Standard state equilibrium constants

Enthalpy changes for biochemical processes can be determined experimentally by measuring the heat absorbed (or given off) by the process in a calorimeter (Figure 3.2). Alternatively, for any process B at equilibrium, the standard-state enthalpy change for the process can be determined from the temperature dependence of the equilibrium constant ... 

The value of the standard free energy AG depends on the choice of reference state, as does the equilibrium constant. Thus it would be safer to write the equilibrium constant K for a gaseous reaction as... 

The standard-state electrochemical potential, E°, provides an alternative way of expressing the equilibrium constant for a redox reaction. Since a reaction at equilibrium has a AG of zero, the electrochemical potential, E, also must be zero. Substituting into equation 6.24 and rearranging shows that... 

Balance the following redox reactions, and calculate the standard-state potential and the equilibrium constant for each. Assume that the [H3O+] is 1 M for acidic solutions, and that the [OH ] is 1 M for basic solutions. 

Preparation and chemistry of chromium compounds can be found ia several standard reference books and advanced texts (7,11,12,14). Standard reduction potentials for select chromium species are given ia Table 2 whereas Table 3 is a summary of hydrolysis, complex formation, or other equilibrium constants for oxidation states II, III, and VI. 

Equilibrium Constants For practical application, Eq. (4-336) must be reformulated. The initial step is elimination of the in favor of fugacities. Equation (4-74) for species i in its standard state is subtracted from Eq. (4-77) for species i in the equilibrium mixture, giving... 

The equilibrium constant can be determined at any temperature from standard state information on reactants and product. Considering the synthesis of NH3, the equilibrium conversion can be determined for a stoichiometric feed of Hj and Nj, at the total pressure. These conversions are determined by the number of moles of each species against conversion X by taking as a basis, 1 mole of N2. 

The numerical values of AG and A5 depend upon the choice of standard states in solution kinetics the molar concentration scale is usually used. Notice (Eq. 5-43) that in transition state theory the temperature dependence of the rate constant is accounted for principally by the temperature dependence of an equilibrium constant. 

In any of these forms, this relationship allows the standard-state free energy change for any process to be determined if the equilibrium constant is known. More importantly, it states that the equilibrium established for a reaction in solution is a function of the standard-state free energy change for the process. That is, AG° is another way of writing an equilibrium constant. 

For biochemical reactions in which hydrogen ions (H ) are consumed or produced, the usual definition of the standard state is awkward. Standard state for the ion is 1 M, which corresponds to pH 0. At this pH, nearly all enzymes would be denatured, and biological reactions could not occur. It makes more sense to use free energies and equilibrium constants determined at pH 7. Biochemists have thus adopted a modified standard state, designated with prime ( ) symbols, as in AG°, AH°, and so on. For values determined... 

Through all these calculations of the effect of pH and metal ions on the ATP hydrolysis equilibrium, we have assumed standard conditions with respect to concentrations of all species except for protons. The levels of ATP, ADP, and other high-energy metabolites never even begin to approach the standard state of 1 M. In most cells, the concentrations of these species are more typically 1 to 5 mM or even less. Earlier, we described the effect of concentration on equilibrium constants and free energies in the form of Equation (3.12). For the present case, we can rewrite this as... 

Hexokinase catalyzes the phosphorylation of glucose from ATP, yielding glncose-6-P and ADR Using the values of Table 3.3, calculate the standard-state free energy change and equilibrium constant for the hexokinase reaction. 

You may wonder why the equilibrium constant, 11, has no units. The reason is that each term in the reaction quotient represents the ratio of the measured pressure of the gas to the thermodynamic standard state of one atmosphere. Thus the quotient (f3No2)2/f>N2o4 in Experiment 1 becomes... 

Redox reactions, like all reactions, eventually reach a state of equilibrium. It is possible to calculate the equilibrium constant for a redox reaction from the standard voltage. To do that, we start with the relation obtained in Chapter 17 ... 

The determination of ArG° for a chemical reaction is very useful in predicting the course of the reaction. Qualitatively, we will show in Chapter 5 that with ArC°<0, the reaction is spontaneous, at least when products and reactants are in their standard state condition. Quantitatively, we will see in Chapter 9 that ArG° can be used to calculate the equilibrium constant for the reaction, from which the final equilibrium conditions can be determined. 

Alternative forms of the equilibrium constant can be obtained as we express the relationship between activities, and pressures or concentrations. For example, for a gas phase reaction, the standard state we almost always choose is the ideal gas at a pressure of 1 bar (or 105 Pa). Thus... 

It is important to note that, for any given temperature, the [thermodynamic] equilibrium constant is directly related to the standard change in free energy. Since, at any given temperature, the free energy in the standard state for each reactant and product, G°, is independent of the pressure, it follows that the standard change in free energy for the reaction, AfG°, is independent of the pressure.g Therefore, at constant temperature, the equilibrium constant K. .. is also independent of the pressure. That is,... 

In the introductory chapter we stated that the formation of chemical compounds with the metal ion in a variety of formal oxidation states is a characteristic of transition metals. We also saw in Chapter 8 how we may quantify the thermodynamic stability of a coordination compound in terms of the stability constant K. It is convenient to be able to assess the relative ease by which a metal is transformed from one oxidation state to another, and you will recall that the standard electrode potential, E , is a convenient measure of this. Remember that the standard free energy change for a reaction, AG , is related both to the equilibrium constant (Eq. 9.1)... 

