Problems in Undergraduate Kinetics & Reactor Design

Jason Haugh

Department of Chemical & Biomolecular Engineering
North Carolina State University

These problems are provided as a study tool for students taking undergraduate-level kinetics, and as a "short-cut" for fellow instructors.  More problems will be added as they become available.

The majority of these problems are designed to be done relatively quickly (10-20 minutes), to be used for in-class test or example problems.  Others were designed as homework/final exam problems and therefore take longer; most if not all of those can be shortened as well.  Nearly all of the problems are purposely general, in that the reactant and product molecules are not identified; feel free to put molecule names to "A", "B", etc.  Within each category, problems are listed roughly in order of difficulty.  Problems denoted with an asterisk (*) are more easily solved using Excel, MATLAB, or a powerful calculator.

Stoichiometry/Reaction Fundamentals

1. Consider the hydrogenolysis of ethane to methane:

CH₃CH₃ + H₂ ==> 2 CH₄

0.25 mole ethane and 0.50 mole H₂ are added to a batch reaction.  At the end of the process, you recover 0.20 mole methane.

a. Calculate the expected amounts of ethane and H₂ remaining at the end of the reaction.

b. Comment on the likelihood that the reaction, as written, is elementary.

2. A feed containing 6.00 mol% propane (C₃H₈) in air enters a continuous combustion reactor at 30.0 moles/minute. The exit stream is analyzed and found to contain

5.88 mol% O₂
7.26 mol% CO₂

a. Show that these measurements are not consistent with the stoichiometry of the combustion reaction.

b. Assuming that the measurements are reasonably accurate, how might this finding be explained?

3. Consider the homogeneous, gas-phase reaction,

A + 2B ==> R + S.

The feed to a continuous reactor contains 40 mol% A and 60 mol% B.   At steady state, the exit stream is found to contain 20 mol% A.

a. For the reactor run cited above, calculate the fractional conversion of A.

b. For this feed, what is the maximum possible fractional conversion of A?

(see Ideal PFR Design for a tougher variation of this problem.)

4. The following data are taken for the reaction    A ==> products:

a. Determine the apparent order of the reaction with respect to A.

b. Calculate the apparent activation energy E_a in kJ/mol (gas constant R = 8.314 J/mol-K).

Rate Law from Chemical Mechanism or Batch Kinetic Data

1. Consider the following overall reaction           A + B ==> R.

The following mechanism has been proposed:

P* and Q* are highly reactive intermediates with very low concentrations compared to [A], [B], and [R].

a. Using the pseudo-steady state approximation, derive expressions for [P*] and [Q*] in terms of major species concentrations ([A], [B], [R]) and rate constants.

b. Continuing with the pseudo-steady state approximation, derive a simplified expression for -r_A in terms of major species concentrations and rate constants.

2. Consider the overall reaction                A + B ==> R + S,

which proceeds according to the following chemical mechanism:

a. Derive an expression for -r_A assuming that reaction (3) is the rate-limiting step.

b. Derive an expression for -r_A assuming that reaction (2) is the rate-limiting step.

3. Consider the following chemical mechanism, where P* and Q* are highly reactive intermediates:

                   k₁
(1)        A <==> R + P*
                   k_-1

                          k₂
(2)        P* + A ==> Q*

                     k₃
(3)        2Q* ==> S

You do not have any indication as to which of these reactions, if any, might be a rate-limiting step.

a. Derive a (simplified) rate law in terms of major species concentrations and rate constants.

b. If the first reaction was the rate-limiting step, predict the apparent rate law.

4. Consider the following chemical mechanism for the production of R and S from reactants A and B (P* and Q* are highly reactive intermediates that do not accumulate):

a. Derive a rate expression for r_R, the rate of R formation, in terms of major species concentrations and rate constants.

b. The initial rate of R formation is measured for various values of C_A and C_B, the concentrations of A and B, respectively.  In Data Set #1, C_B is held constant and C_A is varied, whereas in Data Set #2, C_A is held constant and C_B is varied.  Describe how you would plot each of these data sets to test the rate law derived in part a, and sketch the expected results for both (label the values of the slope and intercept).  If you do not have an answer for part a, assume a rate law and proceed from there.

5. A catalyzed reaction is carried out in a well-mixed batch reactor (fluid volume V, catalyst weight W).  Reactant A, present at concentration C_A,0 initially, is consumed according to the following mechanism:

A + S <==> A•S                       Equilibrium constant K_eq,A.

