FFR: Multiple Reactions

In all this chapter, we have only looked at situations where we have one reaction progressing in one direction. What do we do, then, if more than one reaction is progressing in the same system?

If you recall, we actually did cover this already, though it was fairly well hidden. In our example problem last section involving Calcium Carbonate formation, there were multiple reactions occurring at the same time. Can you identify what reactions we had in that system? There were three in total.

Ca²⁺(aq)+CO₃²⁻(aq)→CaCO₃(s) describes the net precipitation. Solid CaCl₂ and Na₂CO₃ can dissolve to produce ions, but salts already supplied as aqueous solutions are already represented by dissolved ions. Neither a net dissolution nor a precipitation equation automatically describes an elementary step.

What does that mean for us? Well, let’s look at the overall reactions, starting from the salts and ending with the Calcium carbonate.

For mixed aqueous solutions, the complete ionic description has Na⁺ and Cl⁻ as spectators and the net equation Ca²⁺(aq)+CO₃²⁻(aq)→CaCO₃(s). A mechanistic rate model needs evidence and explicit conditions; do not assign exponents solely from these equations.

A sequential chemical mechanism consists of proposed elementary steps. Identify intermediates and cancel them when summing steps. The measured overall rate law tests a mechanism; a balanced overall equation alone does not specify it.

Let’s take a step back from chemistry for the time being and just look at a real world example of this situation.

Let’s say that the plastic formation can create up to 100Mg of plastic per day from petroleum. The bottle factories have 1242 trucks that can move up to 2000kg of plastic per day. The bottle factories can together make up to 300,000 bottles per day, with each bottle using 1.2kg of plastic to make.

The syrup plants can produce 400L of syrup per day. Each bottle requires 1mL of syrup to reach the desired concentration. The bottling plants can fill up to 500,000 bottles per day.

We want to produce more bottled products each day, and we have only have enough money to improve one aspect of the production. Which aspect should we focus on improving?

To figure this out, we have to understand which aspect of production is slowing us down the most. In order to do so, we need to be able to compare every aspect on an single basis of comparison. Perhaps the easiest is to look at each aspect in terms of the bottles it can support per day.

The plastic formation creates 100Mg of plastic per day. We know that each bottle uses 1.2kg of plastic in its production, so that means the plastic formation can support up to 83,333 bottles per day.

The trucks can move up to 2000kg of plastic per day. With 1242 trucks, that means 2484Mg of plastic moved per day, which can make 2,070,000 bottles per day.

The bottle factories have a strict limit of 300,000 bottles per day.

If the syrup plants produce 400L, and you only need 1mL per bottle, that means 400,000 bottles per day can be supported.

The bottling plants have their limit at 500,000 bottles per day.

Now that we know what the capacities of each aspect is in bottles per day, we know what the slowest point of production is. That would be the plastic formation. Therefore, we should focus on improving our plastic formation capabilities. This is because if we were to improve some other aspect, we would not be improving the overall rate of production, as we can only produce so much plastic per day.

The factory analogy compares capacities on one basis: bottles per day. A chemical rate constant is not a capacity on that basis. Step flux depends on concentrations as well as k, and different-order constants can have different units.

A slow step can constrain a reaction under stated conditions, but selecting it needs a justified mechanism, rate law and concentration or energy-barrier information. Comparing the numerical values of unrelated k values is not enough.

Reviewed diagnostic, Example3.4.1: The source supplied four schematic steps with constants0.1353,0.00052 and0.022 L mol⁻¹ s⁻¹, and0.04 for a unimolecular BX→B+X step. If that last step is explicitly elementary first order, its k must have units s⁻¹. Explain why these constants alone cannot establish a rate-limiting step or overall empirical rate law. Species concentrations, a chemically consistent elementary mechanism and appropriate kinetic assumptions are missing.

For a fast pre-equilibrium followed by a slower elementary step, use the equilibrium relation to eliminate an intermediate from the slow-step law. The pre-equilibrium approximation must be justified by relative forward/reverse rates and the small disturbance made by the following step. Reversibility alone does not prove that a step stays at equilibrium.

Another situation you might encounter is reactions that use the same reactant, which proceed simultaneously. That would look something like:

$A+B \rightarrow AB$ $k_1$
$C+B\rightarrow CB$ $k_2$

For two independent1:1 sinks of B, add their species disappearance contributions. If kinetic evidence or explicit elementary-step assumptions establish r₁=k₁[A][B] and r₂=k₂[C][B], then −d[B]/dt=r₁+r₂. Merely knowing each total order is two does not establish first order in each reactant.

