The Geochemistry of Rocks and
Natural Waters
Course no. 210301
Introduction to Thermodynamics and
Kinetics
A. Koschinsky
The Gibbs Free Energy
J. Willard Gibbs used the ideas of enthalpy, entropy, and spontaneity in a concept
called free energy (G). Free energy refers to the maximum amount of energy
free to do useful work. It is related to enthalpy (H), temperature (T), and entropy
(S) by Equation
G = H – T S
Free energy is also a measure of spontaneity. Negative values of G indicate a
spontaneous or forward (reactants make products) reaction. Positive values of
G indicate a nonspontaneous or reverse (products make reactants) system. If
G = 0, the system is in equilibrium. At equilibrium, the composition of the
system (amount of products and reactants) is constant.
The free energy of a sum of a series of equations is the sum of the free energies of
those equations. One form of this is Equation
G°rxn =  n G°f,product –  m G°f,reactant
where G°f refers to the free energy of the formation reaction.
The Gibbs Free Energy
Exercise 1
Calculate G° for the following reaction:
1/2 O2 (g) + Mn2+ + H2O (l) = MnO2 (s, Pyrolusit) + 2 H+
G°f (H+) = 0 kJ mol-1
G°f (O2) = 0 kJ mol-1
G°f (MnO2, s) = -465.1 kJ mol-1
G°f (H2O, l) = -237.18 kJ mol-1
G°f (Mn2+) = -228.0 kJ mol-1
G° = -465.1 - (-237.18 + (-228.0)) = +0.08 kJ mol -1
Exercise 2
Calculate G° for the following reaction:
MnCO3 (s) = Mn2+ + CO32-
The Gibbs Free Energy
Homework: Exercise 3
Calculate G° for the following reaction:
SO42- + 9 H+ + 8 e- = HS- + 4 H2O (l)
G° = -194.2 kJ mol-1
The Law of Mass Action
In a chemical reaction not all the reactants become products. Reactions are reversible. At
the point where the rate of the forward reaction is the same as the reverse reaction,
the concentrations of products and reactants are constant. This point is chemical
equilibrium. At equilibrium, the concentrations of reactant and product are constant,
but not equal. Individual molecules of reactants and products still react, but the overall
amount does not change.
The law of mass action states that any reaction mixture eventually reaches a state
(equilibrium) in which the ratio of the concentration terms of the products to the
reactants, each raised to a power corresponding to the stoichiometric coefficient for
that substance in the balanced chemical equation, is a characteristic value for a given
temperature. For the reaction
aA + bB <--> cC + dD
the lowercase letters represent stoichiometric coefficients, A and B represent reactants,
and C and D represent products. The ratio described by law of mass action is a
constant, called the equilibrium constant (K):
KC = [C]c[D]d
[A]a[B]b
The C in KC represents concentration.
The Law of Mass Action
For gaseous reactions, partial pressures can be used instead of concentration values:
KP =
PCc PDd
Paa PBb
The relationship between KC and KP can be derived from the ideal gas law.
KP = KC(RT) n
where n is the difference in the number of moles of products (sum of their stoichiometric
coefficients) and moles of reactants, T is the temperature, and R is the universal gas
constant = 8.31 J mol-1 K-1.
The concentration of a solid or pure liquid is regarded as a constant. This is normally
combined with the equilibrium constant rather than being included as part of the
equilibrium constant expression.
The way the reaction is written affects the value of the equilibrium constant. For example,
the equilibrium constant of the reverse reaction is the reciprocal of the equilibrium
constant of the forward reaction.
The Law of Mass Action
Reactions move toward equilibrium from either the products or the reactants. If
nonequilibrium concentrations (or pressures) are used in the mass action expression,
the value is called the reaction quotient (Q). If the value of Q is smaller than K, the
reaction must go in a forward or spontaneous (–G) direction to reach the final value
(K). If the value of Q is larger than K, products must react to reach the final value (K).
