Chapter 8: Multiple Reactions
Writing Net Rates of Formation
The reactions:
(1) \( A + 2B \rightarrow 2C \)
(2) \( 2C + \frac{1}{2} B \rightarrow 3D \)
are elementary. Write the net rates of formation for A, B, C and D.\( k_{1A} = 0.1 \left( \text{dm}^3/\text{mol} \right)^2 / \text{min} \)
\( k_{2D} = 2 \left( \text{dm}^3/\text{mol} \right)^{3/2} / \text{min} \)
Hint 1: What is the net rate of formation of A?
$ (a) r_{A} = k_{1A}C_{A}{C_{B}}^2$
$ (b) r_{A} = -k_{1A}C_{A}{C_{B}}^2$
Hint 2: What is the net rate of formation of B?
$ (a) r_{B} = -k_{1A}C_{A}{C_{B}}^2-k_{2D}{C_{c}}^2{C_{B}}^{\frac{1}{2}}$
$ (b) r_{B} = -k_{1A}C_{A}{C_{B}}^2-6k_{2D}{C_{c}}^2{C_{B}}^{\frac{1}{2}}$
$ (c) r_{B} = -k_{1A}C_{A}{C_{B}}^2-\frac{1}{6}k_{2D}{C_{c}}^2{C_{B}}^{\frac{1}{2}}$
Hint 3: What is the net rate of formation of C?
$ (a) r_{C} = 2k_{1A}C_{A}{C_{B}}^2-\frac{2}{3}k_{2D}{C_{c}}^2{C_{B}}^{\frac{1}{2}} $
$ (b) r_{C} = 2k_{1A}C_{A}{C_{B}}^2-\frac{3}{2}k_{2D}{C_{c}}^2{C_{B}}^{\frac{1}{2}} $
$ (c) r_{C} = k_{1A}C_{A}{C_{B}}^2-\frac{2}{3}k_{2D}{C_{c}}^2{C_{B}}^{\frac{1}{2}} $
Hint 1
A.
\( r_A = r_{1A} + r_{2A} = r_{1A} + 0 \)
\( r_{1A} = -k_{1A} C_A C_B^2 \)
\( \boxed{ r_A = -k_{1A} C_A C_B^2 } \)
Hint 2
B.
\( r_B = r_{1B} + r_{2B} \)
\(\frac{r_{1B}}{-2} = \frac{r_{1A}}{-1} \)
\( r_{1B} = 2r_{1A} = -2k_{1A} C_A C_B^2 \)
\(\frac{r_{2B}}{-1/2} = \frac{r_{2D}}{3} \)
\( r_{2B} = -\frac{1}{6} r_{2D} \)
\( r_{2D} = k_{2D} C_B^{1/2} C_C^2 \)
\( r_{2B} = -\frac{1}{6} k_{2D} C_B^{1/2} C_C^2 \)
\( \boxed{ r_B = -2k_{1A} C_A C_B^2 - \frac{1}{6} k_{2D} C_B^{1/2} C_C^2 } \)
Hint 3
C.
\( r_C = r_{1C} + r_{2C} \)
\(\frac{r_{1C}}{2} = \frac{r_{1A}}{-1} \)
\( r_{1C} = 2k_{1A} C_A C_B^2 \)
\(\frac{r_{2C}}{-2} = \frac{r_{2D}}{3} \)
\( r_{2C} = -\frac{2}{3} k_{2D} C_C^2 C_B^{1/2} \)
\( \boxed{ r_C = 2k_{1A} C_A C_B^2 - \frac{2}{3} k_{2D} C_C^2 C_B^{1/2} } \)
The reactions:
(1) \( A + 2B \rightarrow 2C \)
(2) \( 2C + \frac{1}{2} B \rightarrow 3D \)
are elementary. Write the net rates of formation for A, B, C and D.\( k_{1A} = 0.1 \left( \text{dm}^3/\text{mol} \right)^2 / \text{min} \)
\( k_{2D} = 2 \left( \text{dm}^3/\text{mol} \right)^{3/2} / \text{min} \)
Solution
A.
\( r_A = r_{1A} + r_{2A} = r_{1A} + 0 \)
\( r_{1A} = -k_{1A} C_A C_B^2 \)
\( \boxed{ r_A = -k_{1A} C_A C_B^2 } \)
B.
\( r_B = r_{1B} + r_{2B} \)
\(\frac{r_{1B}}{-2} = \frac{r_{1A}}{-1} \)
\( r_{1B} = 2r_{1A} = -2k_{1A} C_A C_B^2 \)
\(\frac{r_{2B}}{-1/2} = \frac{r_{2D}}{3} \)
\( r_{2B} = -\frac{1}{6} r_{2D} \)
\( r_{2D} = k_{2D} C_B^{1/2} C_C^2 \)
\( r_{2B} = -\frac{1}{6} k_{2D} C_B^{1/2} C_C^2 \)
\( \boxed{ r_B = -2k_{1A} C_A C_B^2 - \frac{1}{6} k_{2D} C_B^{1/2} C_C^2 } \)
C.
\( r_C = r_{1C} + r_{2C} \)
\(\frac{r_{1C}}{2} = \frac{r_{1A}}{-1} \)
\( r_{1C} = 2k_{1A} C_A C_B^2 \)
\(\frac{r_{2C}}{-2} = \frac{r_{2D}}{3} \)
\( r_{2C} = -\frac{2}{3} k_{2D} C_C^2 C_B^{1/2} \)
\( \boxed{ r_C = 2k_{1A} C_A C_B^2 - \frac{2}{3} k_{2D} C_C^2 C_B^{1/2} } \)
D.
\( r_D = r_{1D} + r_{2D} = r_{2D} \)
\( \boxed{ r_D = k_{2D} C_C^2 C_B^{1/2} } \)
These net rates of reaction are now coupled with the appropriate mole balance of A, B, C, and D and solved using a numerical software package.
For example for a PFR:
\(\frac{dF_A}{dV} = -k_{1A} C_A C_B^2 \)
\(\frac{dF_B}{dV} = -2k_{1A} C_A C_B^2 - \frac{1}{6} C_B^{1/2} C_C^2 \)
etc.