Dr Cooper's Comments: Matt - can you number the equations? Good work!
Chapter 7: Chemical Equilibrium
Black type: Matt Kershis
Introduction
When chemical species react to form a set of products, the shift in the chemical conditions of the system results in what has been classically defined as a “dynamic equilibrium”. All this is saying is that there is a point in a chemical reaction where the processes of reactants going to products and products going to reactants occur at the same rate. Thus there is still a great deal of activity in the system but there is no longer a net change in the amounts of products and reactants. The goal of this chapter is to use thermodynamic principles to quantify the amounts of reactants and products at equilibrium and to understand how changing variables such as pressure and temperature alters this composition.
Reaction Gibbs Energy and Spontaneous Chemical Reactions
Thermodynamics has shown that spontaneous processes (at constant temperature and pressure) are those that result in a net decrease in the Gibbs energy. For a reacting system, there is a point over the course of the reaction where Gibbs energy is at a minimum. The point at which the Gibbs energy is minimized is also the point at which the system is in its equilibrium state. By identifying this value for G, it is possible to learn about the chemical composition of the system at this point.
To begin to understand how this is accomplished, it is necessary to introduce a new quantity, ξ, which is called the extent of the reaction. As it has units of moles, it can be thought of as the amount of reactants lost in the conversion to products or, conversely, the amount of products created. As this quantity is usually understood in differential form, here is a mathematical summary of the previous discussion:
(1)
(2)
(3)
This concept of the extent of the reaction can be incorporated in the search for the minimum Gibbs energy (also the reaction Gibbs energy) by defining the reaction Gibbs energy as the partial derivative of Gibbs energy (of the system) with respect to the extent of the reaction at constant pressure and temperature:
(4)
Figure 1 shows where the minimum Gibbs energy exists on a plot of G vs. extent of reaction.
Figure 1: Gibbs Energy vs. Extent of Reaction
Another important consequence of reaction Gibbs energy is what it says about the chemical potential of the products and reactants. By taking a previous definition of Gibbs energy and making appropriate substitutions with equations 2 and 3, the following relationships result:
(5)
(6)
(7)
In a plot of Gibbs energy versus extent of reaction, the minimum Gibbs energy is where the slope (or the reaction Gibbs energy) is zero. Thus it becomes clear that at this point, the composition of the equilibrium reaction mixture is such that the chemical potential of the products equals the chemical potential of the reactants.
Finally, in discussing the spontaneity of chemical reactions in terms of Gibbs energy, it is necessary to define a couple of new terms. When , the forward reaction (i.e. reactants --> products) is spontaneous and the reaction is said to be exergonic. In an exergonic process, there is additional energy that becomes available to do work or drive another process. Carbohydrate metabolism is an example of an exergonic process that is a necessary precursor to other biosynthetic processes.
In thinking about how processes, especially in the biosynthetic sense, can drive others, think of two weights joined by a rope that is draped over a pulley. As the heavier weight falls due to gravitational force, it pulls the lighter weight upward. If the two weights were not linked, then they would both fall. Just like joining two weights together by a rope, coupling two reactions together can allow a process to occur that would not occur spontaneously by itself.
Figure 2: Illustration of the weight analogy
When , the reverse reaction is spontaneous (i.e. products --> reactants) and the reaction is said to be endergonic. The forward process does not occur spontaneously as a result, and the only way to drive the forward process is by adding energy to the system. Examples of useful endergonic reactions include the recharging of a battery and the electrolysis of liquid water to produce gaseous hydrogen and oxygen.
Equilibrium
Now that the thermodynamic criteria for chemical equilibrium have been defined, the equilibrium composition of a system can be defined by applying previously studied equations. In chapter 5, it was shown that that the chemical potential of a component in a system can be defined by the following when the system components are ideal gases:
(6)
Where pA is the partial pressure of species A. By combining this equation with the equations introduced here, the result is an expression that relates reaction Gibbs energy to composition under a given set of conditions (not necessarily equilibrium conditions).
(7)
(8)
(9)
The quantity Q is simply called the reaction quotient and is a unit-less number that is simply the ratio of the partial pressure of the products to the partial pressure of the reactants. It can be seen that Q = 0 when pB = 0 and Q --> ∞ as pA --> 0. The reaction quotient is simply a quantitative way of determining whether the reactants or products are favored in a given reaction.
The reaction Gibbs energy of the standard state of the system is given by the difference between the chemical potentials of A and B in their standard states. Likewise, the standard reaction Gibbs energy can be given by the standard Gibbs energy of formation for B minus the standard Gibbs energy of formation for A.
(10)
(11)
Under equilibrium conditions, the reaction quotient becomes K, the equilibrium constant for the reaction. K is still equal to the ratio of the partial pressures, except that now the partial pressures are those that exist under equilibrium conditions. Since it is known that at equilibrium, the following equations may be derived:
(12)
Blue type: Christopher Kazepis
A chemical reaction of known stoichiometry can be written as:
(Eq. 1)
For the reaction products Y and Z the numbers y and z are known as the stoichiometric numbers nY and nZ, for Y and Z respectively. For the reactants, the stoichiometric numbers are the negatives of the numbers appearing in the equation. Basically, the stoichiometric number nA for the reactant A is -a. This means that the stoichiometric numbers are positive for products and negative for reactants.
