## F 5X00

This resulting integral may be evaluated to obtain the time required to achieve an X = 0.5. One procedure would be to employ the trapezoidal rule. However, the Runge-Kutta method would produce more accurate results.

The reader is left the exercise of showing that the required time is approximately 820 min.

KIN.12 COMPLEX SYSTEMS

The following liquid-phase reaction takes place in a constant volume, isothermal batch reactor:

The initial reaction rates were measured at various initial concentrations at 0°C. The following data were obtained (Castellan, Physical Chemistry, Addison-Wesley, Boston, 1964):

Run |
CA0 x 104 |
CB0 x 102 |
Cco x 102 |
Initial Reaction Rate x 107 |

(gmol/L) |
(gmol/L) |
(gmol/L) |
[gmol/(Lmin)] | |

1 |
0.712 |
2.06 |
3.0 |
4.05 |

2 |
2.40 |
2.06 |
3.0 |
14.60 |

3 |
7.20 |
2.06 |
3.0 |
44.60 |

4 |
0.712 |
2.06 |
1.8 |
0.93 |

5 |
0.712 |
2.06 |
3.0 |
4.05 |

6 |
0.712 |
2.06 |
9.0 |
102.00 |

7 |
0.712 |
2.06 |
15.0 |
508.00 |

8 |
0.712 |
2.06 |
3.0 |
4.05 |

9 |
0.712 |
5.18 |
3.0 |
28.0 |

10 |
0.712 |
12.50 |
3.0 |
173.0 |

—rA — kACAaCB^Ccy | |||||

1. |
Estimate the |
order of the reaction with respect |
to |
A, |
a. |

2. |
Estimate the |
order of the reaction with respect |
to |
B, |
ß- |

3. |
Estimate the |
order of the reaction with respect |
to |
c, |
y- |

4. Estimate the overall order of the reaction.

5. Outline how to calculate the reaction velocity constant, kA.

6. Outline how to calculate the above (parts 1-5) numerically by regressing the data.

### Solution

The application of any interpretation of kinetic data to reaction rate equations containing more than one concentration term can be somewhat complex. Consider the reaction iA + /B products where dC, dt

For the differentiation technique, this equation is written log(-= log kA + i log CA +j log CB

Three unknowns appear in this equation and can be solved by regressing the data; a trial-and-error graphical calculation is required. Using the integration technique is also complex. If one assumes i = 1 and j = 1, then

A constant value of kA for the data would indicate that the assumed order (/ = 1,7 = 1) is correct.

The isolation method has often been applied to the above mixed reaction system. The procedure employed here is to set the initial concentration of one of the reactants, say B, so large that the change in concentration of B during the reaction is vanishingly small. The rate equation may then be approximated by

Either the differentiation or integration method may then be used. For example, in the differentiation method, a log-log plot of (—dCA/dt) vs. CA will give a straight line of slope i. A similar procedure is employed to obtain j.

The initial rate of reaction, — rA0, in terms of the initial concentrations CA0, CB0, and Cco for the problem at hand is which may be integrated to give where kA = kACJB ~ constant

Linearizing the equation by taking logs, log(—rA0) = log (*) + a log(CA0) + jSlog(CB0) + ylog(Cco) (1) (2) (3) (4)

One may also use In instead of log in the above equation.

For runs (1H3), the concentrations of B and C are constant (for each run at the initial conditions specified). Therefore, terms (1), (3), and (4) above may be treated as constants. Thus, log(-rA0) = log/: + alog(CA0)

where f_ rf

To obtain a graphically, plot

l°g(->A0) |
l°gCA0 |

-6.3925 |
-4.1475 |

-5.8356 |
-3.6197 |

-5.3506 |
-3.1426 |

Use runs (8)—(10) in order to establish the order of the reaction with respect to B:

log(->-Ao) |
log CB0 |

-6.3925 |
-1.6861 |

-5.5528 |
-1.2856 |

-4.7619 |
-0.9039 |

Use runs (4)-(7) to estimate the order the reaction with respect to C:

log(—rA0) |
log cco |

-7.0315 |
-1.7440 |

-6.3925 |
-1.5228 |

-4.9913 |
-1.0457 |

-4.2941 |
-0.8239 |

The overall order of the reaction, n, is therefore n=a+P+y

To calculate kA, use the a, ft, and y values above and calculate 10 values of kA. Sum the results and average. The units are L5/(gmol5 • min).

In order to calculate a, /?, y, and k by regressing the data, the equation (with logs) is of the form y = a0 + a,x, + a2x2 + a3x3

where y |
= log(-rA0) |

A |
= log A: |

A, |
= a |

a2 |
= ß |

A3 |
= y |

= log(CA0) | |

x2 |
= log(CB0) |

= log(Cco) |

The method of least squares may now be used to generate the a's.

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