First_Order_Rev_Relax_v7

First_Order_Rev_Relax_v7 - Relaxation Kinetics : Reversible...

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Unformatted text preview: Relaxation Kinetics : Reversible First Order Reaction Dr. Kalju Kahn UC Santa Barbara, 2004-2010 We continue to investigate relationships between thermodynamics and kinetics in a reversible first order reaction. The rate equations for the first-order reversible reaction A B are: dA dt dB dt k1 [A] - k2 [B] k1 [A] k2 [B] If the reaction goes on long enough, it will reach an equilibrium, where the concentrations of A and B have reached their equilibrium values A B A eq eq eq and B eq . At equilibrium dA dt = dB dt 0, and: K eq= = k1 k2 We now ask what happens if the system is perturbed from equilibrium. One way to perturb the system is to add A or B. Another way to perturb the equilibrium is to change the environmental conditions (temperature, pressure, pH) such that the equilibrium constant will change significantly. We will first consider a rapid change in temperature as a perturbing factor. In practice, it is possible to raise a system's temperature rapidly by several Centigrade by three methods: Joule heating, microwave heating, and laser heating. Joule heating converts electrical energy stored in a high-voltage capacitor into heat energy by discharging the capacitor through a reaction mixture that contains electrolytes. Joule heating requires a moderately conductive medium, and cannot be used with most non-aqueous solvents. Microwave heating allows to heat any molecule with a permanent dipole moment or by heating the solution with microwave radiation. Laser heating uses coherent high energy-density to heat water molecules (using IR energy) or dissolved dyes (visible light). Modern instruments allow temperature jumps of tens of degrees within tens of nanoseconds. The temperature change causes a change of the equilibrium constant according to van't Hoff's law: d lnK eq dT H0 RT 2 We see that the equilibrium constant, and thus the equilibrium concentration of B, increases with increasing temperature if H 0 is positive (the reaction is endothermic). The equilibrium constant and the equilibrium concentration of B decrease with increasing temperature if H 0 is negative (the reaction is exothermic). This is an example of Le Chatelier's principle, which states that the new direction of equilibrium is such that the reaction partially offsets the perturbation. No change in concentrations occurs if H 0 is zero. Thus, isothermic reactions, such as racemizations, cannot be studied by the temperature-jump method. A large magnitude in the change of the equilibrium constant heat any molecule with a permanent dipole moment or by heating the solution with microwave radiation. Laser heating uses coherent high energy-density to heat water molecules (using IR energy) or dissolved dyes (visible 2 First_Order_Rev_Relax_v7.nb light). Modern instruments allow temperature jumps of tens of degrees within tens of nanoseconds. The temperature change causes a change of the equilibrium constant according to van't Hoff's law: d lnK eq dT H0 RT 2 We see that the equilibrium constant, and thus the equilibrium concentration of B, increases with increasing temperature if H 0 is positive (the reaction is endothermic). The equilibrium constant and the equilibrium concentration of B decrease with increasing temperature if H 0 is negative (the reaction is exothermic). This is an example of Le Chatelier's principle, which states that the new direction of equilibrium is such that the reaction partially offsets the perturbation. No change in concentrations occurs if H 0 is zero. Thus, isothermic reactions, such as racemizations, cannot be studied by the temperature-jump method. A large magnitude in the change of the equilibrium constant occurs if the reaction enthalpy is large. In order to monitor the progress of a reaction to the new equilibrium, the concentrations of species must change detectably. A change in the equilibrium constant translates to a significant change in the reactant concentration if the reaction is not too far from equilibrium. If the reaction follows first order kinetics, then the progress to a new equilibrium is described by the same differential rate laws as any first order reaction. We designate the difference between the instantaneous concentration and the new equilibrium concentration for