The reaction goes in a reverse or nonspontaneous (+G) direction. The relationship
between free energy and the equilibrim constant is
G = G° + RT ln(Q)
R = gas constant = 8.314 J/mol•K
T = temperature (standard conditions 298 K)
At equilibrium, the rate is neither forward nor reverse, so G is zero. However, the
equilibrium constant K can be determined from the free energy at standard state.
G° = –RT ln(K)
A system at equilibrium can be perturbed by changing conditions. Le Chatelier's principle
states that if a stress (perturbation) is applied to a system at equilibrium, the system
will adjust to minimize that stress. Consequently, if reactant is added, the reaction
must go in a forward reaction to use up that reactant and minimize the stress.
The Law of Mass Action
Exercise 4
Calculate the equilibrium constant K for the following reaction:
1/2 O2 (g) + Mn2+ + H2O (l) = MnO2 (s, Pyrolusit) + 2 H+
ln K = -G° / RT = -0.08 x 1000 / (8.314 x 298) = -0.0323
K = 0.97
Exercise 5
Calculate K (equilibrium constant = solubility product) for the following reaction;
what is K for the respective precipitation reaction?
MnCO3 (s) = Mn2+ + CO32K = 3 x 10-11 or ln K = -24.2
K (precipitation) = 1/K (dissolution) = 3.5 x 1010
The Law of Mass Action
Homework: Exercise 6
Calculate K for the following reaction:
SO42- + 9 H+ + 8 e- = HS- + 4 H2O (l)
K = 1034 or ln K = 78
Homework: Exercise 7
An aqueous solution contains 10-4 M CO32- and 10-3 M Ca2+. The concentration or activity,
respectively, of a solid is defined as 1. Will the reaction
Ca2+ + CO32- = CaCO3 (s)
Q = 1/(10-3 x 10-4) = 107
Q < K --> CaCO3 will precipitate.
with K = 108.1 take place?
Review:
Free Energy and Equilibrium Constant
Review:
Free Energy and Equilibrium Constant
Review:
Free Energy and Equilibrium Constant
Review:
Free Energy and Equilibrium Constant
Review:
Free Energy and Equilibrium Constant
Kinetics
Kinetics is a term that relates to how fast a reaction occurs. Whereas thermodynamics is
concerned with the ultimate equilibrium state and not concerned with the pathway to
equilibrium, kinetics concerns itself with the reaction pathway. Very often, equilibrium in
the Earth is not achieved, or achieved only very slowly, which naturally limits the
usefulness of thermodynamics. Kinetics helps to understand why equilibrium is
occasionally not achieved.
While the rate of the forward reaction is equal to the rate of the reverse reaction at
equilibrium state, equilibrium constant expressions are not a measurement of rate. The
expression is determined from the overall reaction rather than from the ratedetermining step. The concentrations at equilibrium give no information on how long it
takes to reach that equilibrium. Catalysts will help the reaction reach equilibrium faster
but will not affect the equilibrium concentration. Instead, equilibrium concentrations
(and equilibrium constants) are related to thermodynamic parameters like G and H.
The rate of reaction is measured as the change in concentration of a product or reactant
([X]) over a given time (t). The rate of reaction for reactants is negative, since reactants
are disappearing, and positive for products, which are appearing. Rate can be
measured as average rate using the equation
Kinetics
Rate decreases over time. Therefore instantaneous rate, the rate at any given time, is
sometimes used. The instantaneous rate can be determined from a tangent line at the
relevant instant of time on a graph of concentration versus time. The instantaneous
rate at the start of the reaction (t = 0) is called the initial rate.
The relationship between concentration and rate is called the rate law. The rate is
proportional to the product of the concentration of reactants raised to some exponent:
rate = k[A]m[B]n
The proportionality constant (k) of this equation is called the rate constant. The exponents
on the reactant concentration are called the order. With the form given, m is the order
in A and n is the order in B. The sum of the exponents (m + n) is called the overall
order. The order of the reaction is normally an integer or simple fraction.