2 A•S ==> products + 2 S         Rate constant k (kg/mol-min.).

The total density of active sites on the catalyst is S_Tot (mol/kg) = S + A•S.  The active site density for fresh catalyst is S_Tot,0.

Unfortunately, unoccupied sites are susceptible to deactivation by an apparently spontaneous mechanism:

S ==> "dead" site                    Rate constant k_d (min.^-1).

a. Set up differential equations for the rates of change of C_A and S_Tot in terms of C_A, S_Tot, and constant parameters.

b. Suppose now that K_eq,AC_A,0 << 1.  Solve for C_A(t).

6. The following reaction occurs in a constant-volume batch reactor:   A + B ==> R,

with apparent rate law     -r_A = (1.8 M^-1min^-1) C_AC_B.

The initial concentrations of A and B are 0.2 mM and 50 mM, respectively.

a. Calculate X_A, the fractional conversion of A, after 10 minutes.

b. Calculate X_B, the fractional conversion of B, after 10 minutes.

7. Consider the reaction       2A ==> R + S,

for which the following rate law has been proposed:

Your goal is to test this rate law, and so you decide to run the reaction in a constant-volume batch reactor, initially charged with a known value of C_A,0 and no R or S.   The concentration of A is measured at periodic time intervals until the reaction is near completion.

a. Specify which method you would use to test this rate law and explain why (one sentence should be sufficient).

b. With the data in hand, what plot would you construct to test this rate law?

Design of Continuous, Isothermal Reactors

1. Consider a liquid-phase reaction with the proposed rate law

-r_A = k C_Aⁿ ,

where the order of the reaction and rate constant value are unknown.  You decide to ascertain these quantities by varying the flow rate through a CSTR.  Steady-state measurements are made at two different flow rates, with the same reactor temperature:

V = 10.0 L,       C_A,0 = 0.10 M

Flow rate (L/s)             C_A (M)

      0.5                           0.027
      1.5                           0.059

a. Determine the apparent order of the reaction.

b. Estimate the value of the rate constant, k, given in proper units, at this temperature.
(If you couldn't get an answer to part a, assume one.)

2. Pure A (gas) is fed at 50.0 L/s (25°C, 1.0 atm) into a well-mixed CSTR (1000 L), maintained at 200°C, 2.0 atm, in which the following reaction occurs:

A ==> 2R.

The disappearance of A follows first-order kinetics, and the exit stream contains 15 mol% A.

Estimate (a) the rate of reaction, -r_A (mol/L-s), and (b) the apparent first-order rate constant, k (s^-1).

3. A gas feed (60°C, 1.0 atm) containing 50.0 mol% A and no R enters a 20.0 L CSTR, in which the following reaction occurs:

A ==> 2R.

The disappearance of A follows second order kinetics.

For a feed rate of 5.0 L/min., the conversion of A is 80%.  Estimate the flow rate that will achieve 90% conversion of A.

4. The decomposition of ozone (O₃) to produce oxygen (O₂) observes the following stoichiometry:   2 A ==> 3 R.

The apparent rate law (derived using the pseudo-equilibrium approximation) is

The reaction is carried out in a 2.0 L CSTR at constant temperature and pressure.  When pure A is fed at 1.00 L/min., the flow rate out of the reactor is 1.30 L/min.

a. Calculate X_A, the fractional conversion of A, under these conditions.

b. Derive expressions for C_A and C_R in terms of X_A and constant parameters.

c. Estimate the value of the rate constant k and give its units.

5. An aqueous feed of A (0.12 M) enters a 10.0 L plug flow reactor, in which A is consumed with second order kinetics.

For a feed rate of 1.0 L/min., the conversion of A is 75%.

a. Estimate the value of the apparent second order rate constant, k, and give its units.

b. Predict the conversion of A when the feed rate is increased to 3.0 L/min.

6. Consider the homogeneous, gas-phase reaction,

A + 2B ==> R + S.

The rate of reaction is first-order with respect to A and zero-order with respect to B.

The feed contains 40 mol% A and 60 mol% B and is fed to a 10.0 L PFR at 1 L/s, 25 °C, 1.0 atm.  The PFR is operated at 150 °C, 1.0 atm.  At steady state, the exit stream is found to contain 20 mol% A.

a. For the PFR run cited above, calculate the fractional conversion of A.

b. For this feed, what is the maximum possible fractional conversion of A?

c. How much larger would the PFR need to be to achieve the maximal conversion determined in part b?