We will see functions like the one occurring under the logarithm operator quite often. For efficiency, this is generally written as In(Products)/(Reactants), where (Products) and (Reactants) denote the partial pressures of the species relative to the standard state pressure raised to a power that is equal to the stoichiometric coefficients. Kp is the equilibrium constant in terms of pressures. Since all pressures are in the same units, Rp is dimensionless. Note that in some literature there may be a combination of some power of P with Kp to obtain an equilibrium constant with pressure units. In this case. 

These four equations are perfectly adequate for equilibrium calculations although they are nonsense with respect to mechanism. Table 7.2 has the data needed to calculate the four equilibrium constants at the standard state of 298.15 K and 1 bar. Table 7.1 has the necessary data to correct for temperature. The composition at equilibrium can be found using the reaction coordinate method or the method of false transients. The four chemical equations are not unique since various members of the set can be combined algebraically without reducing the dimensionality, M=4. Various equivalent sets can be derived, but none can even approximate a plausible mechanism since one of the starting materials, oxygen, has been assumed to be absent at equilibrium. Thermodynamics provides the destination but not the route. 

Enzymes accelerate reaction rates by lowering the activation barrier AGp. While they may undergo transient modification during the process of catalysis, enzymes emerge unchanged at the completion of the reaction. The presence of an enzyme therefore has no effect on AG for the overall reaction, which is a function solely of the initial and final states of the reactants. Equation (25) shows the relationship between the equilibrium constant for a reaction and the standard free energy change for that reaction ... 

Although not stated explicitly, each concentration in a reaction quotient and in an equilibrium constant expression has been divided by standard concentration (1 bar for gases, 1 M for solutes) to make the equilibrium constant dimensionless. For example,... 

All in aqueous solution at 25°C standard states are 1 M ideal solution with an infinitely dilute reference state, and the pure liquid for water equilibrium constants from reference 100, except as noted. 

Here, the a s refer to the activities in the chosen arbitrary state. The concept of activity is presented separately in a later section. For the present, the activity of a species in a system may just be considered to be a function of its concentration in the system, and when the species is in a pure form (or in its standard state), its activity is taken to be unity. The activities ac, aD, aA, aB given above correspond to the actual conditions of the reaction, and these may or may not correspond to the state of equilibrium. Two special situations can be considered. In the first, the arbitrary states are taken to correspond to those for the system at equilibrium. Q would then become identical to the equilibrium constant K and, according to the Van t Hoff isotherm, AG would then be zero. In the second situation, all the reactants and the products are considered to be present as pure species or in their standard states, and aA, aB, ac, and aD are all equal to 1. Then (7=1 and the free energy change is given by... 

The above important relationship now allows evaluation of the thermodynamic driving force of a redox reaction in terms of a measurable cell emf. Moreover, it is possible to utilize the relationship between the standard state potential and the standard state free energy to arrive at an expression for the equilibrium constant of a redox reaction in terms of the emf. Thus... 

For a heterogeneous reaction, the state of all components is not uniform, for example, a reaction between a gas and a liquid. This requires standard states to be defined for each component. The activity of a solid in the equilibrium constant can be taken to be unity. 

In this expression, Ka is the equilibrium constant at T), and Ka2 is the equilibrium constant at l2. AH0 is the standard heat of reaction (kJ) when all the reactants and products are at standard state, given by ... 

Figure 8. van t Hoff plots of equilibrium constant K, standard state 1 atm., for 0,1 equilibrium H3N(CH2)pNH2++ H20H3N(CH2)oNH3(H20)2+. Values of chain length p given beside each plot in the figure. Equilibrium constant K decreases as p is increased. From Klassen, J. A. Blades, A. T. Kebarle, P. ) Am. Chem. Soc. 1996, with permission. 

As equation 2.4.8 indicates, the equilibrium constant for a reaction is determined by the temperature and the standard Gibbs free energy change (AG°) for the process. The latter quantity in turn depends on temperature, the definitions of the standard states of the various components, and the stoichiometric coefficients of these species. Consequently, in assigning a numerical value to an equilibrium constant, one must be careful to specify the three parameters mentioned above in order to give meaning to this value. Once one has thus specified the point of reference, this value may be used to calculate the equilibrium composition of the mixture in the manner described in Sections 2.6 to 2.9. 

The equilibrium constant Ka is independent of pressure for those cases where the standard states are taken as the pure components at 1 atm. This case is the one used as the basis for deriving equation 2.6.9. Tjie effect of pressure changes then appears in the terms KfjP and ps + t+ b c . The influence of pressure on KfjP is quite small. However, for cases where there is no change in the total number of gaseous moles during the reaction, this is the only term by which pressure changes affect the equilibrium yield. For these... 

The feed stream consists of 60 mole percent hydrogen, 20% nitrogen, and 20% argon. Calculate the composition of the exit gases, assuming equilibrium is achieved in the reactor. Make sure that you take deviations from the ideal gas law into account. The equilibrium constant expressed in terms of activities relative to standard states at 1 atm may be assumed to be equal to 8.75 x 10 3. The fugacity of pure H2 at 450 °C and 101.3 MPa may be assumed to be equal to 136.8 MPa. 

At 900 °F the equilibrium constant for this reaction is 5.62 when the standard states for all species are taken as unit fugacity. If the reaction is carried out at 75 atm, what molal ratio of steam to carbon monoxide is required to produce a product mixture in which 90% of the inlet CO is converted to C02 ... 

The equilibrium constants are based on a standard state of unit fugacity for the gaseous species and on a standard state corresponding to the pure solid for carbon. 

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