Rate law and stoichiometry

Chemical Reaction Engineering

Rate law

Reactor mole balance in terms of $X$

Reactor	Mole balance
Batch	$N_{A0}\frac{dX}{dt} = r_AV$
CSTR	$V = \frac{F_{A0}X}{-r_A}$
PFR	$F_{A0}\frac{dX}{dV} = -r_A$
PBR	$F_{A0}\frac{dX}{dW} = -r'_A$

Rate law

The algebraic expression for the reaction rate equation, $r_j$ .

$-r_A = f \left[ \begin{array}{c} \text{temperature} \\ \text{dependent} \\ \text{terms} \end{array} , \begin{array}{c} \text{concentration} \\ \text{dependent} \\ \text{terms} \end{array} \right] = f(T, C)]$
How to derive an equation for $–r_A$ in terms of $X_A$
- Relate all $r_j$ to $C_j$ $\Rightarrow$ Rate law
- Relate all $C_j$ to $V$ or $\upsilon$ $\Rightarrow$ Stoichiometry
- Relate $V$ or $\upsilon$ to $X_A$ $\Rightarrow$ Stoichiometry
- Put together
  
  $r_A = f(T, X_A)$

Molecularity of reaction

Number of atoms, ions, or molecules involved (colliding) in a reaction step
Unimolecular: One molecule colliding in one reaction step
- radioactive decay
  
  alpha decay of Uranium-238 emitting an alpha particle (a helium nucleus) and formation of Thorium-234.
  
  $\ce{ _{92}U^{238} -> _{90}Th^{234} + _{2}He^{4} }$
- rate of disappearance of U $\Rightarrow -r_U = kC_U$
Bimolecular: Two molecule colliding in one reaction step
- Only true bimolecular reactions are those involving collision with free radicals
  
  initiation step of the free radical bromination of ethane
  
  $\ce{ Br. + C2H6 -> HBr + C2H5.}$
- $-r_{\ce{Br.}} = kC_{\ce{Br.}}C_{\ce{C2H6}}$
Trimolecular: Three molecule colliding in one reaction step
- The probability of trimolecular reaction is almost non-existent
- In most instances the reaction pathway follows a series of bimolecular reactions.

Relative rates of reaction

How fast one species appears or disappears relative to the other species in a given reaction.
Can be obtained from the ratio of stoichiometric coefficients
Consider a reaction

$\ce{aA + bB -> cC + dD}$
For every mole of A consumed $\frac{c}{a}$ moles of C appear

$r_C = \left[ \begin{array}{c} \text{rate of} \\ \text{formation of C}\end{array} \right] = \frac{c}{a} \left[ \begin{array}{c} \text{rate of} \\ \text{disappearance of A}\end{array} \right] = \frac{c}{a} (-r_A)$

$r_C = \frac{-c}{a} r_A$ Similarly, $r_C = \frac{c}{d} r_D$

$\begin{align*} \ce{A} & \quad + \quad & \ce{\frac{b}{a} B} &\quad \ce{->} \quad & \ce{\frac{c}{a} C} &\quad + \quad & \ce{\frac{d}{a} D} \\ \frac{-r_A}{a} & \quad = \quad & \frac{-r_B}{b} & \quad = \quad & \frac{r_C}{c} & \quad = \quad & \frac{r_D}{d} \end{align*}$

Rate law

The rate of reaction depends on collision frequency of the molecules.
Concentration and temperature
- Molecular collision frequency $\propto$ concentration
  
  $\Rightarrow$ Rate of reaction $\propto$ concentration. $-r_A = f(C_A, C_B, ...) \quad \text{for const. T}$
- As temperature increases, collision frequency increases
  
  $\Rightarrow$ Rate of reaction $\propto$ temperature. $-r_A = f([T],[C_A, C_B, ...])$
For many reactions

$-r_A = k_A(T) f(C_A, C_B, ...)$
Specific rate of reaction (rate constant), $k_A(T)$
- Proportionality constant in the rate equation
- Depends on the temperature (following the Arrhenius equation), the presence of a catalyst, and other environmental conditions but is independent of the reactant concentrations.

