The Gibbs Free Energy

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 EnergyExercise 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


MnCO3 (s) = Mn2+ + CO32-

The Gibbs Free EnergyHomework: Exercise 3


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.


For gaseous reactions, partial pressures can be used instead of concentration values:

KP = PCc PD

d

Paa PB

b

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.


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 ActionExercise 4

Calculate the equilibrium constant K for the following reaction:

1/2 O2 (g) + Mn2+ + H2O (l) = MnO2 (s, Pyrolusit) + 2 H+

Exercise 5

Calculate K (equilibrium constant = solubility product) for the following reaction; what is K for the respective precipitation reaction?

MnCO3 (s) = Mn2+ + CO32-

ln K = -G° / RT = -0.08 x 1000 / (8.314 x 298) = -0.0323

K = 0.97

K = 3 x 10-11 or ln K = -24.2

K (precipitation) = 1/K (dissolution) = 3.5 x 1010

The Law of Mass ActionHomework: Exercise 6

Calculate K for the following reaction:

SO42- + 9 H+ + 8 e- = HS- + 4 H2O (l)

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) with K = 108.1 take place?

K = 1034 or ln K = 78

Q = 1/(10-3 x 10-4) = 107

Q < K --> CaCO3 will precipitate.

Review: Free Energy and Equilibrium Constant

KineticsKinetics 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 rate-determining 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

KineticsExercise 8

KineticsExercise 9

KineticsHomework: Exercise 10

KineticsReaction 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.

KineticsReaction order

KineticsHomework:

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

KineticsThe 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.

KineticsBonds 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




The Gibbs Free Energy

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