It is common knowledge that chemical reactions occur more rapidly at higher temperatures. Everyone knows that milk turns sour much more rapidly if stored at room temperature rather than in a refrigerator, butter goes rancid more quickly in the summer than in the winter, and eggs hardboil more quickly at sea level than in the mountains. For the same reason, coldblooded animals such as reptiles and insects tend to be noticeably more lethargic on cold days.
It is not hard to understand why this should be. Thermal energy relates direction to motion at the molecular level. As the temperature rises, molecules move faster and collide more vigorously, greatly increasing the likelihood of bond cleavages and rearrangements. Whether it is through the Collision Theory, Transition State Theory, or just common sense, chemical reactions are typically expected to proceed faster at higher temperatures and slower at lower temperatures.
By 1890 it was common knowledge that higher temperatures speed up reactions, often doubling the rate for a 10degree rise, but the reasons for this were not clear. Finally, in 1899, the Swedish chemist Svante Arrhenius (18591927) combined the concepts of activation energy and the Boltzmann distribution law into one of the most important relationships in physical chemistry:
Take a moment to focus on the meaning of this equation, neglecting the A factor for the time being.
First, note that this is another form of the exponential decay law we discussed in the previous section of this series. What is "decaying" here is not the concentration of a reactant as a function of time, but the magnitude of the rate constant as a function of the exponent –E_{a} /RT. And what is the significance of this quantity? If you recall that RT is the average kinetic energy, it will be apparent that the exponent is just the ratio of the activation energy E^{a} to the average kinetic energy. The larger this ratio, the smaller the rate (hence the negative sign.) This means that high temperature and low activation energy favor larger rate constants, and thus speed up the reaction. And because these terms occur in an exponent, their effects on the rate are quite substantial.
The two plots below show the effects of the activation energy (denoted here by E^{‡}) on the rate constant. Even a modest activation energy of 50 kJ/mol reduces the rate by a factor of 10^{8}.
The logarithmic scale in the righthand plot leads to nice straight lines, as described under the next heading below.
Looking at the role of temperature, we see a similar effect. (If the xaxis were in "kilodegrees" the slopes would be more comparable in magnitude with those of the kiloJoule plot at the above right.)
The Arrhenius equation
\[k = A e^{E_a/RT}\]
can be written in a nonexponential form which is often more convenient to use and to interpret graphically. Taking the logarithms of both sides and separating the exponential and preexponential terms yields
which is the equation of a straight line whose slope is \(–E_a /R\). This affords a simple way of determining the activation energy from values of k observed at different temperatures; we just plot \(\ln k\) as a function of \(1/T\).
Example 1: Isomerization of Cyclopropane  

Thus for the isomerization of cyclopropane to propene the following data were obtained (calculated values shaded in pink):
From the calculated slope, we have – (E_{a}/R) = –3.27 × 10^{4} K E_{a}=– (8.314 J mol^{–1} K^{–1}) (–3.27 × 10^{4} K) = 273 kJ mol^{–1} Comment: This activation energy is rather high, which is not surprising because a carboncarbon bond must be broken in order to open the cyclopropane ring. (C–C bond energies are typically around 350 kJ/mol.) This is why the reaction must be carried out at high temperature. 
Since the ln kvs.1/T plot yields a straight line, it is often convenient to estimate the activation energy from experiments at only two temperatures. To see how this is done, consider that
(... in which we have made the lnA term disappear by subtracting the expressions for the two lnk terms.) Solving the expression on the right for the activation energy yields
Example 1 

A widely used ruleofthumb for the temperature dependence of a reaction rate is that a tenC° rise in the temperature approximately doubles the rate. (This is obviously not generally true, especially when a strong covalent bond must be broken.) But for a reaction that does show this behavior, what would the activation energy be? SOLUTION We will center our tendegree interval at 300 K. Substituting into the above expression yields = (8.314)(0.693) / (.00339  0.00328) = (5.76 J mol^{–1} K^{–1}) / (0.00011 K^{–1}) = 52400 J mol^{–1} = 52.4 kJ mol^{–1}