7. You have two reactors available, a CSTR (100 L) and a PFR (150 L), to carry out a liquid reaction.  To achieve the highest conversion possible, you decide to put both reactors in series.  The reactors are to operate at the same constant temperature, and the rate law at that temperature is known:

-r_A (M/s) = 2.0 C_A^1.7 , with C_A in M.

You need to process 25 L/s of feed, and C_A,0 = 0.10 M.

a. What is the optimal order in which to place the two reactors?  Justify your answer (one sentence should suffice).

b. Estimate the % conversion you will achieve with this configuration.

Multiple Reactions

1. Consider the following liquid reaction:

A ==> R,  r_R = 0.5 C_A  [mol/(L-min.)]

Unfortunately, an undesired side reaction also occurs:

A ==> S,  r_S = C_A²  [mol/(L-min.)]

You are asked to design a flow reactor to process a feed containing 2 mol/L A and no R or S.

a. Which reactor type (CSTR or PFR) should you choose?  State the reasoning behind your selection.

b. For the reactor type you chose in part a, calculate the space time τ required to produce 1.50 mol/L of R.

2. Consider the following reactions, which are all occurring in a continuous, isothermal reactor.  The rate of each reaction (mol/L-min.) is written above the arrow.

a. Perform a qualitative analysis considering each of the following possibilities:

i) you wish to maximize the Yield of R.

ii) you wish to maximize the Yield of S.

iii) you wish to maximize the Yield of T.

In each case, decide whether you would choose a CSTR or a PFR, and whether or not there will be a finite value of τ that maximizes yield.

b. The reactions are carried out in a CSTR with τ = 10 min. and C_A,0 = 1 mol/L (C_R,0 = C_S,0 = C_T,0 = 0).   Calculate the concentrations of A, R, S, and T leaving the reactor.

3. Consider the liquid reactions in series:

(both reactions are elementary).

Your goal is to maximize production of R from a feed containing 1 mol/L A (and no R or S), but you don't know the rate constants.  You decide to do a quick and dirty batch experiment.  When some of the feed stock is incubated at the process temperature for 1 minute and then analyzed, it is found that 0.7 mol/L R and 0.1 mol/L S are present.

a. Calculate k₁ and k₂.

b. For a flow rate of 100 L/min., design a reactor to maximize the production of R; state whether you would choose a CSTR or PFR and specify the reactor volume (If you couldn't do part a, show how you would obtain the volume given k₁ and k₂).

Reactor Design with Energy Balance

1. Consider the aqueous reaction, which follows second-order kinetics in a well-mixed CSTR (100 L):

A ==> R;    ΔH_r° = -50 kJ/mol

Feed: T₀ = 35°C; C_A,0 = 1.0 M; υ = 1.0 L/s; ρC_p = 4.3 kJ/(L-°C).

The reactor is to be maintained at a temperature T = 90°C, at which the value of the rate constant is k = 0.20 L/(mol-s)

a. Calculate the fractional conversion of A.

b. Will the reactor need to be heated or cooled?

2. Consider the aqueous reaction, which occurs in a well-mixed and well-insulated CSTR (100 L):

A ==> R;    ΔH_r° = -500 kJ/mol

-r_A = (10⁶ s^-1)exp[-6000/T(K)] C_A

Feed: T₀ = 40°C; C_A,0 = 1.0 M; ρC_p = 4.0 kJ/(L-°C) .

a. Calculate the maximum possible reactor temperature.  Under what conditions is it achieved?

b. Calculate the flow rate (L/s) that will give 90% conversion of A.

c. In order to increase production of R, the operator slowly increases the flow rate through the reactor.  At some point, however, the conversion of A abruptly drops to an unacceptably low level.  The operator then decides to slowly decrease the flow rate in an effort to regain conversion. What will happen?  State how you would explain this behavior to the operator.

3. Consider a first-order liquid reaction with A as the reactant.   The properties of the feed are:

C_A,0 = 0.1 M, T₀ = 50°C.

When the reaction was run in a well-insulated CSTR, the steady-state reactor temperatures were measured for the following space times:

τ (min.)                     T(°C)             

2                                60

4                               140

VLT                         200    (VLT = "very long time")

4. A well-mixed CSTR (V = 1000 L) is used to carry out the combustion of ethane gas:

C₂H₆ + (7/2) O₂ ==> 2 CO₂ + 3 H₂O;    ΔH_r° = -1428 kJ/mol.

