Temperatures and Rates
For many chemical reactions, the rate constant increases exponentially with increasing temperature.
The rate equation and the value of the rate constant for a reaction are deduced from measurements of the rate of reaction at a fixed temperature. If experiments are performed at several different temperatures, generally the concentration dependence exhibited in the rate equation is unchanged, but the value of the rate constant is very temperature dependent.
The temperature dependence of the reaction rate shows up according to
Rate = k(T) × (concentration dependent term) in k(T), the rate “constant”.
Often this term has the temperature dependence, the rate increases rapidly with increasing temperature. Other circumstances dependencies e.g. those of b and c, also show up in the special circumstances but most chemical reactions have the temperature dependence.
In 1889 Arrhenius recognized that this typical temperature dependence indicates an exponential increase in the rate, or rate constant, with temperature. This empirical relation can be conveniently written as
k = Ae^{-Ea/(RT)} where A is called the preexponential factor and Ea is known as the activation energy. With this notation one writes the logarithmic form.
In k = Ea/RT + In A
The data shown on a logarithmic k-versus 1/T plot in the empirical constants Ea and A can be deduced from the slope and intercept of such a graph.
Most reactions that proceed at a moderate rate, i.e. occur appreciably in minutes or hours, have values of Ea of 50 to 100 kJ. For such reactions one can use to verify the photographer’s guide that reactions go 2 or 3 times as fast when the temperatures increases by 10˚C.
Some reactions that appear to follow the temperature dependence expressed are found on closer inspection to have a somewhat temperature dependent A term. If this term contains a power of T as a factor, an equation of the form will contain an In T term with a coefficient indicating the power to which T is raised in the preexponential A factor. There are only a few reactions for which such a temperature dependent A term has been established.
Example: use the results to obtain an expression for the temperature dependence of the rate of hydrolysis of tert-butyl bromide.
Solution: assume an equation of the form k = Ae^{-Ea/(RT)} or In k = -Ea/(RT) + In A. the values of the parameters Ea and Acan be obtained from k values at two temperatures by subtracting the logarithmic equations written for the two conditions. The general result is
In d(T_{2})/k(T_{1}) = -(Ea/R) (1/T_{2} – 1/T_{1}) = Ea(T_{2} – T_{1}/RT_{1}T_{2})
Here we have In (2.37 × 10^{-4})/(0.137 × 10^{-4}) = 2.85 = Ea(25K)/[8.314 J K^{-1} mol^{-1})(298 K)(323 K)]. This gives Ea = 91.2 kJ mol^{-1}. With this value we can return to the original rate constant expression and obtain A = 1.38 × 10^{11} s^{-1}.
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