Example 2 

It takes about 3.0 minutes to cook a hardboiled egg in Los Angeles, but at the higher altitude of Denver, where water boils at 92°C, the cooking time is 4.5 minutes. Use this information to estimate the activation energy for the coagulation of egg albumin protein. SOLUTION The ratio of the rate constants at the elevations of LA and Denver is 4.5/3.0 = 1.5, and the respective temperatures are 373K and 365K. With the subscripts 2 and 1 referring to LA and Denver respectively, we have Comment: This rather low value seems reasonable because protein denaturation involves the disruption of relatively weak hydrogen bonds; no covalent bonds are broken. 
It is now time to focus in on the preexponential term A in the Arrhenius equation. We have been neglecting it because it is not directly involved in relating temperature and activation energy, which is the main practical use of the equation. But since A multiplies the exponential term, its value clearly contributes to the value of the rate constant and thus of the rate.
Recall that the exponential part of the Arrhenius equation expresses the fraction of reactant molecules that possess enough kinetic energy to react, as governed by the MaxwellBoltzmann law. This fraction can run from zero to nearly unity, depending on the magnitudes of E_{a} and of the temperature.
If this fraction were unity, the Arrhenius law would reduce to
k = A
In other words, A is the fraction of molecules that would react if either the activation energy were zero, or if the kinetic energy of all molecules exceeded E_{a} — admittedly, an uncommon scenario.
So what would limit the rate constant if there were no activation energy requirements? The most obvious factor would be the rate at which reactant molecules come into contact. This can be calculated from kinetic molecular theory and is known as the frequency or collision factor Z.
In some reactions, the relative orientation of the molecules at the point of collision is important, so we can also define a geometrical or steric factor (commonly denoted by \(\rho\) (Greek lower case rho). In general, we can express A as the product of these two factors:
\[A = Z\rho\]
Values of ρ are generally very difficult to assess; they are sometime estimated by comparing the observed rate constant with the one in which A is assumed to be the same as Z.
The "Arrhenius Equation" was physical justification and interpretation in 1889 by Svante Arrhenius, a Swedish chemist. Arrhenius performed experiments that correlated chemical reaction rate constants with temperature. After observing that many chemical reaction rates depended on the temperature, Arrhenius developed this equation to characterize the temperaturedependent reactions.
\[ \large k=Ae^{^{\frac{E_{a}}{k_{B}T}}} \]
or
\[\large \ln k=\ln A  \frac{E_{a}}{k_{B}T} \]
With the following terms:
k: Chemical reaction rate constant
A: The preexponential factor or frequency factor
E_{a}: The activation energy is the threshold energy that the reactant(s) must acquire before reaching the transition state.
R: The gas constant.
T:The absolute temperature at which the reaction takes place.
The exponential term in the Arrhenius Equation implies that the rate constant of a reaction increases exponentially when the activation energy decreases. Because the rate of a reaction is directly proportional to the rate constant of a reaction, the rate increases exponentially as well. Because a reaction with a small activation energy does not require much energy to reach the transition state, it should proceed faster than a reaction with a larger activation energy.
In addition, the Arrhenius Equation implies that the rate of an uncatalyzed reaction is more affected by temperature than the rate of a catalyzed reaction. This is because the activation energy of an uncatalyzed reaction is greater than the activation energy of the corresponding catalyzed reaction. Since the exponential term includes the activation energy as the numerator and the temperature as the denominator, a smaller activation energy will have less of an impact on the rate constant compared to a larger activation energy. Hence, the rate of an uncatalyzed reaction is more affected by temperature changes than a catalyzed reaction.
To eliminate the constant A, there must be two known temperatures and/or rate constants. With this knowledge we write:
\[ \ln k_{1}=\ln A  \frac{E_{a}}{k_{B}T_1} \]
at T_{1} and
\[ \ln k_{2}=\ln A  \frac{E_{a}}{k_{B}T_2} \]
at T_{2} . By rewriting the second equation:
\[ \ln A = \ln k_{2} + \frac{E_{a}}{k_{B}T_2} \]
and substitute for ln A into the first equation:
\[ \ln k_{1}= \ln k_{2} + \frac{E_{a}}{k_{B}T_2}  \frac{E_{a}}{k_{B}T_1} \]
This simplifies to:
\[ \ln k_{1}  \ln k_{2} = \frac{E_{a}}{k_{B}T_1} + \frac{E_{a}}{k_{B}T_2} \]
\[ \ln \frac{k_{1}}{k_{2}} = \frac{E_{a}}{k_{B}} \left (\frac{1}{T_1}\frac{1}{T_2} \right ) \]
If we look at the Arrhenius equation more carefully, one notices that the natural logarithm form of the Arrhenius equation is in the form of y = mx + b. In other words, it is similar to the equation of a straight line.
\[ \ln k=\ln A  \frac{E_{a}}{k_{B}T} \]
So if one were given a data set of various values of k, the rate constant of a certain chemical reaction at varying temperature T, one could graph ln(k) versus 1/T. From the graph, one can then determine the slope of the line and realize that this value is equal to Ea/R. One can then solve for the activation energy by multiplying through by R, where R is the gas constant.
From:http://www.youtube.com/watch?v=mM7cszLD6yw&feature=player_embedded by user mkseery
1. E_{a} is the factor the question asks to be solved. Therefore it is much simpler to use
To find E_{a}, subtract ln A from both sides and multiply by RT.
This will give us:
E_{a}= (ln A  ln k)RT
2. Substitute the numbers into the equation:
lnk = (200 X 1000) / (8.314)(289) + ln9
k = 6.37X10^{36 }M^{1}s^{1} ^{ }
3. Use the equatioin ln(k_{1}/k_{2})=Ea/R(1/T_{1}1/T_{2})
ln(7/k_{2})=[(900 X 1000)/8.314](1/3701/310)
k_{2}=1.788X10^{24 }M^{1}s^{1}
4. Use the equation k = Ae^{Ea}^{/RT}
12 = 15e^{Ea}^{/(8.314)(22)}
Ea = 40.82J/mol
5. Use the equatioin ln(k_{1}/k_{2})=Ea/R(1/T_{1}1/T_{2})
ln(15/7)=[(600 X 1000)/8.314](1/T_{1}  1/389)
T_{1} = 390.6K
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