Power law model

Dependence of $-r_A$ on concentration of species present ( $f(C_j)$ ) is almost always determined by experimental observations.
Functional dependence on concentration may be postulated by theory

$\Rightarrow$ Experiments are required to confirm the proposed form
Power law model is most common general forms of the rate law

$-r_A = k_A C_A^\alpha C_B^\beta$
Order of reaction: the powers to which the concentrations are raised
- The reaction is $\alpha$ order with respect to A and $\beta$ order with respect to B.
- Overall order of reaction $n = \alpha + \beta$ .
$k_A$ : Specific reaction rate

$\text{Units of } k_A = \frac{\text{(conc)}^{1-n}}{\text{time}}$

Order of reaction

$\ce{A -> Products}$

Reaction order	Rate equation	Units of $k_A$
$0^{th}$ order	$-r_A = k_A$	$\{k\} = \frac{mol}{volume \ s}$
$1^{st}$ order	$-r_A = k_A C_A$	$\{k\} = \frac{1}{s}$
$2^{nd}$ order	$-r_A = k_A C_A^2$	$\{k\} = \frac{volume}{mol \ s}$
$3^{rd}$ order	$-r_A = k_A C_A^3$	$\{k\} = \frac{(volume/mol)^2}{s}$

Elementary reaction

A reaction involving single step is called an elementary reaction.
Stoichiometric coefficients in the reaction are equal to the powers in the rate law.

$\ce{2NO + O2 -> 2NO2}$

This reaction is not elementary, but under some conditions it follows an elementary rate law

$-r_{NO} = k_{NO} C_{NO}^2 C_{O_2}$

$\Rightarrow$ Elementary as written

$\ce{CO + Cl2 -> COCl2}$

This reaction is non-elementary

$-r_{CO} = k_{CO} C_{CO} C_{Cl_2}^{3/2}$

$1^{st}$ order with respect to $\ce{CO}$ and 3/2 order with resect to $\ce{Cl2}$ (5/2 order overall)

$\Rightarrow$ Non-elementary reaction

Complex rate expression

$\ce{2N2O -> 2N2 + O2} \qquad -r_{N_2O} = \frac{ k_{N_2O} C_{N2O}}{1 + k' C_{O_2}}$

Rate expression cannot be separated into solely temperature dependent and concentration dependent terms.
Overall reaction order cannot be stated
Only undrer limiting circumstances we can speak of reaction order
- for $1 >> k'C_{O_2}; -r_{N_2O} = k_{N_2O} C_{N_2O}$ $\Rightarrow$ ‘Apparent’ first order reaction
- for $1 << k'C_{O_2}; -r_{N_2O} = k_{N_2O} \frac{C_{N_2O}}{C_{O_2}}$ $\Rightarrow$ ‘Apparent’ $0^{th}$ order reaction
  
  Reaction is $-1^{th}$ order with $\ce{O2}$ and $1^{st}$ order with $\ce{N2O}$ .

This kind of rate expression is common for liquid and gaseous reactions promoted by solid catalysts.

Heterogeneous reactions

Historically for many gas-solid catalyzed reactions it is customary to write rate laws in terms of partial pressures rather than concentrations.
Weight of catalyst is important rather than reactor volume ( $\Rightarrow -r'_A$ )
Hydromethylation of toluene

$\ce{C6H5CH3 + H2 ->[{cat}] C6H6 + CH4}$

$-r'_T = \frac{k P_{H_2} P_T}{1 + K_B P_B + K_T P_T}$
- ’ (prime): indicates typical units are in /g-cat
- $P$ : partial pressures ( $kPa$ or $atm$ )
- $K$ : adsorption constant ( $1/kPa$ or $atm^{-1}$ )
- $[K] = \frac{mol \ toluene}{kg\ cat \cdot s \cdot kPa^2}$
Use ideal gas law to express the reaction in terms of concentrations: $P_i = C_i RT$

Reversible reaction

All rate laws must reduce to thermodynamic relationships relating the reacting species concentrations at equilibrium
At equilibrium rate of reaction is zero for all the species ( $-r_A \equiv 0$ )
For a reaction $\ce{aA + bB <=> cC + dD}$

$K_c = \frac{C_{C_e}^c C_{D_e}^d}{C_{A_e}^a C_{B_e}^b} \qquad [K_c] = \left(\frac{mol}{volume}\right)^{c +d -a -b}$
For elementary reaction $\ce{A <=>[{k_f}][{k_r}] B}$

$-r_A = k \left( C_A - \frac{C_B}{K_c} \right); K_c = \frac{k_f}{k_r}$

In class exercise: rate law

Determine the rate law for the reaction described in each of the cases below involving species A, B, and C. The rate laws should be elementary as written for reactions that are either of the form $\ce{A -> B}$ or $\ce{A + B -> C}$ .

