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TOPIC 8 –Determine the order of reactions by graphically methods

Additional Note

Rate Laws from Graphs of Concentration

Versus Time (Integrated Rate Laws)

In order to determine the rate law for a reaction from a set of data consisting of concentration (or the

values of some function of concentration) versus time, make three graphs.

• [A] versus t (linear for a zero order reaction)

• ln [A] versus t (linear for a 1st order reaction)

• 1 / [A] versus t (linear for a 2nd

order reaction)

The graph that is linear indicates the order of the reaction with respect to A. Then, you can choose the

correct rate equation:

For a zero order reaction, rate = k (k = - slope of line)

For a 1st order reaction, rate = k[A] (k = - slope of line)

For a 2nd

order reaction, rate = k[A]2 (k = slope of line)

Examples

For a zero order reaction, as shown in the following figure, the plot of [A] versus time is a straight

line with k = - slope of the line. Other graphs are curved for a zero order reaction.

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For a first order reaction, as shown in the following figure, the plot of the logrithm of [A] versus

time is a straight line with k = - slope of the line. Other graphs are curved for a first order reaction.

For a second order reaction, as shown in the following figure, the plot of 1/[A] versus time is a

straight line with k = slope of the line. Other graphs are curved for a second order reaction.

Exercise 8a

1. The following data were obtained on the rate of hydrolysis of 17 % sucrose in 0.99 mol L-1

HCl aqueous

solution at 35 oC.

t / min 9.82 59.60 93.18 142.9 294.8 589.4

Sucrose remaining, % 96.5 80.3 71.0 59.1 32.8 11.1

What is the order of the reaction with respect to sucrose, and what is the value of the rate constant k?

Solving Method

Sucrose is the reactant. Assume this is first order reaction and we need to prove by plot a suitable graph.

From the integrated rate equation for first-order reaction: ln[A] = ln[A]o – kt , by assuming

[A] = concentration of sucrose = % sucrose remaining and

[A]o = initial concentration of sucrose = 100%

So calculate ln [A] at each times, t and then plot graph ln [A] as y-axis versus t (min)as x-axis. Use graph

paper. If the graph is linear or straight line graph, it shows that the reaction is first-order. Calculate the

slope of the graph; the value is equal to the rate constant, k. Give the unit of k.

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2. Methy acetate is hydrolyzed in approximately 1 M HCl at 25 oC. Aliquots of equal volume are removed

at intervals and titrated with a solution of NaOH. Calculate the first-order rate constant k, from the

following volumes of NaOH required to neutralize the aliquot:

t /s 339 1242 2745 4546 ∞

VNaOH /mL 26.34 27.80 29.70 31.81 39.81

Solving Method

The volumes reading are show the amount of the product of the reaction. We must change the data to the

reactant reading as follows;

Lets the reaction is: A → P

At initial t = 0 [A]o 0

At any time, t [A]o-x x = Vt

At ∞ (reaction completed) 0 [A]o = V∞

So recalculate the data :

t /s-1

339 1242 2745 4546

[A] = V∞-Vt 39.81-26.34 39.81-27.80 39.81-29.70 39.81-31.81

ln[A] = ln (V∞-Vt)

Then plot ln[A] ( y-axis) versus t (x-axis) on a graph paper. The graph is linear with the slope equal to the

the rate constant, k.

3. At 504 oC dimettyl ether undergoes the decomposition

(CH3)2O (g) → CH4 (g) + H2 (g) + CO (g)

The total pressure of the reacting system, measured as function of time, t as below:

t /(s) 0 390 777 1195 3155 ∞

P /(torr) 312 408 488 562 779 931

(a) Calculate the partial pressures of (CH3)2O and CH4 at each time, t.

(b) Determine the order and the rate constant, k for the decomposition.

Solving Method

By assuming Po= initial pressure, and x are the reduced pressures during the composition at any time, t.

You will find the equation that relate Po with P (Total pressure of the system)

(CH3)2O → CH4 + H2 + CO

At t = 0 Po 0 0 0

At t = t Po-x x x x

Total pressure, P = Po-x+3x = Po+2x

x = (P-Po) / 2

Find the values of x, which is gives the partial pressures of CH4, and calculate P(CH3)2O, which is equal

to Po-x, put the data in the table below. Then assume the order of the reaction is first-order, plot ln

P(CH3)2O versus t on graph paper, find the slope, m = k ( rate constant) .

t /(s) 0 390 777 1195 3155 ∞

P /(torr) 312 408 488 562 779 931

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)(CH

o PPP

x =

−

= / (torr) 0

482

)312408(=

−

)/(23 )( torrPxP OCHo =− 0 312-48=264

ln (Po-x) 0 5.576

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4. The following total pressure measurements as a function of time were obtained for the dimerization of 1,

3-butadiene (C4H6) under constant volume conditions at 326 oC.

t/min 0 3.25 12.18 24.55 42.50 68.05

P/ Torr 632.0 618.5 584.2 546.8 509.3 474.6

Determine the order of the reaction and evaluate the rate constant.

Solving Method

By assuming Po= initial pressure, and x are the reduced pressures during the composition at any time, t.

You will find the equation that relate Po with P (Total pressure of the system)

Lets the reaction is: 2A → A2

At initial t = 0 Po 0

At any time, t Po-2x x

Total pressure, P = Po-2x+x = Po-x

x = (Po-P)

PA = Po-2x = Po-2(Po-P) = 2P-Po

Calculate the values of 2P-Po, which is gives the partial pressures of C4H6, and put the data in the table

below. Then assume the order of the reaction is second-order, plot 1/P(C4H6) versus t on graph paper, find

the slope, m = k ( rate constant) .

t/min 0 3.25 12.18 24.55 42.50 68.05

P/ Torr 632.0 618.5 584.2 546.8 509.3 474.6

P(C4H6)=2P-Po/Torr

ln P(C4H6)