The feed (10 L/s, 2 atm, 25°C) consists of 5.0 mol% ethane in air.  The reactor operates at 1 atm, and the maximum permissible reactor temperature is 600°C.

Previous trials have shown that the decomposition of ethane is described by

-r_ethane(M/s) = 5.15x10⁶ exp[-9500/T(K)] [C₂H₆] [O₂], concentrations in M

Other info: see Felder & Rousseau, Tables B.2 & B.8.

a. Derive expressions for [C₂H₆] and [O₂] in terms of X_ethane, the fractional conversion of ethane.

Notes: Are the number of moles changing?  Are the temperatures and pressures of the reactor and feed different?

b. Calculate the maximum conversion of ethane.

c. To achieve the conversion calculated in part b, will the reactor need to be heated or cooled?

5*. Consider the aqueous reaction, which occurs in a well-mixed and well-insulated CSTR:

A ==> R;    ΔH_r° = -50 kJ/mol

-r_A (M/s) = 10⁹ exp[-7000/T(K)] C_A², with C_A in M.

Feed: T₀ = 35°C; C_A,0 = 1.0 M; υ = 1.0 L/s; ρC_p = 4.3 kJ/(L-°C).

Estimate (a) the fractional conversion of A and (b) the temperature of the reactor at steady state.

6. An adiabatic PFR is used to carry out the following liquid reaction:

A ==> R;   ΔH_r° = 100 kJ/mol;     -r_A = (1.0x10⁶ s^-1) exp[-5000/T(K)] C_A

The feed contains C_A,0 = 2 mol/L, and T₀ = 80°C.

Liquid properties:

density = 900 g/L

heat capacity = 4.0 J/(g-°C)

a. The reactor is to be operated such that the exit temperature T = 40°C.  Calculate the conversion of A under those conditions.

7. An adiabatic PFR (100 L) is used to carry out the following endothermic liquid reaction:

A ==> R;   ΔH_r° = 100 kJ/mol;

-r_A (M/s) = 1.70x10⁶ exp[-5000/T(K)] C_A², with C_A in M.

Feed properties: 10 L/s, C_A,0 = 1.0 M, 65°C

Liquid properties:

density = 950 g/L

heat capacity = 3.9 J/(g-°C)

Calculate the temperature of the exit stream and the conversion of A at steady state.

8. Consider a first-order liquid reaction with A as the reactant, and rate constant

k (s^-1) = (1.0x10³)exp[-4000/T(K)].

The properties of the feed are:

C_A,0 = 0.1 M, T₀ = 325 K.

Based on a calorimetry experiment, you find that the adiabatic temperature rise is 150 K.

Determine which reactor type, CSTR or PFR (both well-insulated), will achieve 80% conversion more efficiently, i.e. with the smaller value of τ.

Heterogeneous Catalysis

1. The reaction A + B ==> product

in an isothermal, constant density fluid is carried out in a continuous, fluidized bed reactor containing solid catalyst (well-mixed).  The reaction observes the following rate law:

-r_A = k C_A C_B   (mol/kg-cat/s);

however, the reaction mechanism is not known.  You have at your disposal two catalyst varieties; the physical and chemical properties of the two catalysts are identical, except that Catalyst # 2 has exactly double the density of active sites as Catalyst #1.  The following steady-state data are obtained:

a. Determine the apparent rate constants k for each of the runs, in appropriate units.

b. Based on your answer to part a and knowledge of the two catalysts, provide a plausible mechanism for how the active sites of the catalyst facilitate the reaction (If you can not completely answer part a, describe how the answer to part a would address this question).

2. A first-order reaction is carried out in a well-mixed, fluidized bed reactor.  The rate constant in the presence of fresh catalyst is 3.0 L/kg-min. The feed is a constant-density fluid containing 1 mol/L of A that enters the reactor at 10 L/min.

a. What catalyst weight is required to yield 80% conversion of A with fresh catalyst?

3. The first order reaction  A ==> products

in an isothermal liquid is carried out using a solid, permeable catalyst in a packed-bed, tubular reactor.  The fluid flow past each particle is in the turbulent regime (Re ~ 10⁴).  Steady state runs are made with different average particle diameters d_p:

a. Based on the data provided, comment on the apparent extent to which intraparticle diffusion resistance influences the observed kinetics.

b. Do we need to consider the influence of external mass transfer resistance?  Be quantitative in your reasoning.

