# 12.4 Integrated rate laws  (Page 5/7)

 Page 5 / 7

The half-life of a reaction is the time required to decrease the amount of a given reactant by one-half. The half-life of a zero-order reaction decreases as the initial concentration of the reactant in the reaction decreases. The half-life of a first-order reaction is independent of concentration, and the half-life of a second-order reaction decreases as the concentration increases.

## Key equations

• integrated rate law for zero-order reactions: $\left[A\right]=\text{−}kt+{\left[A\right]}_{0},$ ${t}_{1\text{/}2}=\phantom{\rule{0.1em}{0ex}}\frac{{\left[A\right]}_{0}}{2k}$
• integrated rate law for first-order reactions: $\text{ln}\left[A\right]=\text{−}kt+\text{ln}{\left[A\right]}_{0},\phantom{\rule{0.2em}{0ex}}\text{}{t}_{1\text{/}2}=\phantom{\rule{0.1em}{0ex}}\frac{0.693}{k}$
• integrated rate law for second-order reactions: $\frac{1}{\left[A\right]}\phantom{\rule{0.1em}{0ex}}=kt+\phantom{\rule{0.2em}{0ex}}\frac{1}{{\left[A\right]}_{0}},$ ${t}_{1\text{/}2}=\phantom{\rule{0.1em}{0ex}}\frac{1}{{\left[A\right]}_{0}k}$

## Chemistry end of chapter exercises

Describe how graphical methods can be used to determine the order of a reaction and its rate constant from a series of data that includes the concentration of A at varying times.

Use the data provided to graphically determine the order and rate constant of the following reaction: ${\text{SO}}_{2}{\text{Cl}}_{2}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\phantom{\rule{0.2em}{0ex}}\text{SO}}_{2}+{\text{Cl}}_{2}$

 Time (s) 0 5.00 $×$ 10 3 1.00 $×$ 10 4 1.50 $×$ 10 4 [SO 2 Cl 2 ] ( M ) 0.100 0.0896 0.0802 0.0719 Time (s) 2.50 $×$ 10 4 3.00 $×$ 10 4 4.00 $×$ 10 4 [SO 2 Cl 2 ] ( M ) 0.0577 0.0517 0.0415

Plotting a graph of ln[SO 2 Cl 2 ] versus t reveals a linear trend; therefore we know this is a first-order reaction:

k = −2.20 $×$ 10 5 s −1

Use the data provided in a graphical method to determine the order and rate constant of the following reaction:
$2P\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}Q+W$

 Time (s) 9.0 13.0 18.0 22.0 25.0 [P] ( M ) 1.077 $×$ 10 −3 1.068 $×$ 10 −3 1.055 $×$ 10 −3 1.046 $×$ 10 −3 1.039 $×$ 10 −3

Pure ozone decomposes slowly to oxygen, ${\text{2O}}_{3}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{3O}}_{2}\left(g\right).$ Use the data provided in a graphical method and determine the order and rate constant of the reaction.

 Time (h) 0 2.0 $×$ 10 3 7.6 $×$ 10 3 1.00 $×$ 10 4 [O 3 ] ( M ) 1.00 $×$ 10 −5 4.98 $×$ 10 −6 2.07 $×$ 10 −6 1.66 $×$ 10 −6 Time (h) 1.23 $×$ 10 4 1.43 $×$ 10 4 1.70 $×$ 10 4 [O 3 ] ( M ) 1.39 $×$ 10 −6 1.22 $×$ 10 −6 1.05 $×$ 10 −6

The plot is nicely linear, so the reaction is second order.
k = 50.1 L mol −1 h −1

From the given data, use a graphical method to determine the order and rate constant of the following reaction:
$2X\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}Y+Z$

 Time (s) 5 10 15 20 25 30 35 40 [ X ] ( M ) 0.099 0.0497 0.0332 0.0249 0.02 0.0166 0.0143 0.0125

What is the half-life for the first-order decay of phosphorus-32? $\left({}_{15}^{32}\text{P}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{}_{16}^{32}\text{S}+{\text{e}}^{-}\right)$ The rate constant for the decay is 4.85 $×$ 10 −2 day −1 .

