Mass–action Ratio
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The mass–action ratio, often denoted by \Gamma, is the ratio of the product concentrations, p, to reactant concentrations, s. The concentrations may or may not be at equilibrium. \Gamma = \frac This assumes that the stoichiometric amounts are all unity. If not, then each concentration must be raised to the power of its corresponding stoichiometric amount. If the product and reactant concentrations are at equilibrium then the mass–action ratio will equal the equilibrium constant. At equilibrium: \Gamma = K_ The ratio of the mass–action ratio to the equilibrium constant is often called the disequilibrium ratio, denoted by the symbol \rho. \rho = \frac and is a useful measure for indicating how far from equilibrium a given reaction is. The ratio is always greater than zero, and at equilibrium, the ratio is one: \rho = 1. When the reaction is out of equilibrium, \rho \neq 1. When \rho < 1, the reaction is out of equilibrium with a forward rate higher than the reverse rate, and the reaction has a negative free energy (i.e., a spontaneous, exergonic reaction), as explained below. For a uni-molecular reaction such as A \rightleftharpoons B, where the net reaction rate is given by the reversible mass-action ratio: v = k_1 A - k_2 B = v_f - v_r At thermodynamic equilibrium the rate equals zero, that is 0 = k_1 A_ - . Rearranging gives: \frac = \frac = K_ but \rho = \frac, therefore \rho = \Gamma\frac and therefore \rho = \frac \frac = \frac In other words, the disequilibrium ratio is the ratio of the reverse to the forward rate. When the reverse rate, v_r is less than the forward rate, the ratio is less than one, \rho < 1, indicating that the net reaction is from left to right.


Relationship to Free Energy

The thermodynamic equation of the
chemical equilibrium In a chemical reaction, chemical equilibrium is the state in which both the Reagent, reactants and Product (chemistry), products are present in concentrations which have no further tendency to change with time, so that there is no observable chan ...
states that :\Delta_\mathrmG_= \Delta_\mathrmG^ + RT \ln \Gamma where ''R'' is the
universal gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature, temperature ...
and ''T'' the
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
. The standard Gibbs free energy, \Delta_\mathrmG^ , can also be expressed in function of the equilibrium constant: \Delta_\mathrmG^ = -R T \ln K_ Introducing this in the previous equation, one obtains: \Delta_\mathrmG_= -R T \ln K_ + RT \ln \Gamma Substituting the mass-action ratio by the disequilibrium ratio times the equilibrium constant, this leads to: \Delta_\mathrmG_= -R T \ln K_ + RT \ln (\rho K_) Therefore \Delta_\mathrmG_= -R T \ln K_ + RT \ln (\rho) + RT \ln (K_) and \Delta_\mathrmG_= RT \ln (\rho) This shows that the disequilibrium ratio is just an alternative way of expressing the free energy of a reaction and as such gives a more intuitive interpretation of free energy. That is if the free energy for a reaction is less than zero then it indicates that \rho < 1 and hence v_r < v_f, i.e the net reaction rate is from left to right.


References


Other sources

* Atkins, P.W. (1978). ''Physical Chemistry''
Oxford University Press Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books ...
*Trevor Palmer (2001) Enzymes: biochemistry, biotechnology and clinical chemistry Chichester Horwood Publishing {{DEFAULTSORT:Mass-Action Ratio Physical chemistry