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Wize University Chemistry Textbook > Kinetics

The Rate Law

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Rate Laws

  • We can use inital reaction rates (using initial concentrations of reactants) to get an expression of the reaction rate called the rate law.
aA + bB → cC + dDaA\ +\ bB\ \rightarrow\ cC\ +\ dD

rate=k[A]x[B]yrate=k[A]^x[B]^y

Wize Concept
It's important to be able to recognize the rate law!

  • The rate law only includes reactants/products:
    reactants
  • k is the rate constant

Watch Out!
We use x and y here because x and y are NOT the coefficients in the balanced equation!

𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑟𝑑𝑒𝑟 =x+y\text{𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑟𝑑𝑒𝑟 =}x+y

The reaction order can't be determined from the overall balanced equation

a) We could determine the reaction order from experimental data

OR

b) We could determine the reaction order from a reaction mechanism (that shows elementary steps)


  • We will take a look at both of these ways to figure out the order of the reaction next!

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Determine the order of each reactant and the overall order for the following rate laws.
  1. v=k[NO][C]2v=k[\rm NO][C]^2
  2. v=k[O3]1v=k[\rm O_3]^1
  3. v=k[O2]1/2[CH4]v=k[\rm O_2]^{1/2}[CH_4]
  4. v=kv=k

  1. 1st order in NO\rm NO, 2nd order in C\rm C, 3rd order overall
  2. 1st order in O3\rm O_3, 1st order overall
  3. half order in O2\rm O_2, 1st order in CH4\rm CH_4, 1.5 order overall
  4. 0 order reaction
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Determining the Rate Law from Experimental Data

The reaction order can't be determined from the overall balanced equation

a) We could determine the reaction order from experimental data

OR

b) We could determine the reaction order from a reaction mechanism (that shows elementary steps)


Wize Tip
This method of finding the rate law is found on tests quite often!

Part 1: Use the rate data provided to determine an experimental rate law for the following reaction:


BrO3−(aq)+5Br−(aq)+6H+(aq)−>3Br2(aq)+3H2O(l)BrO_3^-\left(aq\right)+5Br^-\left(aq\right)+6H^+\left(aq\right)->3Br_2\left(aq\right)+3H_2O\left(l\right)


1) We'll write the general rate law expression:

rate=k[BrO3 −]x[Br−]y[H+]zrate=k\left[BrO_3^{\ -}\right]^x\left[Br^-\right]^y\left[H^+\right]^z


2) Let's figure out what x would be (exponent for [BrO3-]):

(it doesn't matter which exponent we choose to solve for first!)
It makes sense to look at trials 1 and 2 because in these trials we have the concentration of this reactant doubling while the other reactants' concentrations stay the same. This means that whatever happens to the rate is because of [BrO3-]
[BrO3-] gets multiplied by 2 from trials 1 to 2.
The rate gets multiplied by 2 as well.
rate=k[BrO3-}x
2=k[2]x
Therefore x=1
rate=k[BrO3-][Br-]y[H+]z

3) Let's figure out what y would be (exponent for [Br-]):

It makes sense to look at trials 2 and 3 here because the concentrations of the other reactants aren't changing. As a result we know that any change in the concentration of Br- is solely responsible for any change in the rate of the reaction.
From trial 2 to 3 [Br-] gets multiplied by 2.
From trial 2 to 3, the rate gets multiplied by 2.
This means that y=1
rate=k[BrO3-][Br-] [H+]z

4) Let's figure out what z would be (exponent for [H+]):

Here, the easiest way to solve this problem would be to consider trials 1 and 4. This is because in trials 1 and 4 the concentration of the other reactants don't change, but the concentration of H+ doubles so any change we see in the rate is due to the change in concentration of H+!
From trial 1 to 4 we see that the rate gets multiplied by 4!
rate=k[H+]z
4=k[H+]z
This means z=2

5) Finally, we can write out the completed rate law:

rate=k[BrO3−][Br−][H+]2rate=k\left[BrO_3^-\right]\left[Br^-\right]\left[H^+\right]^2

Part 2: What is the overall reaction order?

To figure out the overall reaction order all we need to do is add up the exponents in the rate law:
1+1+2=4
Therefore the overall reaction order is 4!