14.3 d

What is the half-life for the first-order decay of carbon-14? $\left({}_{\phantom{\rule{0.5em}{0ex}}6}^{14}\text{C}⟶{}_{\phantom{\rule{0.5em}{0ex}}7}^{14}\text{N}+{\text{e}}^{-}\right)$ The rate constant for the decay is 1.21 $×$ 10 −4 year −1 .

What is the half-life for the decomposition of NOCl when the concentration of NOCl is 0.15 M ? The rate constant for this second-order reaction is 8.0 $×$ 10 −8 L/mol/s.

8.3 $×$ 10 7 s

What is the half-life for the decomposition of O 3 when the concentration of O 3 is 2.35 $×$ 10 −6 M ? The rate constant for this second-order reaction is 50.4 L/mol/h.

The reaction of compound   A to give compounds   C and   D was found to be second-order in   A . The rate constant for the reaction was determined to be 2.42 L/mol/s. If the initial concentration is 0.500 mol/L, what is the value of t 1/2 ?

0.826 s

The half-life of a reaction of compound A to give compounds D and E is 8.50 min when the initial concentration of A is 0.150 mol/L. How long will it take for the concentration to drop to 0.0300 mol/L if the reaction is (a) first order with respect to A or (b) second order with respect to A ?

Some bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of 3 $×$ 10 4 g/mol that converts penicillin into inactive molecules. Although the kinetics of enzyme-catalyzed reactions can be complex, at low concentrations this reaction can be described by a rate equation that is first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: 1.0 L of a solution containing 0.15 µg (0.15 $×$ 10 −6 g) of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant.

[Penicillin] ( M ) Rate (mol/L/min)
2.0 $×$ 10 −6 1.0 $×$ 10 −10
3.0 $×$ 10 −6 1.5 $×$ 10 −10
4.0 $×$ 10 −6 2.0 $×$ 10 −10

The reaction is first order.
k = 1.0 $×$ 10 7 mol −1 min −1

Both technetium-99 and thallium-201 are used to image heart muscle in patients with suspected heart problems. The half-lives are 6 h and 73 h, respectively. What percent of the radioactivity would remain for each of the isotopes after 2 days (48 h)?

There are two molecules with the formula C 3 H 6 . Propene, ${\text{CH}}_{3}\text{CH}={\text{CH}}_{2},$ is the monomer of the polymer polypropylene, which is used for indoor-outdoor carpets. Cyclopropane is used as an anesthetic:

When heated to 499 °C, cyclopropane rearranges (isomerizes) and forms propene with a rate constant of
5.95 $×$ 10 −4 s −1 . What is the half-life of this reaction? What fraction of the cyclopropane remains after 0.75 h at 499.5 °C?

4.98; 20% remains

Fluorine-18 is a radioactive isotope that decays by positron emission to form oxygen-18 with a half-life of 109.7 min. (A positron is a particle with the mass of an electron and a single unit of positive charge; the equation is ${}_{518}^{\phantom{\rule{1em}{0ex}}9}\text{F}⟶{}_{18}^{\phantom{\rule{0.5em}{0ex}}8}\text{O}+{\text{e}}^{-}.\right)$ Physicians use 18 F to study the brain by injecting a quantity of fluoro-substituted glucose into the blood of a patient. The glucose accumulates in the regions where the brain is active and needs nourishment.

(a) What is the rate constant for the decomposition of fluorine-18?

(b) If a sample of glucose containing radioactive fluorine-18 is injected into the blood, what percent of the radioactivity will remain after 5.59 h?

(c) How long does it take for 99.99% of the 18 F to decay?

Suppose that the half-life of steroids taken by an athlete is 42 days. Assuming that the steroids biodegrade by a first-order process, how long would it take for $\frac{1}{64}$ of the initial dose to remain in the athlete’s body?

252 days

Recently, the skeleton of King Richard III was found under a parking lot in England. If tissue samples from the skeleton contain about 93.79% of the carbon-14 expected in living tissue, what year did King Richard III die? The half-life for carbon-14 is 5730 years.

