27

TOPIC 7 - THE RATE OF CHEMICAL REACTIONS

Examples of Solved Problems

1. The rate of formation of C in the reaction 2A + 3B → C + 4D is 2.7 mol dm-3

s-1

. State

the rates of formation and consumption of the participants.

Answer:

1313

1313

1313

13

8.10)7.2(4][

D, offormation of rate The

1.8)7.2(3][

4.5)7.2(2][

2][

AofnconsumptioofrateThe

][

4

1][][

3

1][

2

1

7.2dt

d[C]C offormation of rate The

−−−−

−−−−

−−−−

−−

==

==−

===−

==−=−=

==

sdmmolsdmmoldt

Dd

sdmmolsdmmoldt

Bd

sdmmolsdmmoldt

Cd

dt

Ad

dt

Dd

dt

Cd

dt

Bd

dt

AdRate

sdmmol

2. The rate law for reaction A + 2B → 3C + 4D was reported as

d[C]/dt= kr[A][B][C]. Express the rate law in terms of the reaction rate; what is

the units for kr in each case?

Answer:

162

33

13

)(]][][[

;

]][][[][

4

1][

3

1][

2

1][

−−

−

−−

===

===−=−=

sdmmoldmmol

sdmmol

CBA

Ratek

kofUnit

CBAkdt

Dd

dt

Cd

dt

Bd

dt

AdRate

r

r

r

3. At 518 oC, the rate of decomposition of a sample of gaseous acetaldehyde,

initially at a pressure of 363 Torr, was 1.07 Torr s-1

when 5.0 percent had reacted

and 0.76 Torr s-1

when 20.0 per cent had reacted. Determine the order of the

reaction.

Answer:

( )

reactionordersecond00.299.10746.0

1486.0

1875.1log

4079.1log

1875.1log4079.1log1875.180.0

95.04079.1

)20.0(

)05.0(

%)20(

%)5(

76.0

07.1

2

1

)()(Pr][

1

1

≈===

==

=

−

−=

−

−==

−=∝∝

−

−

n

n

PPk

PPk

Pk

Pk

sTorr

sTorr

rate

rate

xPkrateAessureARate

n

n

n

oo

n

oo

n

o

n

o

n

o

nn

28

4. At 518 oC, the half-life for the decomposition of a sample of gaseous

acetaldehyde, initially at 363 Torr was 410 s. When the pressure was 169 Torr,

the half-life was 880 s. Determine the order of the reaction.

Answer:

reactionordersecond00.2999.1999.01

999.03320.0

3317.0

1479.2log

4659.0log)1(

1479.2log)1(4659.0log

)1479.2()169(

)363(4659.0

880

410

)2(

)1(

1

][

1,

1

1

1

2/1

2/1

1

1

0

12/1

≈=+=

−=−

==−

−=

====

∝∝∝−

−

−

−

−

−−

n

n

n

C

C

s

s

t

t

PPA

tlifeHalf

n

n

n

n

onn

o

5. A second-order reaction of the type A + B →P was carried out in a solution that

was initially 0.075 mol dm-3

in A and 0.060 mol dm-3

in B. After 1.0 h the

concentration of A had fallen to 0.020 mol dm-3

. Calculate the rate constant.

113510754.166326

2231.03863.12231.0663263863.1

060.0

075.0ln)60601()060.0075.0(

055.0060.0

020.0ln

055.0020.0075.0][][][][

][

][ln][][

][

][ln

][

][ln:lawratedintergrateThe

,:typereactionsecFor

−−−×=

−=+=

+××−=−

=−=−=−=

+−=−

−=

→+−

smoldmkk

sk

BBAAx

B

AktBA

xB

xA

B

A

PBAorderond

oo

o

o

oo

o

o

6. A reaction 2B →P has a second–order rate law with k = 4.30×10-4

dm3 mol

-1 s

-1.

Calculate the time required for the concentration of B to change from 0.210 mol

dm-3

to 0.010 mol dm-3

.