Part 3: What is the rate constant?

To find k we can just substitute data in directly from the table with the experimental data.
We need the completed rate law and we can choose any trial to take #s from (choose the trial with the lowest numbers for easier calculations!)
Here we'll use trial 1:

rate=k[BrO3−][Br−][H+]2rate=k\left[BrO_3^-\right]\left[Br^-\right]\left[H^+\right]^2

According trial 1, rate=8.0x10-14, [BrO3-]=0.1M, [Br-]=0.1M, [H+]=0.1M

Now rearrange to solve for k (the rate constant)!

k=rate[BrO3−][Br−][H+]2k=\frac{rate}{\left[BrO_3^-\right]\left[Br^-\right]\left[H^+\right]^{^2}}

k=(8x10−4)[0.1][0.1][0.1]2k=\frac{\left(8x10^{-4}\right)}{\left[0.1\right]\left[0.1\right]\left[0.1\right]^2}

k=8

What about the units for k?

Recall rate has units:
M/s
And concentration has units:
M
k=Ms[M][M][M2]k=\frac{M}{s\left[M\right]\left[M\right]\left[M^2\right]}

Therefore, k has units M-3 s-1

Part 4: If we were asked to solve for the rate but were given k and the reactant concentrations for a trial, how would we do that?

rate=8[BrO3-][Br-][H+]2
Plug in the k value and the concentration values for a trial into the rate law to solve for the rate!

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Shortcut to Find Units of k Quicker


For 0th order: k has units M s-1

For 1st order: k has units s-1

For 2nd order: k has units M-1 s-1

Recognize a pattern?

What would the units for k be for a 3rd order reaction?
For 3rd order: k has units M-2 s-1

In summary, units of k will be: M1-n s-1

Low yield:
What if the answer options don't have M but instead have moles and L?
What would the units be for a 2nd order reaction for example?

Well M=mol/L
So for a 2nd order reaction, we know the units would be M-1 s-1
This is the same as writing: 1Ms\frac{1}{Ms}
Now let's substitute M for mol/L:
1Lmol(s)\frac{1L}{mol\left(s\right)}
Which could be written as: Lmol-1s-1
Use the rate data provided to determine an experimental rate law for the following reaction. Solve for the rate constant, k, as well.
CH3MgBr+H2O→CH4+HOMgBrCH_3MgBr + H_2O \to CH_4 +HOMgBr

The following data was gathered for the reaction:
2A+B+2D→E+2F2A+B+2D\to E+2F
Find the rate law exponents and the rate constant for this reaction. Report your exponents as integers and your rate constant as a decimal to two significant figures without units in the answer field.
v=k[A]x[B]y[D]zv=k[A]^x[B]^y[D]^z