reactant and product as [A] and [B], respectively. The movement of a system from its original equilibrium toward the new equilibrium can be described d dt d dt d dt B B B k1 [A] - k2 [B] k1 ( A eq + [A]) - k2 ( B eq + [B]) k1 A eq - k2 B eq + k1 [A] - k2 [B] Recall that at equilibrium d B dt dB = 0, and thus k1 A dt eq - k2 B eq = 0 k1 [A] - k2 [B] B process concentration changes are linked such that [A] = - [B] Recall also that for the A d B dt k1 [B] - k2 [B] = (k1 + k2 ) [B] Next we solve this differential equation subject to a boundary condition that [B] at time zero is constant. The function DSolve in Mathematica can solve differential equations and determine unknown constants if both the form of the differential equation, and initial conditions, are specified. The following few commands clear the memory from symbols that may have been defined previously and turn off spell-checking, which otherwise would produce warnings about some variables having similar names. Recall that at equilibrium d B dt dB = 0, and thus k1 A dt eq - k2 B eq = 0 First_Order_Rev_Relax_v7.nb 3 k1 [A] - k2 [B] B process concentration changes are linked such that [A] = - [B] Recall also that for the A d B dt k1 [B] - k2 [B] = (k1 + k2 ) [B] Next we solve this differential equation subject to a boundary condition that [B] at time zero is constant. The function DSolve in Mathematica can solve differential equations and determine unknown constants if both the form of the differential equation, and initial conditions, are specified. The following few commands clear the memory from symbols that may have been defined previously and turn off spell-checking, which otherwise would produce warnings about some variables having similar names. In[402]:= In[403]:= Out[403]= In[404]:= Out[404]= In[406]:= Remove "Global` " tjump DSolve deltaB ' t deltaB0 . tjump k1 k2 t k1 k2 deltaB t , deltaB 0 deltaB0 , deltaB t , t deltaB t dB deltaB t deltaB0 Plot dB . AxesLabel B 1.0 k1 k2 t deltaB0 1.0, k1 0.693 , k2 "time ", " B " , PlotStyle 0.12 , t, 0, 7 , Orange , Thickness 0.008 0.8 0.6 Out[406]= 0.4 0.2 1 2 3 4 5 6 7 time The solution for the concentration change during relaxation is: [B] = B 0 k1 k2 t = B 0 t Τ where Τ = 1 k1 k2 The sum of the forward and reverse rate coefficient, k 1 + k 2 , can be obtained by determining the rate of decay to a new equilibrium or determining the observed half-life Τ. Individual values of k 1 and k 2 can be obtained if we are also able to determine the new equilibrium constant. This would require the measurement of equilibrium concentrations of A and B. Alternatively, the equilibrium constant can determined in an independent experiment. Relaxation techniques are not limited to temperature jumps. Any method that can rapidly perturb the equilibrium can be used to create a non-equilibrium system that then relaxes into a new equilibrium state. Another commonly used perturbation is pressure; increase in pressure will shift the equilibrium toward state with smaller volume. 4 First_Order_Rev_Relax_v7.nb The solution for the concentration change during relaxation is: [B] = B 0 k1 k2 t = B 0 t Τ where Τ = 1 k1 k2 The sum of the forward and reverse rate coefficient, k 1 + k 2 , can be obtained by determining the rate of decay to a new equilibrium or determining the observed half-life Τ. Individual values of k 1 and k 2 can be obtained if we are also able to determine the new equilibrium constant. This would require the measurement of equilibrium concentrations of A and B. Alternatively, the equilibrium constant can determined in an independent experiment. Relaxation techniques are not limited to temperature jumps. Any method that can rapidly perturb the equilibrium can be used to create a non-equilibrium system that then relaxes into a new equilibrium state. Another commonly used perturbation is pressure; increase in pressure will shift the equilibrium toward state with smaller volume. If the perturbations are significant one should remember that the rate coefficients and the equilibrium constant obtained correspond to the conditions of the perturbed state. For example, if the system is perturbed from 25 °C to 40 °C, the rate coefficients obtained are the rate coefficients at 40 °C. If one wishes to know what are the rate coefficients at 25 °C, the system should be perturbed from a lower temperature to 25 °C. ...
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