Problems, Problems, and More Problems: Kinetics through Examples

A first order
reaction has a rate constant of 1.00 s^{-1}.
What is the half life of the reaction?

---------------------

*Solution:*

For a first order reaction, and for a **first order reaction
only**, the half life is a constant and is proportional to the
inverse of the rate constant. By definition, at the time t = t_{1/2},
the concentration is equal to exactly one half of what it was when you started, i.e. [A] = [A]_{0}.
Thus, the relationship between the half life and the rate constant of a reaction can easilty be derived for any type of rate law. In this case we have a first order reaction. Remembering the integrated rate law for a first order reaction:

Then

[A]/[A]_{0} = 1/2 = exp( - k t_{1/2} )

taking the **ln **of both sides

ln( 1/2 ) = - ln 2 = - k t_{1/2}

or

t_{1/2} = ( ln 2 ) / k

Thus, the halflife of the reaction is (ln 2)/1.00 seconds or

t_{1/2} = 0.6931 s

For any order other than first order, the halflife of the reaction is

An elementary reaction, **2 A +
C --> D**, is second order in A and first order
in C. The rate of this reaction is 2.5 x 10^{-1} M/s
when the concentrations of A, C, and D are all 1.00 mM. What is the
rate constant for the reaction?

-------------------------

*Solution:*

The rate law for the reaction is

rate = k [A]^{2}[C]

Since we know the instantaneous rate of the reaction at known concentrations, we can directly evaluate the rate constant

2.5 x 10^{-1} M/s = k
(1.00 x 10^{-3} M)^{2}
(1.00 x 10^{-3} M)

k = 2.5 x 10^{8} M^{-2}s^{-1}

This is the appropriate units of a rate constant for a reaction that is third order overall.

The rate constant for the reaction

H_{2}(g)
+ I_{2}(g) ---> 2HI(g)

is 5.4 x 10^{-4 }M^{-1}s^{-1}
at 326 ^{o}C.
At 410 ^{o}C the rate constant was
found to be 2.8 x 10^{-2} M^{-1}s^{-1}.

Calculate the

**a) activation energy**

- and

- for this reaction.

-------------------------

*Solution:*

All reactions are activated processes. Rate constant is exponentially dependent on the Temperature

We know the rate constant for the reaction at two
different temperatures and thus
we can calculate the **activation energy** from the above relation.
First, and always, convert
all temperatures to Kelvin, an **absolute** temperature scale. Then simply
solve for E_{a} in units of R.

ln(5.4 x 10^{-4 }M^{-1}s^{
-1}/
2.8 x 10^{-2} M^{-1}s^{-1}) =
(-E_{a} /R ){1/599 K - 1/683 K}

-3.9484 = - E_{a}/R {2.053 x 10^{-4
}K^{-1}}

E_{a}= (1.923 x 10^{4} K) (8.314
J/K mol)

E_{a}= 1.60 x 10^{5} J/mol

Now that we know E_{a}, the pre-exponential factor, **A**,
(which is the largest rate constant that the reaction can possibly have) can be
evaluated from any measure of the absolute rate constant of the reaction.

so

5.4 x 10^{-4 }M^{
-1}s^{-1}
=

A exp{-(1.60 x 10^{5} J/mol)/((8.314 J/K mol)(599K))}

(5.4 x 10^{-4 }M^{-1}s^{-1})
/ (1.141x10^{-14}) = 4.73 x 10^{10 }M^{-1}s^{-1}

The infinite temperature rate constant is 4.73 x 10^{10 }M^{-1}s^{-1}

Consider the first order reaction

which has a rate constant, k = 2.95 X 10

---------------

*Solution:*

For a first order reaction, the integrated rate law is:

The fraction remaining is the concentration divided by
the initial concentration or **[A]/[A] _{0 }**thus

fraction remaining = exp{-(2.95 X 10^{-3} s^{-1})
(150 s)} = 0.642

or

64.2%

The reaction A --> products is **second order**
in A. Initially [A]_{0} = 1.00 M; and after 25 mins,
[A] = 0.25 M. What is the rate constant for this reaction?

-----------------

*Solution:*

The integrated rate law for a second order reaction is:

Since we know the concentration initially and at some later time, we can simply solve for k.

