<< Graphics`Graphics`
<< Statistics`LinearRegression`

This example illustrates the use of the differential method to determine the order of a reaction (with respect to a selected reactant). 
This is Problem 3-16 in Levenspiel, the same one that Wilhelmy studied 150 years ago (before Guldberg and Waage had formulated the law of mass action and before Michaelis and Menten had come up with their enzyme-catalyzed reaction rate expression).

x[1] = 1;
y[1] =  0.84;
x[2] = 2;
y[2] =  0.68;
x[3] = 3;
y[3] =  0.53;
x[4] = 4;
y[4] =  0.38;
x[5] = 5;
y[5] =  0.27;
x[6] = 6;
y[6] =  0.16;
x[7] = 7;
y[7] =  0.09;
x[8] = 8;
y[8] =  0.04;
x[9] = 9;
y[9] =  0.018;
x[10] = 10;
y[10] =  0.006;
x[11] = 11;
y[11] = 0.0025;

xPoints1 = Table[x[i], {i, 1, 11}]
yPoints1 = Table[y[i], {i, 1, 11}]
data1 = Table[{x[i], y[i]}, {i, 1, 11}]

dataPlot1 = 
        ListPlot[data1, PlotStyle -> PointSize[0.02], 
          AxesOrigin -> {0, 0}];
concVsTime = 
      Regress[data1, Table[t^i, {i, 0, 3}], t, 
        RegressionReport -> RSquared]
fittedConcVsTime = Fit[data1, Table[t^i, {i, 0, 3}], t]
Plot[%, {t, 0, 11}, AxesOrigin -> {0, 0}];

Show[%, dataPlot1];

dCdT = Dt[fittedConcVsTime, t]

ypoints2 = Table[-dCdT, {t, 0, 9. , 0.5}]

xpoints2 = Table[fittedConcVsTime, {t, 0, 9. , 0.5}]

Plot[-dCdT, {t, 0, 9. }];

data2 = 
      Table[{xpoints2[[i]], ypoints2[[i]]}, {i, 1, 19}]

dataPlot2 = 
        ListPlot[data2, 
          PlotStyle -> PointSize[0.02]];
rateVsConc = 
      Regress[data2, Table[conc^i, {i, 0, 3}], conc, 
        RegressionReport -> RSquared]
fittedRate = Fit[data2, Table[conc^i, {i, 0, 3}], conc]
Plot[%, {conc, 0, 1}];

Show[%, dataPlot2];

LogLogListPlot[data2];

data3 = 
      Table[{Log[xpoints2[[i]]], Log[ypoints2[[i]]]}, {i, 1, 
          19}]

logRateVsLogConc = 
      Regress[data3, Table[logConc^i, {i, 0, 1}], logConc, 
        RegressionReport -> {BestFit, RSquared}]

So the reaction is indeed NOT first-order with respect to sucrose concentration. The slope of the log(rate) vs. log(concentration) plot is n=0.5. It is between the orders of zero and one, as it should be because these are the two extremes in the Michaelis-Menten equation, eq. 57: at low pressure (or at low cA), 1>>k2'cA and the equation is of 1st order; at high cA, k2'cA>>1, and the equation is of order zero (independent of concentration).

So we follow eq. 61 (pp. 64-65), because this reaction is now known to obey the Michaelis-Menten equation:

data4 = 
      Table[{1/fittedConcVsTime, -1/dCdT}, {t, 1. , 7.5, 0.5}]

dataPlot4 = 
        ListPlot[data4, 
          PlotStyle -> PointSize[0.02]];
newRateVsConc = 
      Regress[data4, Table[recipConc^i, {i, 0, 3}], recipConc, 
        RegressionReport -> RSquared]
fittedRate = 
      Fit[data4, Table[recipConc^i, {i, 0, 1}], recipConc]
Plot[%, {recipConc, 1, 16}];

Show[%, dataPlot4];

This is in good agreement with the results in #3.15, but such agreement was obtained only when the selected concentration range was narrowed to 1 ≤ t ≤ 7.5 hours. If the entire interval 1 ≤ t ≤11 had been selected, the agreement would have been much worse (please verify!)...