The units of the specific reaction rate are $k = \left[\frac{dm^3}{mol \ h} \right]$ .
The units of the specific reaction rate are $k = \left[\frac{mol}{kg-cat \ h \ (atm)^2} \right]$ .
The units of the specific reaction rate are $k = \left[\frac{1}{h} \right]$ .
The units of a nonelementary reaction rate are $k = \left[\frac{mol}{dm^3 \ h} \right]$ .

The reaction rate constant ( $k_A$ )

$\Rightarrow$ Not a constant. Just independent of concentration

Arrhenius equation

$k_A(T) = A e^{\frac{-E}{RT}} \Rightarrow k_0 e^{-E/RT}$
- $k_0, A$ : Preexponential factor/ frequency factor
- $E$ : Activation energy (J/mol) or (cal/mol)
- $R$ : Gas constant (= 8.314 J/mol K)
- $T$ : Absolute temperature (K)
Empirically varified over a large temperature range

Determining activation energy

Experimentally measuring reaction rate at different temperatures
$k_A(T) = A e^{-E/RT}$

Taking natural log

$\ln k_A = \ln A -\frac{E}{R} \left( \frac{1}{T} \right)$

$k(T) = k(T_0) e^{\frac{E}{R} \left( \frac{1}{T_0} - \frac{1}{T} \right)}$

Rule of thumb: Reaction rate doubles every 10 °C rise in temperature

Arrhenius plot

In class exercise: activation energy

The decomposition of benzene diazonium chloride to give chlorobenzene and nitrogen

$\ce{C6H5N2Cl -> C6H5Cl + N2}$

follows first order kinetics. The rate constant data at different temperatures is given in Table 1. Calculate the activation energy.

Table 1: Rate constant data

$k (s^{-1})$	0.00043	0.00103	0.00180	0.00355	0.00717
$T (K)$	313.0	319.0	323.0	328.0	333.0

Stoichiometric table

Represents stoichiometric relationships between reacting molecules for a single reaction
How many molecules of one species will be formed during a chemical reaction given a number of molecules of another species disappear.
Let us consider general reaction

$\ce{aA +bB -> cC + dD}$
relative rates

$\frac{-r_A}{a} = \frac{-r_B}{b} = \frac{r_C}{c} = \frac{-r_D}{d}$
Basis of calculation: Species A

$\ce{A +\frac{b}{a}B -> \frac{c}{a}C + \frac{d}{a}D}$

Stoichiometric table: batch systems

Moles of B reacted

$moles\ of\ B\ reacted = \frac{mol\ B\ reacted}{mol\ A\ reacted} \times mol\ A\ reacted = \frac{b}{a} N_{A0} X$

Species	Initially (mol)	Change (mol)	Remaining (mol)
A	$N_{A0}$	$-N_{A0}X$	$N_A = N_{A0} - N_{A0}X$
B	$N_{B0}$	$-(b/a)N_{A0}X$	$N_B = N_{B0} -(b/a)N_{A0}X$
C	$N_{C0}$	$(c/a)N_{A0}X$	$N_C = N_{C0} +(c/a)N_{A0}X$
D	$N_{D0}$	$(d/a)N_{A0}X$	$N_D = N_{D0} +(d/a)N_{A0}X$
I	$N_{I0}$	0	$N_I = N_{I0}$
Total	$\mathbf{N_{T0}}$		$\mathbf{N_T = N_{T0} + \delta N_{A0}X}$

$\delta = +\frac{d}{a}+\frac{c}{a}-\frac{b}{a}-1$

Change in total number of moles per mole A reacted

Equations for the concentration

$C_A = \frac{N_A}{V} = \frac{N_{A0}(1-X)}{V}$
$C_B = \frac{N_B}{V} = \frac{N_{B0}-(b/a)N_{A0}X}{V}$
$C_C = \frac{N_C}{V} = \frac{N_{C0}+(c/a)N_{A0}X}{V}$
$C_D = \frac{N_D}{V} = \frac{N_{D0}+(d/a)N_{A0}X}{V}$
For a constant volume batch reactor, $V = V_0$ .
Let $\Theta_i= N_{i0}/N_{A0} = C_{i0}/C_{A0}$ .

