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**Second-order reactions**

**A **

*second-order reaction *

**is one whose rate depends on the **

*square* of the concentration of a single reactant , or on the

*concentrations of two reactants, each raised to the first power*

**For example, for the second order reaction 2A **

** B, **

**Rate**

**k**

**2**

**Or, for A + B **

** C, **

**Rate**

**k**

*We will only concern ourselves with second-order rate laws of the first form (i.e., Rate = k[A]*

*2*

*) *

**Just as we did for the first-order case, we can write an **

*integrated second-order rate law*

**: **

**1**

**kt**

**1**

**0**

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**Again, this equation has the general form of a line: **

*a plot of 1/[A] vs. time will be linear with a slope = k and intercept = 1/[A]*

*0*

* if a reaction is second-order*

*This provides us with a method to determine reaction orders and rate constants experimentally*

*Collect concentration vs time data; if the reaction is firstorder, a plot of ln [ ] vs time will be linear; for a secondorder reaction, a plot of 1/[ ] vs time will be linear *

*What if neither plot is linear?*

*E.g., the following data were obtained at 300 o*

*C for the reaction *

*2NO*

*2*

*(g) *

* 2NO(g) + O*

*2*

*(g) time (s) [NO*

*2*

*]*

*0.0 *

*5.0 *

*10.0 *

*15.0 *

*20.0 *

*0.100*

*0.017*

*0.0090*

*0.0062*

*0.0047*

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*Is the reaction first or second-order in NO*

*2*

*? *

*What is the value of the rate constant? *

*In this case, we must plot the data in the first and secondorder forms.....*

0

-1

-2

-3

-4

-5

-6

0 5 10

**Time (s)**

15 20 25

Does the first-order treatment appear to linearize the data?

60

250

200

150

100

50 y = 10.16x + 9.1984

0

0 5 10

**Time (s)**

15

Find the rate constant from the plot.......

61

20 25

*Temperature dependence of reaction rates*

**The kinetic rate constant **

*k*

** for all reactions is a **

*constant at *

*a given temperature*

**However, k is temperature-dependent: consider the second-order gas-phase reaction **

**CO(g) + NO**

**2**

**(g) **

** CO**

**2**

**(g) + NO(g) **

*What does a plot of k vs. temperature look like for this reaction?*

62

35

30

25

20

15

10

5

0

600 650 700 750

**Temperature (K)**

800 850

**As k increases with T, the reaction rate increases with T**

**Why? **

*How to explain the dependence of k on T?*

**We use the collision model to account for the dependence of reaction rate on temperature **

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**Collision Model **

**Based on kinetic-molecular theory **

*Molecules must collide to react *

*Model helps explain dependence of rate on concentration and temperature *

*As reactant concentration increases, the frequency of collisions increases, and hence the reaction rate increases *

*As temperature increases, average molecular speeds increase *

*As molecules move faster, collisions become more energetic, and reaction rate increases*

*So... the collision model pretty much explains it all, right?*

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*Consider the reaction between hydrogen and iodine to form HI: *

*H*

*2*

*(g) + I*

*2*

*(g) *

* 2HI(g) *

**At STP, each molecule undergoes ~ 10**

**10 collisions/sec -- **

*yet only ~ 1 in 10*

*13 collisions forms products !*

*So, only a very small fraction of collisions actually lead to reaction *

*Why? *

*Simple collisions between reactant molecules aren’t enough to produce a reaction *

*Molecules must collide in the proper orientation for reaction to occur *

*E.g. with H*

*2*

* and I*

*2*

* collisions, what possible orientations are possible? Which would be most efficient at producing product? *

65

*In addition to orientation, we also use the concept of activation energy to explain the temperature dependence of reaction rates*

*Arrhenius: Molecules must have a certain minimum amount of energy in order to react *

*Collision model: this energy comes from the kinetic energies of the colliding molecules *

*Upon collision, KE can be converted into other types of energy, i.e., into vibrational energy which can stretch and break a bond *

*If molecules are moving too slowly (low KE), they bounce off each other when colliding *

*Kinetic energy not converted and molecules bounce off each other without reacting *

*Conclude that colliding molecules must have a total KE *

*some minimum value in order to react *

*This minimum energy required in order to initiate a reaction is known as the activation energy E a*

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*We represent this *

**graphically: **

*energy profile*

** for an exothermic reaction **

**E.g., For the decomposition of H**

**2**

**O**

**2**

**( aq) to H**

*E a*

** = 75.3 kJ/mol and overall **

**2**

**O( l) and O**

**2**

**( g), **

*E *= -98.1 kJ/mol.

**Sketch the energy profile for this reaction **

**Note that the reactants pass through an activated complex at the top of the activation energy barrier **

** Activated complex (transition state): high-energy, transient arrangement of reactants **

*What is E a*

* for the reverse reaction? *

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*How to calculate E a*

*? How to relate k to E a*

* and temperature? *

*From above: k varies nonlinearly with temperature *

*Arrhenius equation: gives the dependence of k on: *

*Activation energy Ea *

*Temperature T *

**k**

**Ae**

**E a**

**RT**

*Here, A= frequency factor *

*– A is essentially constant *

*R = gas constant (8.314 J/mol K) *

*Note the exponential behavior: *

*As E a*

* increases, k decreases: rate decreases *

*As T increases, k increases: rate increases *

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**Arrhenius equation: k**

**Ae**

**E a**

**RT**

*Put in linear form: take ln of both sides..... *

**ln k**

**E a**

**RT**

**ln A**

*In this form: equation of a line........ *

*Plot lnk vs. 1/T: slope = -E a*

*/R......... *

*Now: suppose we know E a*

* and k at some temperature T*

*1*

*, and we wish to find k at some other temperature T*

*2*

*.....*

**We have ln k**

**1**

**E a**

**RT**

**1**

**ln A**

*And *

**ln k**

**2**

**E a**

**RT**

**2**

**ln A**

*Subtract (and assume that E a*

* and A do n’t depend on T)..... *

**ln k**

**1**

**ln k**

**2**

**E a**

**RT**

**1**

**E a**

**RT**

**2**

*Clean this up a little... *

**k ln k**

**2**

**1**

**E a**

**R**

**1**

**T**

**2**

**1**

**T**

**1**

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*Based on their activation energies and *

*E values, and assuming that all frequency factors (A) are the same, which of the following reactions would be fastest and which be slowest? *

*(a) E a*

* = 45 kJ/mol, *

*E = -25 kJ/mol *

*(b) E a*

* = 35 kJ/mol, *

*E = -10 kJ/mol *

*(c) E a*

* = 55 kJ/mol, *

*E = 10 kJ/mol *

*The rate of the reaction *

*CH*

*3*

*COOC*

*2*

*H*

*5*

*(aq) + OH*

*-*

*(aq) *

* CH*

*3*

*COO*

*-*

*(aq) +C*

*2*

*H*

*5*

*OH(aq) *

*Was measured at several temperatures, and the following data collected: *

* T ( o*

*C) k (M*

*-1 s*

*-1*

*) *

* 15 *

* 25 *

*0.0521 *

*0.101 *

* 35 *

* 45 *

*0.184 *

*0.332 *

*Find the value of E a*

* for the reaction. *

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