The reaction A + 2B → products was found to follow the rate law: rate = k[A]2[B]. Predict by what factor the rate of reaction will increase when the concentration of A is doubled, the concentration of B is tripled, and the temperature remains constant.
Suppose a gas-phase reaction has the rate law, rate = k[A][B]2 Which of the following could be the units for the rate constant, k?
Extra Practice
Rate Laws: Concentration
The reaction A + 2B → products was found to follow the rate law: rate = k[A]2[B]. Predict by what factor the rate of reaction will increase when the concentration of A is doubled, the concentration of B is tripled, and the temperature remains constant.
Rate Laws: Relative Rates of Reactions
Consider the balanced chemical equation for the reaction
2CO(g)+O2(g)⟶2CO2(g)Rate=k[CO]22CO(g)+O_2(g)⟶2CO_2(g)\qquad Rate=k[CO]^2
If the rate of the reaction is 3.2 M/s which of the following is TRUE.
A proposed mechanism for the reaction of NO2 with CO is shown below. Which of the given rate laws would accurately describe a reaction with this mechanism?
NO2(g)+NO2(g)→NO3(g)+NO(g)slowNO3(g)+CO(g)→NO2(g)+CO2(g)fastNO2(g)+CO(g)→NO(g)+CO2(g)overall\def\arraystretch{1.5} \begin{array}{cc} NO_2(g)+NO_2(g)\to NO_3(g)+NO(g) & slow \\ NO_3(g)+CO(g)\to NO_2 (g) +CO_2 (g) & fast \\ \hline NO_2(g)+CO(g)\to NO(g)+CO_2(g) & overall \end{array}
Given the following data for the NH4++NO2−→N2+2H2ONH_4^++NO_2^-→N_2+2H_2O reaction, the rate law for the reaction is
Trial[NH4+][NO2–]Rate10.010M0.030M0.020M/s20.0150.0300.03030.0100.0150.005\def\arraystretch{1.5} \begin{array}{c} Trial &[NH_4^+] &[NO_2^–]& Rate \\ \hline 1 & 0.010 M & 0.030 M &0.020 M/s \\ 2& 0.015& 0.030 &0.030 \\ 3&0.010 & 0.015 &0.005 \\ \end{array}
Assume the following reaction occurs by the given reaction mechanism.
3H2+CO→CH4+H2O3H_2 +CO \to CH_4+H_2O
Step 1 H2+CO→H2COH_2 +CO \to H_2CO Slow
Rate Laws: Concentration
The reaction A + 2B → products was found to follow the rate law: rate = k[A]2[B]. Predict by what factor the rate of reaction will increase when the concentration of A is doubled, the concentration of B is tripled, and the temperature remains constant.
Rate Laws
The reaction A + 2B → products has been found to have the rate law, rate = k[A][B]2 . While holding the concentration of A constant, the concentration of B is increased from x to 3x . Predict by what factor the rate of reaction increases.
Rate Laws: Relative Rates of Reactions
Consider the balanced chemical equation for the reaction
2CO(g)+O2(g)⟶2CO2(g)Rate=k[CO]22CO(g)+O_2(g)⟶2CO_2(g)\qquad Rate=k[CO]^2
If the rate of the reaction is 3.2 M/s which of the following is TRUE.
For the following elementary reactions apply the Law of mass balance to write a rate law, What is the overall order of the reaction? (Enter answer as 1st, 2nd, 3rd etc)
a) B2H6→2 BH3\rm B_2H_6\quad\overrightarrow{\hspace{1.5cm}}\quad 2\ BH_3
b) Xe+F2→XeF2\rm Xe\quad+\quad F_2\quad\overrightarrow{\hspace{1.5cm}}\quad XeF_2
Hydrofluoric acid, HF, is a weak acid which exists as a gas. However HF is incredibly corrosive and can dissolve porcelain, metal, wood etc. Aqueous solutions of HF are featured on Breaking Bad as a method for disposing of bodies. If light is shone on HF in the gas phase it rapidly decomposes to H2 and F2 by the following mechanism.