Under those declared laws, −d[B]/dt=k₁[A][B]+k₂[C][B]. Other consumption coefficients would multiply the corresponding event rates.

Reviewed diagnostic, Example3.4.3: The source supplies [CO₃²⁻]=4.00 M, [Ca²⁺]=2.10 M and [OH⁻]=1.20 M, and proposes two sink laws r₁=0.00324[Ca²⁺][CO₃²⁻] and r₂=0.00562[Ca²⁺][OH⁻], with coefficients in L mol⁻¹ s⁻¹, plus a calcium supply0.0332[CaCl₂] M s⁻¹. Treat these as formally supplied empirical laws, not laws inferred from reaction stoichiometry. Write the conditional net disappearance expression and determine what further information a numerical answer needs. The formal molecular CaCl₂ source model is not established for an already dissolved strong electrolyte.

Net process1: Ca²⁺(aq)+CO₃²⁻(aq)→CaCO₃(s). The declared sink law is r₁=(0.00324 L mol⁻¹ s⁻¹)[Ca²⁺][CO₃²⁻]; charge on carbonate is2−.

Net process2: Ca²⁺(aq)+2 OH⁻(aq)→Ca(OH)₂(s). The declared empirical sink law is r₂=(0.00562 L mol⁻¹ s⁻¹)[Ca²⁺][OH⁻]. Its order was supplied, not inferred from the coefficients. This net ionic equation is not declared an elementary step.

A proposed supply term s=(0.0332 s⁻¹)[CaCl₂] requires a defined precursor population and its concentration. Aqueous CaCl₂ is represented as Ca²⁺ and2 Cl⁻, so a first-order reservoir of molecular dissolved CaCl₂ cannot simply be assumed. Without a justified supply model, even this term is conditional.

1. A rate constant is0.0332 s⁻¹. Assuming a conventional concentration-rate power law, with rate in M s⁻¹ and concentrations in M, what is the overall order? Does this identify the reactant powers individually?
2. At constant volume and temperature, a1:1 reaction A+B→P starts with [A]₀=[B]₀=10.0 M and has no other sources or sinks. The following rounded formal data are supplied. Determine the supported standard disappearance model and its first half-life.

   Time(s)	[A](M)
   0	10
   5	9.09
   10	8.333
   15	7.692
   20	7.142
   25	6.666
   30	6.25
   35	5.882

3. Use the corrected table and maintained-equality model from Problem2. Assuming that fitted disappearance law remains valid outside the measured range, estimate when[A] reaches0.230 M.
4. The source describes10 g solid NaHCO₃ added to10 mL of10 M acetic acid, gives k=0.02324 L mol⁻¹ s⁻¹ and asks for CO₂ volume after5 min at1 atm and25 °C. The balanced net stoichiometry is NaHCO₃+CH₃COOH→CH₃COONa+H₂O+CO₂. Explain why no unique time-dependent gas volume follows from these supplied data. Identify the missing kinetic information.
5. Use the reviewed graphs in Gallery3.4.2. Each represents the same supplied model[A](t)=exp(2.4998−0.0230t) M, with t in seconds. Determine the supported order and rate constant, including units.

