Friday, November 7, 2014
Fitting a Simple Catalyzed Reaction
In a catalyzed reaction the catalyst is not used up in the reaction. With the dissociation of sucrose in an acid solution water takes part in the reaction but the quantity of acid and its ions remain constant. The quantity of hydrogen ions in solution is maintained by the dissociation of water. We can take this into account and it simplifies the equations for the constants in the formula for the amount of sucrose as a function of time. We proceed as we did before determining the coefficients for the quadratic equation for the reaction rate but let B remain unchanged. Integrating the reaction rate equation again produces a formula with a hyperbolic tangent function but the constants in it are slightly different due to changes in relative magnitude. We can assume some initial conditions for the sake of argument.
We then generate some random data to work with and allow sufficient time for equilibrium to be reached.
If one checks the initial reaction rate one gets dA/dt = −k0B0A which agrees with the formula that Wilhelmy used so the initial solution is approximately an exponential decay function. To make the change easier to fit we can use the natural log of A instead of A and fit a polynomial equation. F(t) below is a vector function containing the powers of t and is needed by linfit to find the coefficients of the polynomial. We can estimate the initial value for A by averaging the values of A divided by powers of e with x values corresponding to the data points. A check of the standard deviation shows that the rms error is about one unit.
Once we have the fit we can draw a smooth curve through the data points using the empirical power law.
By taking the derivative of the fit polynomial at t = 0 we get the initial rate constant and this gives us an estimate of the first rate constant, k0, on division by B0. To find the second rate constant, k1, we need an estimate of the equilibrium value for A and the equilibrium condition k0B0Aeq = k1(A0 - Aeq)2 which states that the amount of product that is produced is canceled by the reverse reaction.
With the estimates for the rate constants we can now estimate the constants needed to fit the formula to the data. Using an auxiliary constant γ makes it easier to determine an estimate for κ. The fit agrees fairly well with the initial assumptions.