$C_A = C_{A0}(1-X)$
$C_B = C_{A0}(\Theta_B-(b/a)X)$
$C_C = C_{A0}(\Theta_C+(c/a)X)$
$C_D = C_{A0}(\Theta_D+(d/a)X)$

$\Theta_i= \frac{\text{Moles of species 'i' initially}}{\text{Moles of species A initially}}$

Equimolar: $\Theta_B = 1$

Stoichiometric: $\Theta_B = b/a$

Rate expression

$-r_A = k C_A C_B$
$-r_A = k C_{A0}^2 (1-X) (\phi_B-(b/a)X)$
For $C_{A0}= C_{B0}$ , and $b/a = 1$

$-r_A = k C_{A0}^2 (1-X)^2$
Mole balance

$\frac{dC_A}{dt}=-k C_{A0}^2 (1-X)^2$

as $C_A = C_{A0}(1-X)$ ; $dC_A = -C_{A0} dX$
The mole balance can then be written as

$\frac{dX}{(1-X)^2}=k C_{A0} dt$

Stoichiometric table: flow systems

Species	Feed rate to reactor (mol/time)	Change within reactor (mol/time)	Effluent rate from reactor (mol/time)
A	$F_{A0}$	$-F_{A0}X$	$F_A = F_{A0} - F_{A0}X$
B	$F_{B0}$	$-(b/a)F_{A0}X$	$F_B = F_{B0} -(b/a)F_{A0}X$
C	$F_{C0}$	$(c/a)F_{A0}X$	$F_C = F_{C0} +(c/a)F_{A0}X$
D	$F_{D0}$	$(d/a)F_{A0}X$	$F_D = F_{D0} +(d/a)F_{A0}X$
I	$F_{I0}$	0	$F_I = F_{I0}$
Total	$\mathbf{F_{T0}}$		$\mathbf{F_T = F_{T0} + \delta F_{A0}X}$

$\delta = +\frac{d}{a}+\frac{c}{a}-\frac{b}{a}-1$
$\Theta_{i0}= F_{i0}/F_{A0} = C_{i0}/C_{A0}$ .
$C_B = C_{A0}(\Theta_B-(b/a)X)$

For liquid phase systems ( $\upsilon = \upsilon_0$ )

Gas phase reactions

Volumetric flow rate changes during the course of reaction
- Changes in total number of moles
- Changes in pressure and temperature
Variable flow rate

Gas phase reactions that do not have equal number of product and reactant moles

e.g. $\ce{N2 + 3H2 <=> 2NH3}$
- 4 mole reactant give 2 mole products
Stoichiometric tables
- No assumptions made reagarding volume
- The tables are exactly same for constant volume (constant density) and variable volume (variable density) systems

$\Rightarrow$ Only for concentration expressed in terms of conversion density/ volume comes into play.

Flow reactor with variable volumetric flow rate

We will use relationships for total concentration

$C_T = \frac{F_T}{\upsilon} \qquad ... \qquad \frac{\text{total molar flow rate}}{\text{volumetric flow rate}}$
For gases

$C_T = \frac{P}{zRT} \qquad z: \text{compressibility factor (=1 for ideal gas)}$
At the entrance of the reactor

$C_{T0} = \frac{P_0}{z_0RT_0}$
Assuming negligible change in $z$ ( $z = z_0$ )

$\upsilon = \upsilon_0 \left( \frac{F_T}{F_{T_0}} \right) \left( \frac{P_0}{P} \right) \left( \frac{T}{T_0} \right)$

Flow reactor with variable volumetric flow rate

We can now express $C_j (=F_j/\upsilon)$ in terms of $F$ , $P$ , and $T$

$C_j = C_{T0} \left( \frac{F_j}{F_T} \right) \left( \frac{P}{P_0} \right) \left( \frac{T_0}{T} \right)$
$F_T = \sum_{j=1}^{n} F_j = F_A + F_B + F_C + F_D + F_I$

$F_j$ is found by solving mole balance equations
Concentration in terms of conversion: $F_T = F_{T0} + F_{A0} \delta X$
Dividing by $F_{T0}$ and defining $\epsilon = y_{A0} \delta$ , where $y_{A0}$ is the mole fraction of A at inlet

$\frac{F_T}{F_{T0}} = 1 + \epsilon X$

$\epsilon = \frac{ \text{change in total no. of moles for complete conversion}} { \text{total moles fed}}$

Flow reactor with variable volumetric flow rate

Equation for volumetric flow rate $\upsilon = \upsilon_0 (1 + \epsilon X) \left( \frac{P_0}{P} \right) \left( \frac{T}{T_0} \right)$
Molar flow rate $F_j = F_{A0} (\Theta_j + \nu_j X)$
Concentration ( $C_j = F_j /\upsilon$ )