( )

sst

tsmoldmmoldmdmmoldmmol

kktBB

BBBB

dtkBdBB

Bd

Bkdt

BdPBorderond

ot

tt

tB

B

B

B

t

o

t

o

5

113413

33

1

0

112

0

12

0

][

][

2

][

][

2

2

1011.1110742

1060.8)762.4100(210.0

1

010.0

1

022][

1

][

1

1

][

1

][

12

][

12

][

2][][][

][

][2][

:lawrateThe,2:typereactionsecFor

×==

×=−=−

−=−=−

+−

−=

+−−−

+−−

=−=−

=−→−

−−−−

−−

−−+−+−

−

∫∫∫

29

7. Hydrogen peroxide reacts with thiosulfate ion in slightly acidic solution as

follows: H2O2 + 2S2O32-

+2H+

→ 2H2O + S4O62-

This reaction is second order, independent of the hydrogen ion concentration in

the pH range 4 to 6. The rate constant, k = 9.1×10-2

mol-1

L min-1

were obtained at

25 o C and pH 5.0, and initial concentrations: [H2O2] = 0.036 mol L

-1, [ −3

32OS ] =

0.024 mol L-1

. How long will it take for 50% of the thiosulfate ion, −3

32OS to

react?

Answer:

stt

t

xBxbx

baOSBOHA

B

AktBaAb

bxB

axA

BAkdt

AdPbBaA

o

o

o

oo

o

o

06.117405.01068.49163.0

024.0

036.0ln)101.9))(024.0(1)036.0(2(

)100.6(2024.0(

)100.6036.0(ln

100.6024.025.0,]%[502

2,.1,][],[][

][

][ln)][][(

)]([

)]([ln:equationrateedinteragratThe

]][[][

:lawrateThe,:typereactionordersecondFor

3

2

3

3

3

2

3222

=+×=

+×−=×−

×−

×=×===

====

+−=−

−

=−→+−

−

−

−

−

−

−

Exercise 7a

1. If the reaction aA → Products is xth

-order, where x = 0, x = 1, x = 2 and x = n (any

integer ≠1). Write the rate law, derive the integrated rate law, and also derive the

expression for half-life in terms of a, k, n and [A]o for each order.

2. The gas-phase formation of fosgene, CO + Cl2 → COCl2, is ½ order with respect to

CO. Derive the integrated rate equation for ½-order reaction and derive the

expression for the half-life.

3. The rate of the reaction A + 2 B → 3 C + D was reported as 1.0 mol dm−3

s−1

. Which

of the following states of the rates is NOT TRUE.

A. –d[A]/dt =1.0 mol dm−3

s−1

B. –d[B]/dt =2.0 mol dm−3

s−1

C.-d[C]/dt=3.0 mol dm−3

s−1

D.- d[D]/dt= -1.0 mol dm−3

s−1

4. The rate law for the reaction A + 3 B → C + 2 D was found to be v = k[A][B].

Which of the following state(s) is/are true.

I The unit of the rate of reaction is mol-1

L s-1

II The unit of k is mol-1

L s-1

III The reaction is second order reaction IV Rate = ]][[/][ BAkdtCd =−

A. I and III B. II and III C. I, II , and III D. IV only

5. Nitrogen pentoxide gas decomposes according to: 2N2O5 (g) →4 NO2(g) + O2(g) .

At 328 K the rate of reaction v under certain conditions is 3.38 × 10−5

s−1

. Assuming

that none of the intermediates have appreciable concentrations. Which values is true.

A. -d[N2O5]/dt=3.38 × 10−5

s−1

B. –d[NO2]/dt=3.38 × 10−5

s−1

C.-d[O2]/dt= 3.38 × 10−5

s−1

D. d[O2]/dt=3.38 × 10−5

s−1

30

6. In studying the decomposition of ozone: 2O3(g) = 3O2(g)

in a 2-L reaction vessel, it is found that d[O3]/dt = -1.5×10-2

mol L-1

s-1

. What is the

rate of reaction in mol L-1

s-1

?

A.7.5×10−3

B.1.5×10-2

C. -1.5×10-2

D. - 7.5×10−3

7. Under certain conditions, it is found that ammonia is formed from its elements at a

rate of 0.10 mol L-1

s-1

. N2(g) + 3H2(g) = 2NH3(g) . Calculate the value of d[H2]/dt?

A. 0.10 mol L-1

s-1

B. 0.20 mol L-1

s-1

C. 0.30 mol L-1

s-1

D. -0 .30 mol L-1

s-1

8. The decomposition of N2O5: 2N2O5 = 4NO2 + O2 is studied by measuring the

concentration of O2 as a function of time, and it is found that

d[O2]/dt = (1.5 ×10-4

s-1

)[N2O5] at constant temperature and pressure. Under these

conditions the reaction goes to completion to the right. Calculate the half-life of the

reaction under these conditions?

