

[TOP]


7. Empirical Reaction Kinetics (ii) (13) Reactions approaching equilibrium [ TOP ]
Reactions approaching equilibrium have to be treated in a special way to account for the fact that in proximity to equilibrium conditions, the reverse reaction becomes important. Let us consider a first order reaction close to equilibrium: The concentration of A is:
(1) Exactly the same conclusion can be reached (and far more quickly) by noting that at equilibrium, the forward and reverse rates must be equal, i.e.: (2) This equation is a particularly significant as it relates a thermodynamic property (the equilibrium constant) to quantities relating to rates. [ TOP ]
(14) Disturbing equilibria  Relaxation methods
The term relaxation denotes the return of a system to equilibrium, and in chemical kinetics, it is used to indicate the readjusting that a reaction has to make in response to a sudden (normally) disturbance of an equilibrium. Sudden changes could include:
Let us consider the scenario of a sudden temperature increase applied to the equilibrium A B (see Fig. 1). Let k_{a}_{1} and k_{b}_{1} be the rate constants at the initial temperatures T_{1}, that is, prior to the temperature jump, the net rate if change of [A] is given by: which under equilibrium conditions: i.e.: Now let k_{a2} and k_{b2} be the rate constants at the final temperatures T_{2}, that is, after to the temperature jump. As before, once we reach the new equilibrium at temperature T_{2} we have: Note that this state of equilibrium is reached at a rate that depends that depends solely on the new rate constants k_{a2} and k_{b2,} that is from:
we obtain (for x in the region between 0 and '[A]_{eq1}  [A]_{eq2}'): since: Also, since [A] = [A]_{eq2} + x, then: d[A] = dx, i.e.: Thus we have: i.e.: which becomes: i.e. by raising to the power of e: or: where t is known as the relaxation time given by: Note that through the relaxation time, t, and the new equilibrium constant we may derive k_{a2}and k_{b2} by simultaneous equations ( K_{2} = k_{a2}/k_{b2}).Similar equations may also be derived for other types of elementary reactions. For example for A B + C where the forward is first order, and the reverse is second order overall, we have: i.e. at equilibrium 1: whilst in between equilibrium 1 and 2 we have:[A] = [A]_{eq2}^{ }+ xi.e.: where: i.e.: i.e.: which since: we have: where is we assume that x^{2} is approximately zero, then: i.e.: i.e.: or: where:


CH237  Chemical Thermodynamics
and Kinetics
Dr.
Joseph N. Grima, Department
of Chemistry

7. Empirical Reaction Kinetics (ii) (13) Reactions approaching equilibrium [ TOP ]
Reactions approaching equilibrium have to be treated in a special way to account for the fact that in proximity to equilibrium conditions, the reverse reaction becomes important. Let us consider a first order reaction close to equilibrium: The concentration of A is:
(1) Exactly the same conclusion can be reached (and far more quickly) by noting that at equilibrium, the forward and reverse rates must be equal, i.e.: (2) This equation is a particularly significant as it relates a thermodynamic property (the equilibrium constant) to quantities relating to rates. [ TOP ]
(14) Disturbing equilibria  Relaxation methods
The term relaxation denotes the return of a system to equilibrium, and in chemical kinetics, it is used to indicate the readjusting that a reaction has to make in response to a sudden (normally) disturbance of an equilibrium. Sudden changes could include:
Let us consider the scenario of a sudden temperature increase applied to the equilibrium A B (see Fig. 1). Let k_{a}_{1} and k_{b}_{1} be the rate constants at the initial temperatures T_{1}, that is, prior to the temperature jump, the net rate if change of [A] is given by: which under equilibrium conditions: i.e.: Now let k_{a2} and k_{b2} be the rate constants at the final temperatures T_{2}, that is, after to the temperature jump. As before, once we reach the new equilibrium at temperature T_{2} we have: Note that this state of equilibrium is reached at a rate that depends that depends solely on the new rate constants k_{a2} and k_{b2,} that is from:
we obtain (for x in the region between 0 and '[A]_{eq1}  [A]_{eq2}'): since: Also, since [A] = [A]_{eq2} + x, then: d[A] = dx, i.e.: Thus we have: i.e.: which becomes: i.e. by raising to the power of e: or: where t is known as the relaxation time given by: Note that through the relaxation time, t, and the new equilibrium constant we may derive k_{a2}and k_{b2} by simultaneous equations ( K_{2} = k_{a2}/k_{b2}).Similar equations may also be derived for other types of elementary reactions. For example for A B + C where the forward is first order, and the reverse is second order overall, we have: i.e. at equilibrium 1: whilst in between equilibrium 1 and 2 we have:[A] = [A]_{eq2}^{ }+ xi.e.: where: i.e.: i.e.: which since: we have: where is we assume that x^{2} is approximately zero, then: i.e.: i.e.: or: where:

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