aaa2HF→H2+F2aaa‾\underline{\textcolor{white}{aaa}2HF \to H_2+F_2\textcolor{white}{aaa}}
HF→k1⋅H+⋅FHF \xrightarrow{k_1} \cdot H + \cdot F
Which of the following has NO EFFECT on the rate of a zero-order reaction
Write a general rate law for the following elementary reactions
CH4→⋅CH3+⋅H\rm CH_{4}\quad\overrightarrow{\hspace{1.5cm}}\quad \cdot CH_{3}\quad+\quad \cdot H
⋅Br+⋅Br→Br2\rm \cdot Br\quad+\quad \cdot Br\quad\overrightarrow{\hspace{1.5cm}}\quad Br_{2}
The Rate Law
The graph below describes the reaction
A→BA \rightarrow B
What is the slope of this line?
Determine an empirical rate law for the following reaction using the kinetic data collected at 298 K298\ Kand evaluate the rate constant kk. Report your k value with 2 sig figs and no units.
2 Fe+Al2O3+6 HCl→Fe2O3+2 AlCl3+3 H2O\rm 2\ Fe\quad+\quad Al_2O_3\quad+\quad6\ HCl\quad\overrightarrow{\hspace{1.5cm}}\quad Fe_2O_3\quad+\quad2\ AlCl_3\quad+\quad3\ H_2O
Run[Fe]0[Al2O3]0[HCl]0initial rate11.0 M1.0 M1.0 M0.02021.0 M2.0 M1.0 M0.08032.0 M1.0 M1.0 M0.02043.0 M2.0 M2.0 M0.160\begin{array}{|c|c|c|c|c|}\hline \rm Run & \rm [Fe]_0 & \rm [Al_2O_3]_0 & \rm [HCl]_0 &\rm initial\ rate\\\hline 1& 1.0\ M& 1.0\ M& 1.0\ M& 0.020\\\hline 2 &1.0\ M& 2.0\ M& 1.0\ M& 0.080\\\hline 3 &2.0\ M& 1.0\ M& 1.0\ M& 0. 020\\\hline 4 &3.0\ M &2.0\ M& 2.0\ M& 0.160\\\hline \end{array}
Determining Rate Law Exponents
The following data was gathered for the reaction:
2A+B+2D→E+2F2A+B+2D\to E+2F
Find the rate law exponents and the rate constant for this reaction. Report your exponents as integers and your rate constant to two significant figures without units in the answer field.
Determining Reaction Order
Use the rate data provided to determine an experimental rate law for the following reaction.
CH3MgBr+H2O→CH4+HOMgBr\rm CH_3MgBr\quad+\quad H_2O\quad\overrightarrow{\hspace{1.5cm}}\quad CH_4\quad+\quad HOMgBr
Run[CH3MgBr]0[H2O]0initial rate10.4 M0.2 M0.012320.8 M0.2 M0.047830.8 M0.4 M0.0948\begin{array}{|c|c|c|c|}\hline \rm Run &\rm [CH_3MgBr]_0 & \rm [H_2O]_0 & \rm initial\ rate\\\hline 1 & 0.4\ M & 0.2\ M & 0.0123\\\hline 2 & 0.8\ M & 0.2\ M & 0.0478\\\hline 3 & 0.8 \ M & 0.4\ M & 0.0948\\\hline \end{array}
Given the following data for the NH4++NO2−→N2+2H2ONH_4^+ +NO_2^- \to N_2+2H_2O reaction
The rate law for the reaction is
Assume the following reaction
3H2+CO→CH4+H2O3H_2 +CO \to CH_4+H_2O
occurs by the given reaction mechanism.
The gas-phase reaction 2NO+2NH2→N2+2H2O2NO+2NH_2 \to N_2+2H_2O has the following rate law expression, rate = (0.14 L2 mol2 s-2 )[NO]2 [H2]. If the [NO] is 0.95 M and the [H2] is 0.45 M, what rate is expected?
The Rate Law
For the dimerization of butadiene:
2𝐶4𝐻6(𝑔)→𝐶8𝐻12(𝑔)Rate=k[𝐶4𝐻6]2𝐶_4𝐻_6 (𝑔) \to𝐶_8𝐻_{12} (𝑔) \qquad Rate = k [𝐶_4𝐻_6]
The units of the reaction rate are:
The Rate Law
Two reactions are both described by the equation A → products. Reaction 1 is first order with respect to [A], and Reaction 2 is second order with respect to [A]. Which option below correctly describes the initial rates of these reactions when [A] = 0.100 M?
Consider the reaction: 2A + B → C. If we determine experimentally that the overall order of this reaction is 2, which of the following options could be the differential rate law for this reaction? Select all that apply.