A. 4621s B. 2310 s C. 1155 s D. 577 s

9. At 400 K, the t1/2 for the decomposition of a sample of a gaseous compound initially at

55.5 kPa was 340 s. When the P was 28.9 kPa, the t1/2 was 178 s. The order of the

reaction is

A. n = 0 B. n =1 C. n = 2 D. n = 3

10. The composition of HI to H2 + I2 at 508 oC has a t1/2 of 135 min when the initial

pressure of HI is 0.1 atm and 13.5 min when the pressure is 1 atm. What is the order

of the reaction?

A. n = 0 B. n =1 C. n = 2 D. n = 3

11. The rate constant for the first-order decomposition of a compound A in the reaction

2A → P is k = 2.78 × 10−7

s−1

at 25°C. Calculate the half-life of A?

A. 2.78 × 10−7

s−1

B. 1.25 ×10−6

s−1

C. s4105.2 × D. 1.25 × 10

6 s

−1

12. The first-order reaction 2A→ 2B + C is 35% complete after 325 s. How long will it

take for the reaction to be 70% complete?

A. 325 s B. 650 s C. s908 D. 6.63 ×104 s

13. The t1/2 of a first-order reaction A → P is 12 min. What % of A remains after 50 min?

A. 5.6 % B. 6.5 % C. %08.9 D. 16.63 %

14. The decomposition of ammonia at 1000 oC was a zero-order reaction with rate constant,

k = 4.30 mol dm-3

s-1

.What is the t1/2 for this reaction in which the initial concentration,

[NH3]o=0.0560 M?

A. 77 s B. 4.15 s C. 1.30×10−2

s D. 1.61×10−1

s

15. A solution of A is mixed with an equal volume of a solution of B containing the same

number of moles, and the reaction A + B = C occurs. At the end of 30 min. A is 75%

reacted. How much of A be left at the end of 50 min if the reaction is zero order in

both A and B?

A. 85% B. 60% C. 40% D. 15%

31

Exercise 7b

1. The rate law for the reaction 2A →A2 was found to be r = k [A]2. Which of the

following state(s) is / are FALSE.

A The unit of the rate of reaction is mol L-1

s-1

C. The unit of k is mol L-1

s-1

B The reaction is second order reaction D. - 2

2 ][/][ AkdtAd A=

2. A solution of A is mixed with an equal volume of a solution of B containing the same

number of moles, and the reaction A + B = C occurs. If the reaction is first order in

both A and B, what is the rate law?

A. Rate = k[A][B] C. d[B]/dt = k([A]o-x)([B]o-x)

B. d[C]/dt = k[A]2 = k[B]

2 D. -d[B]/dt = k([A]o-x)([B]o-x)

3. The half-life, t1/2 of a reaction is the time required for half of the reactant to disappear.

Which of the following state(s) is / are true?

A. For a zero-order reaction, t1/2 is dependent of the initial concentration.

B. For a first-order reaction, t1/2 is independent of the initial concentration.

C. For a 2rd

-order reaction, t1/2 is inversely proportional to the initial concentration

D. All of the above are true.

4. A dimer is formed in the solution reaction 2A →A2. The rate law is r = 2k[A]2,where k

= 0.015 M-1

s-1

. What is the half-life of A when [A]o = 0.05 M?

A.1333 s B.667 s C. 1.5 s D. 1.5×10−2

s

5. A reaction 2 A → P has a rate law with k = 3.50 ×10−2

dm3 mol

−1 s

−1. Calculate the

time required for the concentration of A to change from 0.26 mol L−1

to 0.11 mol L−1

A. 75 s B. 150 s C. 750 s D. 1333 s

6. A gas reaction 2A → B is second order in A and goes to completion in a reaction

vessel of constant volume and temperature with a half-life of 600 s. If the initial

pressure of A is 1 bar, what are the partial pressure of A at 800 s?

A. 0.328 bar B. 0.430 bar C. 0.504 bar D. 0.845 bar

7. A solution of A is mixed with an equal volume of a solution of B containing the same

number of moles, and the reaction A + B = C occurs. At the end of 30 min A is 65%

reacted. How much of A be left at the end of 50 min if the reaction is first order in

both A and B?