Kinetics
Kinetics
Kinetics
Kinetics
Exercise 8
Kinetics
Exercise 9
Kinetics
Homework: Exercise 10
Kinetics
Homework: Exercise 11
Kinetics
Reaction order
To characterize the affect that changing a particular reactant has on the rate of a
reaction, kineticists use the term "reaction order." When the rate of a reaction is directly
related to the concentration of a substance, it is said to be "first order" in that substance.
This is the case for radioactive decomposition.
The reaction of chlorine atoms and ozone, which has the rate law rate = k[Cl][O3] is first
order in chlorine atoms and first order in ozone. The order of the entire rate law, called
the reaction order, is the sum of all the exponents of the concentrations in the rate law.
For the above reaction, the overall order is 2.
A reaction is of zero order when the rate of reaction is independent of the concentration
of materials. The rate of reaction is a constant. When the limiting reactant is completely
consumed, the reaction stops abruptly.
A zero order reaction obeys the rate law:
-d[A]/dt = k
This type of reaction is important in enzyme catalyzed reactions.
Zero order reactions are also typically found when a material required for the reaction to
proceed, such as a surface or a catalyst, is saturated by the reactants.
Kinetics
Reaction order
Kinetics
Reaction order
Kinetics
Homework: Exercise 12
Kinetics
Homework:
Exercise 12
Kinetics
The rate law can be integrated to get a relationship between time (t) and concentration. For
a first–order reaction with a single reactant (rate = k[X]), the integrated rate law is
ln[X] = –kt + ln[X]0
where [X]0 is the initial concentration of X. The integrated rate law for a second–order
reaction with a single reactant (rate = k[X]2) is
A reaction that is first order in two reactants (rate = k[X][Y]) can be expressed as a
pseudo–first–order reaction if the concentration of one reactant is significantly
greater than that of the other. For example, if [Y] is much greater than [X], the rate
law can be expressed as
rate = k'[X], where k' = k[Y]
It is also possible to have a zero–order reaction (rate = k). For zero–order reactions, the
integrated rate law is
[X] = –kt + [X]0
Kinetics
Another way to express the rate of reaction is with the half–life. Half–life is the time
required for the reactant concentration to decrease to half its initial value ([X] =
1/2[X]0. The integrated rate laws can be used to relate the half–life (t1/2) to rate
constant (k) and initial concentration ([X]0). For a first–order reaction,
A first–order reaction is not dependent on concentration of reactant. All nuclear reactions
are first order reactions and the rates of nuclear reactions are commonly designated
by the half–life.
The half–life of a second–order reaction is
and that for a zero–order reaction is
Kinetics
The rate law is determined experimentally, rather than from the chemical reaction. This is
because the overall chemical reaction does not necessarily reflect the way in which
the reaction occurs. A mechanism is the step–by–step sequence by which a chemical
reaction occurs. Each of these elementary steps goes at a specific rate. The rate law
is determined by the slowest, rate–determining, elementary step rather than by the
overall reaction.
Reactions occur when bonds are broken and formed. The substance formed during this
process, as bonds are breaking and forming, is called an activated complex. In some
steps, an unstable substance (intermediate product) that later undergoes further
reaction is formed.
Kinetics
Bonds breaking and forming usually occur
as a result of a collision. For bond
breakage to occur in the collision, the
molecules must have sufficient kinetic
energy. The energy required to get a
reaction going is called the activation
energy (Ea). The energy difference
between products and reactants is the H
(or G) for the reaction.
The relationship between temperature (T ) and rate constant (k) is described by the
Arrhenius equation
where Ea is the activation energy, R is the gas constant (8.314 J/mol•K) and A is the
frequency factor. The frequency factor is related to how successful the collisions
between molecules are.
One way to increase the rate of a reaction is to add a catalyst. A catalyst increases the rate
of reaction without itself being consumed. It does this by lowering the activation
energy, often by directing the orientation of the colliding molecules.
Kinetics
Kinetics
For a repetition and/or more information, look at:
http://www.louisville.edu/a-s/chemistry/Chapter7.ppt