eaf - C)/RT) / Ar exp (-(Ear - C)/RT))
which obviously yields the same equation:
kf / kr = (Af/Ar) exp (-(Eaf - Ear)/RT)
The equilibrium constant depends only on the difference between the energies of
the reactants and the products as reflected in the difference between Eaf and Ear.
If you go back to the energy diagram for the reaction, the difference between the
energy of activation of the forward and the reverse reaction is exactly equal to
the difference between the energy of the reactants and the energy of the
products. The introduction of a catalyst speeds up both forward and reverse
reactions by the same amount and, thus, has no effect on chemical
equilibrium.
The textbook devotes a great deal of discussion regarding the Haber process:
This process involves the production of ammonia from the elements, nitrogen
and hydrogen:
N2(g) + 3H2(g) <----> 2NH3(g)
Epilogue: Remarks on Science and the Social Order (taken from Leonard K.
Nash (Emeritus Professor of Chemistry at Harvard University) Elements of
Chemical Thermodynamics, Mass.: Addison-Wesley Publishing Company, Inc.,
1962, pp.89-90)
Haber began his studies of the synthesis of ammonia in 1904. On the basis of
calculations like that given above (equations which make use of constant of
equilibrium expressions), confirmed by a great many experiments, he came to
conceive the process for which he took the key patent in 1908. This patent
describes a continuous recirculation of the process gas --- using heat exchangers
but maintaining throughout the same high pressure --- in such fashion that even
at low equilibrium concentration of ammonia is removed continuously, by
condensation, while the remaining process gas is reheated and passed again
over the catalyst. By 1909 Haber was able to demonstrate, to a representative of
the Badische Anilin und Sodafabrik, a tiny pilot system producing about 80 grams
of liquid ammonia per hour. The many remaining problems of the synthesis were
solved in the laboratories of this concern which --- five years later, in 1914 --- had
managed to bring the Haber process into quantity production.
That date is significant. All the high explosives used in the World War of 19141918?absolutely require nitric acid which, at the outbreak of the World War I, was
derived almost exclusively from Chile saltpeter, NaNO3. A country cut off from its
supply of this mineral, as Germany was by the British blockade, could, however,
still obtain nitric acid if only it could fix atmospheric nitrogen as ammonia, since
ammonia can be converted into nitric acid by a series of steps summarized in the
equation
NH3 + 2O2 --> HNO3 + H2O
The Haber process was brought into production not a minute too early for
Germany. Because of the Haber process, Germany avoided early defeat, the war
was prolonged. Is it a blessing so to be saved? Prolongation of the war so
depleted Germany, and so embittered her foes, that the postwar situation may be
thought to have made inevitable the rise of a Hitler. If this be so, the Haber
process was no blessing for Germany, and certainly no blessing for Haber, who,
with the rise of Hitler, became "the Jew Haber" driven to his death in exile.
In the late 19th century there were prophets of disaster who foretold the early
doom of urban civilization, an early resurgence of chronic famine, consequent to
the ultimately inevitable exhaustion of the supply of Chile saltpeter, the major
source of fixed nitrogen for agricultural fertilizer. The Haber process frees us
once and for all from this threat?
?The scientist makes an ethical judgment, and assumes a moral responsibility,
when he elects to participate in the technological exploitation of science for
destructive purposes. "Social demand" may applaud, but cannot justify, such a
decision --- any more than it can the decision of the smith who turns iron into
swords rather than ploughshares. But science, scientific knowledge, is ethically
as neutral as the iron: "evil"; only when men forge it as a sword; "good" when
beaten into a ploughshare. Conceivably there is some knowledge that can lead
only to "evil"; certainly there is none that can lead only to "good"? One can never
deny the possibility of what Sophocles though a certainty, "No great thing ever
enters human life without a curse." Unlike the Greeks, however, we have an
abiding faith in the possibility of ameliorating the human condition which
emboldens us to push on in science, and elsewhere, with Whitehead's conviction
that: "Panic of error is the death of progress."