A proposed mechanism for the reaction of NO2 with CO is shown below. Which of the given rate laws would accurately describe a reaction with this mechanism?
NO2(g)+NO2(g)→NO3(g)+NO(g)slowNO3(g)+CO(g)→NO2(g)+CO2(g)fastNO2(g)+CO(g)→NO(g)+CO2(g)overall\def\arraystretch{1.5} \begin{array}{cc} NO_2(g)+NO_2(g)\to NO_3(g)+NO(g) & slow \\ NO_3(g)+CO(g)\to NO_2 (g) +CO_2 (g) & fast \\ \hline NO_2(g)+CO(g)\to NO(g)+CO_2(g) & overall \end{array}
Given the following data for the NH4++NO2−→N2+2H2ONH_4^++NO_2^-→N_2+2H_2O reaction, the rate law for the reaction is
Trial[NH4+][NO2–]Rate10.010M0.030M0.020M/s20.0150.0300.03030.0100.0150.005\def\arraystretch{1.5} \begin{array}{c} Trial &[NH_4^+] &[NO_2^–]& Rate \\ \hline 1 & 0.010 M & 0.030 M &0.020 M/s \\ 2& 0.015& 0.030 &0.030 \\ 3&0.010 & 0.015 &0.005 \\ \end{array}
Rate Law
For the following elementary reactions apply the Law of mass balance and write a rate law.
Xe+F2→XeF2Xe+F_2\rightarrow XeF_2
Reaction mechanisms: Rate law
The rate law for the overall reaction A + 2B → E was experimentally determined to be: Rate = k[A]2[B]
Which of these proposed mechanisms could explain this observed rate law?
Integrated Rate Laws
Suppose you were measuring the concentration of NOBr during this reaction:
2NOBr→2NO+Br22 NOBr → 2 NO + Br_2
The reaction is first-order with respect to [NOBr]; the initial concentration of [NOBr]0 = 0.50M.
Chemical reaction: Rate Law
Consider the reaction given below and its experientially determined rate law. Which of the following statements is correct?
2H2(g)+2NO(g)→N2(g)+2H2O(g)        Rate=k[H2][NO]22H_2(g)+2NO(g)\to N_2(g)+2H_2O(g)\ \ \ \ \ \ \ \ Rate=k[H_2][NO]^2
The Rate law
For the reaction: BrO3−+5Br−+6H+→3Br2+3H2OBrO_3^-+5Br^-+6H^+\to3Br_2+3H_2O at a particular time, the rate of disappearance of Br- was 7.5 x 10-2 M/s. What is the rate of disappearance of BrO3- at the same instant?
The Rate Law: Half life
Radioactive plutonium Pu-241 has a half-life of 14.4 years and decays by a 1st order process. If 25 kg of Pu-241, waste from a nuclear reactor, is stored at a disposal site, how long will it take for the amount of Pu-241 to be reduced to 1 kg?
The Rate Law: Stoichiometry
If N2 is consumed at 2 M/s, what is the production rate of NH3 and consumption rate of H2 in the reaction?
N2  + 3H2 → 2 NH3N_2\ \ +\ 3H_2\ \rightarrow\ 2\ NH_3
The Rate Law
For a hydrogenation reaction involving H2, plotting 1/[H2] versus time produces a straight line graph. What is the order of hydrogen in this reaction.
Chemical Kinetics
Determine the order of each reactant and the overall order for the following rate laws.
a) 𝑣 = 𝑘[𝑁𝑂][𝐶]2
b) 𝑣 = 𝑘[𝑂3]1
Determining Reaction Order
Use the rate data provided to determine an experimental rate law for the following reaction.
CH3MgBr+H2O→CH4+HOMgBr\rm CH_3MgBr\quad+\quad H_2O\quad\overrightarrow{\hspace{1.5cm}}\quad CH_4\quad+\quad HOMgBr
Run[CH3MgBr]0[H2O]0initial rate10.4 M0.2 M0.012320.8 M0.2 M0.047830.8 M0.4 M0.0948\begin{array}{|c|c|c|c|}\hline \rm Run &\rm [CH_3MgBr]_0 & \rm [H_2O]_0 & \rm initial\ rate\\\hline 1 & 0.4\ M & 0.2\ M & 0.0123\\\hline 2 & 0.8\ M & 0.2\ M & 0.0478\\\hline 3 & 0.8 \ M & 0.4\ M & 0.0948\\\hline \end{array}
Determining Rate Law Exponents
The following data was gathered for the reaction:
2A+B+2D→E+2F2A+B+2D\to E+2F
Find the rate law exponents and the rate constant for this reaction.
Rate Law