6. Quantitative enrichment: a hypothetical first-order disappearance model has A=4.31 s⁻¹, Eₐ=422 kJ mol⁻¹ and T=45.0 °C. Use R=8.314 J mol⁻¹ K⁻¹. Find k and its half-life;[A]₀=4.00 M. Keep the stated422 kJ mol⁻¹ value.
7. Reviewed replacement with an explicit target: A hypothetical undissolved solute population obeys dn_u/dt=−0.320n_u s⁻¹, beginning at4.00 mol. A monitor’s threshold for this remaining population is5.00 mg NaCl-equivalent per kilogram of water; water mass=10.0 kg and equivalent molar mass=58.44 g mol⁻¹. Find the time to the threshold. This changes the source’s ambiguous target; dissolved ions would increase during dissolution, rather than disappear.
8. Quantitative enrichment: two supplied first-order disappearance plots in Gallery3.4.3 have ln([AB]/1 M)=4.85−0.230t at85.0 °C and4.85−0.0500t at35.0 °C, with t in seconds. AB is the defined remaining modeled population, not a measured increase of dissolved ions. Estimate Eₐ and A using the same Arrhenius model at both temperatures and R=8.314 J mol⁻¹ K⁻¹.
9. Three schematic processes have explicitly supplied event rates r₁=0.00325[AB][CD], r₂=0.221[AB] and r₃=0.322[A⁺][B⁻], each in M s⁻¹. AB is consumed once per event in processes1 and2, and produced once per event in3. The bimolecular constants have L mol⁻¹ s⁻¹ units and0.221 has s⁻¹ units. Write net AB disappearance.
10. Reviewed schematic population model: X₁→X₂→X₃→X₄ consists of irreversible1:1 first-order conversions with k₁=0.411, k₂=0.121 and k₃=0.523 s⁻¹. X labels count equivalent conserved units; these are not unbalanced nitrogen chemical equations. Find the half-life of X₁. Which step has the longest isolated first-order lifetime, and why does that alone not determine every instantaneous step flux?
11. Quantitative enrichment: a hypothetical fixed-volume1:1 A+B→P disappearance law is −d[A]/dt=k[A][B]. Both initial concentrations are10.0 M, with no additional sources or sinks. A=0.0324 L mol⁻¹ s⁻¹ and Eₐ=24.0 kJ mol⁻¹. Compare time to[A]=3.00 M at35.0 versus20.0 °C, using R=8.314 J mol⁻¹ K⁻¹. Treat these concentrations as formal model data.
12. A supplied1:1 disappearance law is −d[A]/dt=k[A][B], with k=0.0734 L mol⁻¹ s⁻¹. A reservoir holds[B]=10.0 M constant. Find A’s half-life in this formal model.
13. Two supplied irreversible first-order sinks of A have k₁=0.0003242 and k₂=0.000632 s⁻¹. The following formal concentration data include an additional unidentified sink. Show that the two known sinks are insufficient, and explain why the table alone does not uniquely establish one simple order for the extra process.

    Time(s)	[A](M)
    0	20
    30	14.6
    60	10.5
    90	7.35
    120	4.97
    150	3.14
    180	1.72
    210	0.596

14. The source describes precipitation while a solution warms at5.00 °C min⁻¹ from35.0 °C, and asks when it boils. Explain why rate constants and ion concentrations alone do not establish the boiling temperature. As a separate stipulated temperature-threshold calculation, find time to100.0 °C if the given constant temperature-rise rate continues.
15. The source calls iron oxidation pseudo-first-order but supplies k=3.2421 L mol⁻¹ yr⁻¹ and says oxygen is constant without giving its value. Explain why the time to24% conversion is underdetermined. Give the expression if the law were specifically −d[A]/dt=k[A][O₂].
16. Two formal reactions consume A: A+B→AB has event rate r₁=0.23511[A]², and2A→A₂ has event rate r₂=0.331[A]. Their constants use L mol⁻¹ s⁻¹ and s⁻¹, respectively. Write A disappearance and explain why time to95% consumption and product amounts need more information.
17. A source sequence supplies r₁=0.753[A][BC] and r₂=0.441 for a following step using AB. Initially[A]=[BC]=5.00 M. Explain what is missing before a crossover time can identify a kinetic bottleneck. As a conditional capacity comparison only, state the concentration at which r₁ equals0.441 M s⁻¹ when[A]=[BC].
18. Reviewed replacement: A hypothetical auxiliary reactant R is consumed1:1 with A during a first stage: −d[A]/dt=−d[R]/dt=0.320 M s⁻¹ while R remains. [R]₀=3.00 M and[A]₀=5.00 M. When R reaches zero, A follows −d[A]/dt=0.02421[A] s⁻¹. Find time from the start to[A]=1.00 M. R is a consumed reactant, not a catalyst.
19. Reviewed replacement: A hypothetical remaining material fraction f obeys −df/dt=k p f. k=5.00 atm⁻¹ h⁻¹, and a reservoir keeps the relevant gas partial pressure p=0.210 atm. Find the time for f to fall from1.00 to0.200. This abstract supplied model is not a wood-burning prediction or experimental procedure.
20. Reviewed replacement: A hypothetical ideal-gas reservoir model has A₂+B₂→2 AB. Reservoirs keep x(A₂)=x(B₂)=0.0400 and total molar density C=P/(RT) fixed, with P=1.00 atm,T=298.15 K,R=0.082057 L atm mol⁻¹ K⁻¹. The event rate is r=0.321[A₂][B₂] M s⁻¹; AB initially has zero concentration and accumulates without removal until x(AB)=0.0100. Find the model time. This is an abstract calculation, not a room-exposure scenario.