Le Chatelier's Principle - Chemical equilibrium
(Historical) Law of Mass Action
In 1864, Cato Maximilian Guldberg and Peter Waage postulated the expression
for the equilibrium constant.
aA + bB --> cC + dD
K = ([C]c[D]d)/([A]a[B]b
The system will try to obey the above equation at all times. This was illustrated in
the last lecture in the case of HI being formed from hydrogen and iodine and
vice-versa.
Important considerations:
(1) The equilibrium constant is normally dimensionless eventhough its evaluation
may produce units.
(2) The notation Kc is normally used to denote that the equilibrium constant refers
to the expression in which the amounts of materials are expressed in molar
concentrations.
(3) For a particular reaction, the value of the equilibrium constant varies with
temperature. Remember,
K = kf / kr = (Af/Ar) exp (-(Eaf - Ear)/RT)
(4) The notation Kp is used when the amounts of the materials are expressed as
gas pressures.
(5) For the value of an equilibrium constant to be meaningful, we must specify
how the equilibrium reaction is written, i.e., which species are written on the
product side and which are written on the reactant side. The equilibrium
expression for a reaction written in one direction is the reciprocal of the one for
the reaction written in the reverse direction.
Quantitative considerations:
When K >> 1, formation of products is favored
When K << 1, reactants are favored
Relating Kcand Kp:
The ideal gas law: PV = nRT
rearranges into: P = (n/V)RT
for gases (n/V) will be the molar concentration, [C]:
Thus, P = [C]RT
and Kp = Kc(RT)(ngas in products-ngas in reactants)
Question: What can affect chemical equilibirium?
Le Chatelier's Principle:
NaCl(s) <==> Na+(aq) + Cl-(aq)
In the laboratory, a saturated aqueous solution of NaCl was prepared. A
saturated aqueous solution means that the maximum
amount of NaCl which can be dissolved in a given amount of water is present.
Additional crystals of NaCl when added to a
saturated solution will not dissolve. The constant of equilibrium expression for the
dissolution of NaCl in water is given by:
Kc = [Na+][Cl-]/[NaCl(s)]
The above equation contains the concentration of NaCl in solid NaCl. This
concentration is related to the density of solid NaCl
and its molar mass. Since the density of NaCl or of any pure solid or pure liquid
does not vary with the extent of a reaction, the
concentration of any pure solid or liquid can be regarded as a constant
and, thus, can be further absorbed into the equilibrium constant:
Kc' = [Na+][Cl-]
where Kc' = Kc[NaCl(s)]. For dissolution processes, this constant is given a
special name: Constant of Solubility Product,
and the symbol Ksp.
Ksp of NaCl = [Na+][Cl-]
The constant of solubility product is, in fact, another statement of solubility. The
higher the Ksp value is the higher the solubility.
Ksp gives the highest concentration for dissolved species before crystallization or
precipitation will occur. If the product
[Na+][Cl-] is greater than Ksp of NaCl, NaCl crystals will be formed. Thus, in a
saturated solution Ksp = [Na+][Cl-]. The
dynamic equilibrium can be easily disturbed by changing any one of these
concentrations: either [Na+] or [Cl-].
How does the system respond to a disturbance, the answer is given by Le
Chatelier's Principle:
If a system at equilibrium is disturbed by a change in temperature,
pressure, or the concentration of one of the
components, the system will shift its equilibrium position so as to
counteract the effect of the disturbance.
-Le Chatelier's Principle (1888)
"It is known that in the blast furnace the reduction of iron oxide is produced by
carbon monoxide, according to the reaction:
Fe2O3(s) + 3CO(g) <==> 2Fe(s) + 3CO2(g)
but the gas leaving the chimney contains a considerable proportion of carbon
monoxide,?. Because this incomplete reaction was thought to be due to an
insufficiently prolonged contact between carbon monoxide and the iron ore
(confusing a problem with equilibrium with that of kinetics), the dimensions of the
furnaces have been increased. In England they have been made as high as thirty
meters. But the proportion of carbon monoxide escaping has not diminished, thus
demonstrating, by an experiment costing several hundred thousand francs, that
the reduction of iron oxide by carbon monoxide is a limited reaction.
Acquaintance with the laws of equilibrium would have permitted the same
conclusion to be reached more rapidly and far more economically."