For the following elementary reactions apply the Law of mass balance and write a rate law.
B2H6→2BH3B_2H_6\rightarrow 2 BH_3
The reaction A + 2B → products was found to follow the rate law: rate = k[A]2[B]. Predict by what factor the rate of reaction will increase when the concentration of A is doubled, the concentration of B is tripled, and the temperature remains constant.
By what factor does the rate change in each of the following cases?
A reaction is first order in A, and [A] is doubled
A reaction is second order in B, and [B] is halved
Determine an empirical rate law for the following reaction using the kinetic data collected at 298 K298\ K
and evaluate the rate constant kk.
2 Fe+Al2O3+6 HCl→Fe2O3+2 AlCl3+3 H2O\rm 2\ Fe\quad+\quad Al_2O_3\quad+\quad6\ HCl\quad\overrightarrow{\hspace{1.5cm}}\quad Fe_2O_3\quad+\quad2\ AlCl_3\quad+\quad3\ H_2O
Determining Reaction Order
Use the rate data provided to determine an experimental rate law for the following reaction.
CH3MgBr+H2O→CH4+HOMgBr\rm CH_3MgBr\quad+\quad H_2O\quad\overrightarrow{\hspace{1.5cm}}\quad CH_4\quad+\quad HOMgBr
Run[CH3MgBr]0[H2O]0initial rate10.4 M0.2 M0.012320.8 M0.2 M0.047830.8 M0.4 M0.0948\begin{array}{|c|c|c|c|}\hline \rm Run &\rm [CH_3MgBr]_0 & \rm [H_2O]_0 & \rm initial\ rate\\\hline 1 & 0.4\ M & 0.2\ M & 0.0123\\\hline 2 & 0.8\ M & 0.2\ M & 0.0478\\\hline 3 & 0.8 \ M & 0.4\ M & 0.0948\\\hline \end{array}
Determining Rate Law Exponents
The following data was gathered for the reaction:
2A+B+2D→E+2F2A+B+2D\to E+2F
Choose the correct rate law. For a bonus mark, solve for the rate constant (include units).
Determine an empirical rate law for the following reaction using the kinetic data collected at 298 K and evaluate the rate constant k.
A+B+C⟶Pr⁡oductsA+B+C⟶\Pr oducts
[A]0(M)[B]0(M)[C]0(M)Initial Rate(M/s)0.1510.2130.3980.4800.2510.1050.3250.3560.1510.2130.5251.1020.1510.2500.4800.988\def\arraystretch{1.5} \begin{array}{c} \hline [A]_0 (M) &[B]_0 (M) &[C]_0 (M)&Initial \ Rate (M/s) \\ \hline 0.151 & 0.213 & 0.398 &0.480 \\ 0.251& 0.105& 0.325 &0.356 \\ 0.151&0.213 & 0.525 &1.102 \\ 0.151&0.250 &0.480 &0.988 \\ \hline \end{array}
Determine an empirical rate law for the following reaction using the kinetic data collected at 298 K298\ K
and evaluate the rate constant kk.
2 Fe+Al2O3+6 HCl→Fe2O3+2 AlCl3+3 H2O\rm 2\ Fe\quad+\quad Al_2O_3\quad+\quad6\ HCl\quad\overrightarrow{\hspace{1.5cm}}\quad Fe_2O_3\quad+\quad2\ AlCl_3\quad+\quad3\ H_2O
The following data tracking the concentration of reactant A was collected:
What is the order of this process with respect to A? Estimate the rate constant
Phosgene is a toxic gas prepared by the reaction of carbon monoxide with chlorine:
CO(g)+Cl2(g)→COCl2(g)CO_{(g)}+Cl_{2(g)}\to COCl_{2(g)}
The following data was obtained in a kinetics study of this reaction:
Assume the following reaction 3H2+CO→CH4+H2O3H_2+CO→CH_4+H_2O occurs by the given reaction mechanism. The rate law expression must be rate = ______________.
Step1H2+CO→H2COslowStep2H2+H2CO→CH4+OfastStep3H2+O→H2OfastOverall3H2+CO→CH4+H2O\def\arraystretch{1.5} \begin{array}{c} Step 1 &H_2+CO→H_2 CO&slow \\ Step 2 & H_2+H_2 CO→CH_4+O & fast \\ Step 3& H_2+O→H_2 O& fast \\ \hline Overall&3H_2+CO→CH_4+H_2 O \\ \end{array}
Assume the following reaction
3H2+CO→CH4+H2O3H_2 +CO \to CH_4+H_2O
occurs by the given reaction mechanism.