$C_j = \frac{F_{A0} (\Theta_j + \nu_j X)} {\upsilon_0 (1 + \epsilon X) \left( \frac{P_0}{P} \right) \left( \frac{T}{T_0} \right) }$

Stoichiometric coefficient ( $\nu_j$ )

$\ce{ A + \frac{b}{a} B -> \frac{c}{a} C + \frac{d}{a} D}$

-ve for reactant

$\nu_A = -1$ ; $\nu_B = -b/a$
+ve for products

$\nu_C = c/a$ ; $\nu_D = -d/a$

$C_j = \frac{C_{A0} (\Theta_j + \nu_j X)} {(1 + \epsilon X)} \left( \frac{P}{P_0} \right) \left( \frac{T_0}{T} \right)$

In class exercise: saponification reaction

The saponification for the formation of soap is:

$\ce{3NaOH + (C17H35COO)3C3H5 -> 3C17H35COONa + C3H5(OH)3}$

Letting X represent the conversion of NaOH set up a stoichiometric table expressing the concentration of each species in terms of the initial concentration of NaOH and the conversion of X.

In class exercise: determining $C_j = h_j(X)$

A mixture Of 28% $\ce{SO2}$ and 72% air is charged to a flow reactor in which $\ce{SO2}$ is oxidized.

$\ce{2SO2 + O2 -> 2 SO3}$

First, set up a stoichiometric table using only the symbols (i.e., $\Theta_i, F_i$ ).
Next, prepare a second table evaluating the species concentrations as a function of conversion for the case when the total pressure is 1485 kPa (14.7 atm) and the temperature is constant at 227 °C.
Evaluate the parameters and make a plot of each of the concentrations $\ce{SO2}$ , $\ce{SO3}$ , $\ce{N2}$ as a function of conversion

In class exercise: liquid phase first order reaction

Orthonitroanaline (an important intermediate in dyes—called fast orange) is formed from the reaction of orthonitrochlorobenzene (ONCB) and aqueous ammonia. The liquid-phase reaction is first order in both ONCB and ammonia with $k = 0.0017 \ m^3 /kmol \cdot min$ at $188 \ ^{\circ}C$ with $E = 11273 \ cal/mol$ . The initial entering concentrations of ONCB and ammonia are $1.8 \ kmol/m^3$ and $6.6 \ kmol/m^3$ , respectively.

$\ce{C6H4ClNO2 + 2 NH3 -> C6H6N2O2 + NH4Cl}$

Set up a stoichiometric table for this reaction for a flow system.
Write the rate law for the rate of disappearance of ONCB in terms of concentration.
Explain how parts (a) and (b) would be different for a batch system.
Write $-r_A$ solely as a function of conversion.
What is the initial rate of reaction (X = 0) at $188 \ ^{\circ}C$ , at $25 \ ^{\circ}C$ , and at $288 \ ^{\circ}C$ ?
What is the rate of reaction when X = 0.90 at $188 \ ^{\circ}C$ , at $25 \ ^{\circ}C$ , and at $288 \ ^{\circ}C$ ?
What would be the corresponding CSTR reactor volume at $25 \ ^{\circ}C$ to achieve 90% conversion and at $288 \ ^{\circ}C$ for a feed rate of $2 \ dm^3 /min$

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Rate law and stoichiometry Chemical Reaction Engineering

Rate law and stoichiometry
Table of contents
Rate law
Reactor mole balance in terms of X
Basic definitions
Molecularity of reaction
Relative rates of reaction
In class exercise: relative rates
In class exercise: relative rates
Rate law
Power law model
Order of reaction
Elementary reaction
Complex rate expression
Heterogeneous reactions
Reversible reaction
In class exercise: rate law
In class exercise: rate law
Equilibrium constant
The reaction rate constant (k_A)
Activation energy
Determining activation energy
In class exercise: activation energy
Stoichiometry
Stoichiometric table
Batch systems
Stoichiometric table: batch systems
Equations for the concentration
Rate expression
In class exercise: equilibrium conversion in batch reactor
Flow systems
Stoichiometric table: flow systems
Gas phase reactions
Flow reactor with variable volumetric flow rate
Flow reactor with variable volumetric flow rate
Flow reactor with variable volumetric flow rate
In class exercise: equilibrium conversion in flow reactor
In class exercise: saponification reaction
In class exercise: determining C_j = h_j(X)
In class exercise: liquid phase first order reaction
Summary

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