(Eq. 2)
Equation 2 is a generic equation that can be simplified by subtracting the products from the reactants.
0 = 3C + D - 2A - B (Eq. 3)
Equation 3 is the symbolic expression of Equation 2. By subtracting the reactants from both sides, and then replacing the arrow with an equals sign, the equation changes into:
(Eq. 4)
In Equation 4, J indicates the substances, and the nJ are corresponding stoichiometric numbers in the chemical equation.
In order to express the following equation:
(Eq. 5)
in the form of Equation 4, it must be arranged as:
0 = 2 NH3(g) - N2(g) + 3 H2(g) (Eq. 6)
The stoichiometric numbers are then defined as: nNH3 = +2, nN2 = -1, and nH2 = -3
This yields that, if initially there is 10 mol of N2 present, the extent of reaction changes from x = 0 to 1 mol (Dx = +1 mol), and therefore the amount changes from 10 mol to 9 mol. Additionally, when Dx = +1 mol, then the amount of NH3 changes by +2 mol, and H2 changes by -3. All the N2 is consumed when Dx = +10 mol.
Gibbs energy of reaction can be written as:
(Eq. 7)
The standard reaction Gibbs energy equation is easily calculated through subtracting the reactants from the products, or therefore is just notably:
(Eq. 8)
Q, the reaction quotient is evaluated as:
Q = (activities of products)/(activities of reactants) (Eq. 9)
The symbol Õ is a general expression of Q, used to indicate the product of what follows it. Therefore, Q is defined as:
(Eq. 10)
When considering the reaction quotient,
(Eq. 11)
where nA = -2, nB = -3, nc = +1, and nD = +2, the reaction quotient then becomes:
Q = a-2A (Eq. 12)
At equilibrium, the slope of G is zero. The activites, therefore, have their equlibrium values leading the expression:
(Eq. 13)
Equilibrium values are used for K, whereas for Q, the values at the specified stage of the reaction are used. Thermodynamic equilibrium constant is K expressed in terms of activities. Activities are actually dimensionless numbers, and therefore the thermodynamic equilibrium constant is also dimensionless.
Furthermore, in the activities that occur in Equation 13, or elementary applications, can be replaced by the numerical values of molalities, molar concentrations, or the values of the partial pressures. Thereby, the resulting expressions are merely approximations.
(Eq. 14)
Equation 14 is a prominent thermodynamic relation, for it allows the prediction of the equilibrium composition of the reaction mixture. The value of DrGq, which is identified as a single, standard pressure, influences the equilibrium constant. Thereby making K independent of the pressure.
(Eq. 15)
The relation between equilibrium constants can be determined by utilizing Equation 15, where all four species are solutes. In elementary applications, Kg = 1, and therefore K » Kb.
(Eq. 16)
Equation 16 expresses the independence of the value of DrGq, and therefore K. However, it is vital to consider that although K is independent of the pressure, the equilibrium composition is not necessarily independent of the pressure, because it depends on how the pressure is applied. The pressure within a reaction vessel has the potential to increase by injecting an inert gas. When considering this example, it can be found that the presence of another gas does not change the equilibrium composition because the partial pressure of each reacting gas molecules did not change with the inert gas. Yet, confining the gases to a smaller volume through compression increases the pressure.
(Eq. 17)
(Eq. 18)
Equation 17 is the perfect gas equilibrium, for which the equilibrium constant is Equation 18. PA must increase greatly in order to cancel out the increase in the square of PB, which can therefore keep the right hand side of Equation 18 constant. In order for PA to increase, however, the equilibrium composition must shift in favor of A, inevitably at the cost of B. Therefore, as the volume decreases, the number of A molecules will increase.
When there is an increase in A molecules with the corresponding decrease in B molecules in the equilibrium represented in Equation 17, a special case of Le Chatelier’s principle is demonstrated. The principle states that if a system at equilibrium is compressed, then the reaction will adjust in order to minimize the increasing pressure. This is done by reducing the number of molecules in the gas phase, thereby illustrating, A ¬ 2B.
(Eq. 19)
Equation 19 simplifies into the equation:
(Eq. 20)
In order to treat the effect of compression quantitatively, suppose there is an amount of n of A present initially (without B). At equilibrium, the amount of A is (1-a)n, and the amount of B is 2an, where a is the extent of dissociation of A into 2B. Equation 19 simplifies in Equation 20, following the mole fractions present at equilibrium.
(Eq. 21)
Equation 21 simplifies into:
(Eq. 22)
(Eq. 23)
Equation 22 is the equilibrium constant for the reaction, which then rearranges into Equation 23. The formula demonstrates that although K is independent of pressure, the amounts of A and B do depend on pressure. Furthermore, as according to Le Chatelier’s principle, as p is increases, a decreases.