1/0.25 M = 1/1.00M + k(25 min)

3.00 M^{-1}/(25 min) = k = 1.2 x 10^{-1
}M^{-1}min^{-1}

The decomposition of N_{2}O_{5} is given
by the following equation

**2 N _{2}O_{5}(g) ----> 4
NO_{2}(g) + O_{2}(g) **

At an initial [N_{2}O_{5}]_{0}
= 3.15 M, the initial rate of reaction is 5.45 X 10^{-5}
M/s and when [N_{2}O_{5}]_{0} = 0.78 M, the initial
rate of reaction is 1.35 X 10^{-5} M/s.
What is the rate constant of this reaction?

------------------------

*Solution:*

This is an initial rates problem. There is only one reactant, so our life is simple. We make a dimensionless ratio of two rates and equate that to a rate law with an as yet an undetermined order.

(5.45 X 10^{-5} M/s) / (1.35
X 10^{-5} M/s) = (3.15 M)^{a} / (0.78 M)^{a}

4.037 = (4.039)^{a}

a = 1

Now that we know the order or the reaction is one, we can solve for the rate constant:

5.45 X 10^{-5} M/s = k (3.15
M)

k = 1.73 x 10^{-5}s^{-1}

We can check this by using the other set of conditions

1.35 X 10^{-5} M/s = k (0.78
M)

k = 1.73 x 10^{-5 }s^{-1}

Everything if fine.

The initial rate of the reaction 2A + 2B ---> C + D is determined for different initial conditions, with the results listed in the following table:

Run # | [A]_{0},
M |
[B]_{0},
M |
Initial rate, M/s |

1 | 0.185 | 0.133 | 3.35 x 10^{-4} |

2 | 0.185 | 0.266 | 1.35 X 10^{-3} |

3 | 0.370 | 0.133 | 6.75 X 10^{-4} |

4 | 0.370 | 0.266 | 2.70 X 10^{-3} |

Find the rate law and rate constant for this reaction.

-----------------------

*Solution:*

This, too, is an initial rates problem. There is more
than on species present as a rectant, so we must isolate the effect of
each of the reactants' concentrations on the overall rete of reaction.
Runs number 1 and 3 have the same concentration of B but the concentration
of A doubles. The rate of the reaction doubles too, so the reaction is
first order in A. Runs number 1 and 2 have the concentration of A the same
but the concentration of B doubles. The rate **quadruples** under these
conditions so the reaction is second order in B. Thus the rate law is

rate = k[A][B]^{2}

The rate constant may be evaluated with any set (row) of data. Try run #1

3.35 x 10^{-4} M/s = k(0.185 M)(0.133
M)^{2}

k = 0.102 M^{-2} s^{-1}

As a check, lets use run #4

2.70 x 10^{-3} M/s = k(0.370 M)(0.266
M)^{2}

k = 0.103 M^{-2} s^{-1}

Looks good to me.

Determine the order of the reaction A --> 2B + C from the following data obtained for [A] as a function of time.

time |
[A] |

0 min |
0.80 M |

8 mins |
0.60 M |

24 mins |
0.35 M |

40 mins |
0.20 M |

-----------------

*Solution:*

This, unfortunately is **not** an initial rates problem.
We have not been provided with data spaced close enough to get an instantaneous
rate under known reactant concentrations. So we must directly test the
integrated rate laws for validity.

If the reaction were zeroth order, the change in concentration divided by the elapsed time would be a constant. Let's see if this is true:

(0.80 - 0.60) / 8 = 0.0250

(0.80 - 0.35) / 24 = 0.0188

(0.80 - 0.20) / 40 = 0.0150

Nope, it is not zeroth order.

If the reaction were second order, the change in inverse concentration divided by the elapsed time would be constant.

(1/0.60 - 1/0.80) / 8 = 0.0521

(1/0.35 - 1/0.80) / 24 = 0.0670

(1/0.20 - 1/0.80) / 40 = 0.0938

Nope, it's not second order either.

If the reaction is first order, the change in the natural log of the concentration divided by the elapsed time will be constant.

(ln(0.80) - ln(0.60)) / 8 = 0.0360

(ln(0.80) - ln(0.35)) / 24 = 0.0344

(ln(0.80) - ln(0.20)) / 40 = 0.0347

Looks Good! The reaction is **first order** and the
rate constant is 0.035 min^{-1}