4. Consider the decomposition of A:   A ==> R,

carried out in a tubular, packed bed catalytic reactor.  The reaction follows first-order kinetics:

-r_A (mol/kg-cat/s) = k C_A

a. Calculate the apparent rate constant, kη, in appropriate units.

b. Determine whether the reaction is operating with minimal, intermediate, or strong pore diffusion resistance.

5. The first order reaction    A ==> products

in an isothermal, constant density fluid is carried out in a well-mixed batch reactor containing solid, permeable catalyst.  The catalyst undergoes deactivation with first order, independent kinetics, and the catalyst particle size is such that the Thiele modulus for fresh catalyst is 0.1.  The following kinetic data are obtained:

V = 25 L;   W = 10 kg

a. Estimate the values of the reaction rate constant k (L/kg/min.) and deactivation rate constant k_d (min.^-1).

b. Suppose the particle size is increased such that the Thiele modulus for fresh catalyst is now 5.  How will this change, in relative terms, the observed k and k_d calculated from early time points?

6. The gas-phase reaction    A ==> products

is carried out at constant temperature in a well-mixed, constant-volume batch reactor containing a solid catalyst (fluid volume V = 10 L, catalyst weight W = 5 kg, ρ_cat = 2.0 kg/L).  The catalyst particles are slabs with a mean thickness of 0.4 cm (L = 0.2 cm).  For the concentrations of A encountered, the reaction proceeds with zero-order kinetics.

One can show that the slab geometry, zero-order kinetics case is unique in the following way: as the Thiele modulus φ increases, the effectiveness factor switches from η = 1 to η = 1/φ without an intermediate regime of pore diffusion resistance.

The concentration of A is measured at 2 minute intervals (see plot below, with concentration values given above each symbol).

a. Estimate the value of the rate constant k, with catalyst weight as the basis.

b. Estimate the value of D_A,eff, the effective diffusion coefficient of A, in cm²/s.

7. A reaction is carried out in a continuous, well-mixed fluidized bed reactor, loaded with 10 kg of catalyst.  The constant-density feed stream contains 1.0 M reactant (A) and is fed at a rate of 0.10 L/s.

Under these conditions, the steady-state concentration of A exiting the reactor is found to be 0.20 M (80% conversion).

The catalyst particles are well characterized, with a density of 2.0 kg/L, an effective diffusion coefficient of A of 1.0x10^-6 cm²/s and a mean V_p/A_p ratio of 150 μm (0.015 cm).

Note: The order of the reaction is uncertain; however, based on your knowledge of kinetics you feel confident that n is between 0 and 2.

a. Determine whether the reaction is operating with minimal, intermediate, or strong pore diffusion resistance.

When the flow rate is increased to 0.25 L/s, you find that the steady-state concentration of A in the exit stream is 0.33 M (67% conversion).

8*. Consider a fluidized bed reactor used to make a valuable product R from a less expensive reactant A.  The feed contains 1.0 M A and no R, and the contents of the reactor are well mixed.  The catalyst processes A ==> R with the following kinetics:

-r_A = k C_A;     k = 10 L/kg-cat/s,

and it is deactivated with first order, independent kinetics (k_d = 0.1 s^-1).  The catalyst is recovered from the exit stream and recycled, and a fraction φ of the recycle stream is discarded and replaced with fresh catalyst.

Your goal is to assess the profitability of the process and design it accordingly.  The relevant prices are:

Product price (p_R): $2/mole

Cost of the feed (p_feed): $1/L

Cost of fresh catalyst (p_cat): $0.50/kg

Other costs can be neglected or considered constant for a given reactor size.

Constraints:

The required purity of your product is such that the conversion of A must be at least 95%.

The maximum concentration of catalyst in the reactor (W/V) is 1.0 kg/L.  Any denser, and you will not be able to fluidize the suspension.

a. Derive an expression for the rate of profit per unit reactor volume, p/V, in terms of φ, W/V, X_A, and the space time τ.

b. Design the reactor setup to maximize p/V; specify τ, φ, W/V, and X_A.

c. A sales rep for the catalyst manufacturer informs you that they have developed a new and improved catalyst with double the density of active sites (k = 20 L/kg-cat/s).  The only catch is that it is four times the price ($2/kg).  Do you make the switch?

d. In this model, the rate of profit increases in proportion to the reactor volume if the intensive variables specified in part b are held constant.  It follows that you would want to make your reactor as large as possible to increase profits.  Discuss the possible disadvantages or constraints you might encounter with the process as it becomes increasingly large, and suggest how the profit model could be improved to reflect those constraints.