Nitroglycerine is an extremely sensitive explosive. In a series of carefully controlled experiments, samples of the explosive were heated to 160 °C and their first-order decomposition studied. Determine the average rate constants for each experiment using the following data:

 Initial [C 3 H 5 N 3 O 9 ] ( M ) 4.88 3.52 2.29 1.81 5.33 4.05 2.95 1.72 t (s) 300 300 300 300 180 180 180 180 % Decomposed 52 52.9 53.2 53.9 34.6 35.9 36 35.4

[ A ] 0 ( M ) k $×$ 10 3 (s −1 )
4.88 2.45
3.52 2.51
2.29 2.54
1.81 2.58
5.33 2.35
4.05 2.44
2.95 2.47
1.72 2.43

For the past 10 years, the unsaturated hydrocarbon 1,3-butadiene $\left({\text{CH}}_{\text{2}}=\text{CH}–\text{CH}={\text{CH}}_{2}\right)$ has ranked 38th among the top 50 industrial chemicals. It is used primarily for the manufacture of synthetic rubber. An isomer exists also as cyclobutene:

The isomerization of cyclobutene to butadiene is first-order and the rate constant has been measured as 2.0 $×$ 10 −4 s −1 at 150 °C in a 0.53-L flask. Determine the partial pressure of cyclobutene and its concentration after 30.0 minutes if an isomerization reaction is carried out at 150 °C with an initial pressure of 55 torr.

who are the alchemist?
alchemy science of transmutation. typically it is aim at tranforming lead to or other base metals to gold and the creation of the philosophers stone which in reality isn't a stone it's something priceless something we all need for coming times. don't be fooled
Kendrick
read Corinthians 5 verses 50 to the end of the chapter then read revelations chapter 2 verse 17
Kendrick
The word "Alchemy" comes from the forgotten name for Ancient Egypt, Khemmet. Khem was the name for the Egyptian Empire, but the actual land of Egypt was called Khemmet because the "T" on the end of a word denoted a physical location on Earth and not just an idea.
Michael
Wow!
mendie
What's the mass number of carbon
Akinbola
mass number of carbon is 12.
Nnenna
wat d atomic number of oxygen
safiya
atomic number of oxygen is 8
Nnenna
which quantum number divides shell into orbitals?
azimuthal
Emmanuel
hi
Charlie
azimuthal
reinhard
azimuthal
Charlie
what is atom
an atom is a smallest indivisible part of an element
Henry
an atom is the smallest part of an element that takes part in a chemical reaction
Nana
wat is neutralization
when any acid reacts with base to decrease it's acidity or vice-versa to form salt and solvent.. which is called neutralization
Santosh
explain buffer
Organic
buffer is a solution which resists changes in pH when acid or alkali added to it..
Santosh
hello, who is online
UTHMAN
buffer is the solution which resist the change in pH by addition of small amount of acid or alkali to it
KAUSIK
neutralisation is the process of mixing of a acid and a base to form water and corresponding salt
KAUSIK
how to solve equation on this
what are the elent of ionic and covalent bonding
Princewill
what is gases
Its one of the fundamental sate of matter alone side with liquid, solid and plasma
John
What is chemical bonding
John
To my own definitions. It's a unit of measurement to express the amount of a chemical substance.
What is mole
It's the unit of measurements used to express the amount of chemical substance.
Ozoaniehe
What is pressure
force over area
Jake
force applied per unit area
john
force applied per unit area
Prajapati
Why does carbonic acid don't react with metals
Why does carbonic acid don't react with metal
Some metals will react depending on their Standard Electrode Potential. Carbonic acid is a very weak acid (i.e. a low hydrogen ion concentration) so the rate of reaction is very low.
Paul
sample of carbon-12 has a mass of 6.00g. How many atoms of carbon-12 are in the sample
a sample of carbon-12 has a mass of 6.00g. How many atoms of carbon-12 are in the sample
an object of weight 10N immersed in a liquid displaces a quantity of d liquid.if d liquid displaced weights 6N.determine d up thrust of the object
how human discover earth is not flat
We don't fall off. If set off in any direction in a straight line and keep going. You'll end up back where you started.
earth is spherical
Unique
Also, every other planet is spherical as that is the most energy efficient shape. gravity pulls equally on all areas. Sphere.