A. 84.8 % B. 75.6 % C. 48.8% D. 24.4%

8. Equal molar quantities of A and B are added to a liter of a suitable solvent. At the end

of 500 s one-half of A has reacted according to the reaction A + B = C. How much of

A will be reacted at the end of 800 s if the reaction is second order with respect to A

and zero order with respect to B?

A. 61.5 % B. 57.6 % C. 38.5% D. 26.0%

9. The second-order rate constant for an alkaline hydrolysis of ethyl formate in 85%

ethanol (aqueous) at 29.86 o

C is 5.43 L mol-1

s-1

. If the reactants are both present at

0.01 mol L-1

, calculate the t1/2.

A.0.13 s B. 5.43 s C. 18.42 s D. 85 s

32

10. A second-order reaction of the type A + B → P was carried out in a solution that

was initially 0.050 mol dm−3

in A and 0.080 mol dm−3

in B. After1.0 h the

concentration of A had fallen to 0.020 mol dm−3

. Calculate the rate constant?

A. 0.00413 mol-1

Ls-1

B.0.0 413mol-1

Ls-1

C. 0.248 mol-1

Ls-1

D. None of the above

11. A second-order reaction of the type A + 2 B → P was carried out in a solution that

was initially 0.75 mol dm−3

in A and 0.80 mol dm−3

in B. After1.0 h the concentration

of A had fallen to 0.65 mol dm−3

. Calculate the rate constant.

A. 8.03×10

-4 mol

-1Ls

-1 B. 5.74×10

-5 mol

-1Ls

-1 C. 1.34×10

-5 mol

-1Ls

-1 D. None of the above

12. The second-order rate constant for the reaction 2A + B → C is 0.21 dm3 mol

−1 s

−1.

What is the concentration of C after 10s when the reactants are mixed with initial

concentrations of [A] = 0.15 mol dm−3

and [B] = 0.25 mol dm−3

?

A.0.184 mol dm−3

B. 0.046 mol dm−3

C. 0.021 mol dm−3

D. 0.07 mol dm−3

13. The reaction CH3CH2NO2 + OH- → H2O + CH3CHNO −

2

is second order, and k at 20

o C is 9.1×10

-2 mol

-1 L min

-1. An aqueous solution is

0.006 molar in nitroethane and 0.007 molar in NaOH. How long will it take for

10% of the nitroethane to react?

A. 1.44×10-3

s B. 15.82 s C. 15.82 min D. 9.1×102

min

14. The second-order rate constant for the reaction:

CH3COOC2H5 (aq) + OH−

(aq) → CH3CO2−

(aq) + CH3CH2OH(aq)

is 0.11 dm3 mol

−1 s

−1. What is the concentration of ester (ROOR) after 10 s when

ethyl acetate is added to NaOH so that the initial concentrations are [NaOH] = 0.050

mol dm−3

and [CH3COOC2H5] = 0.100 mol dm−3

?

A. 0.100 mol dm−3

B. 0.095 mol L-1

C. 0.050 mol dm−3

D. 0.045 mol L-1

15. The rate constant, k, for second-order decomposition of HBr at 1000 K is 6.2×10-3

dm3mol

-1 s

-1. In a reaction in which the initial concentration, [HBr]o is 0.65 mol dm

-3,

at what time [HBr]= 0.12 mol dm-3

?

A. 18 .27 s B. 109 s C. 182 s D. 1.1×103

s

16. The following table gives kinetic data the following reaction at 25 oC: A+ B + C = D

[A] /molL-1

[B] /molL-1

[C] /molL-1

d[D]/dt /10-4

mol L-1

s-1

0.01 0.005 4.00 1.75

0.01 0.004 4.00 1.75

0.01 0.003 8.00 3.50

0.001 0.002 4.00 0.55

What is the rate law for the reaction?

A. ][][ 2/1 CAkr = B. ][][][ 12/1 CBAkr −= C. ]][[ CAkr = D. ]][][[ CBAkr =

17. A second-order reaction of the type A + B → P , the reaction is 60 % complete in

60 seconds when [A]o= 0.1 M and [B]o= 0.5 M. What is the rate constant for this

reaction?

A. 0.0041 mol-1

Ls-1

B. 0.041mol-1

s-1

C. 0.248 mol-1

L s-1

D. None of the above