0.5 Reaction rates  (Page 5/10)

An interesting point in the reaction is the time at which exactly half of the original concentration of $A$ has been consumed. We call this time the half life of the reaction and denote it as $t_{\frac{1}{2}}$ . At that time, $\left[A\right]=\frac{1}{2}{\left[A\right]}_0$ . From and using the properties of logarithms, we find that, for a first orderreaction

These conclusions are only valid for first order reactions. Consider then a second order reaction, such as thebutadiene dimerization discussed above . The general second order reaction $A\to \mathrm{products}$ has the rate law

The half-life of a second order reaction differs from the half-life of a first order reaction. From , if we take $\left[A\right]=\frac{1}{2}{\left[A\right]}_0$ , we get

Observation 3: temperature dependence of reaction rates

It is a common observation that reactions tend to proceed more rapidly with increasing temperature. Similarly,cooling reactants can have the effect of slowing a reaction to a near halt. How isthis change in rate reflected in the rate law equation, e.g. ? One possibility is that there is a slight dependence on temperature ofthe concentrations, since volumes do vary with temperature. However, this is insufficient to account for the dramatic changesin rate typically observed. Therefore, the temperature dependence of reaction rate is primarily found in the rate constant, $k$ .

Consider for example the reaction of hydrogen gas with iodine gas at high temperatures, as given in . The rate constant of this reaction at each temperature can be found using the method of initial rates,as discussed above, and we find in that the rate constant increases dramatically as the temperature increases.

As shown in , the rate constant appears to increase exponentially with temperature. After a littleexperimentation with the data, we find in that there is a simple linear relationship between $\ln k$ and $\frac{1}{T}$ .