with the proper orientation to react and is the probability that any given collision will result in a reaction. It can be seen that either increasing the temperature or decreasing the activation energy (for example through the use of catalysts) will result in an increase in rate of reaction. Given the small temperature range of kinetic studies, it is reasonable to approximate the activation energy as being independent of the temperature. Similarly, under a wide range of practical conditions, the weak temperature dependence of the pre-exponential factor is negligible compared to the temperature dependence of the factor; except in the case of "barrierless" diffusion-limited reactions, in which case the pre-exponential factor is dominant and is directly observable. With this equation it can be roughly estimated that the rate of reaction increases by a factor of about 2 or 3 for every 10°C rise in temperature. The term denotes the fraction of molecules with energy greater than or equal to . Arrhenius plot Taking the natural logarithm of Arrhenius equation yields: Rearranging yields: This has the same form as an equation for a straight line: where x is the reciprocal of T. So, when a reaction has a rate constant that obeys the Arrhenius equation, a plot of ln k versus T−1 gives a straight line, whose gradient and intercept can be used to determine Ea and A . This procedure has become so common in experimental chemical kinetics that practitioners have taken to using it to define the activation energy for a reaction. That is the activation energy is defined to be (−R) times the slope of a plot of ln k vs. (1/T): Modified Arrhenius equation The modified Arrhenius equation makes explicit the temperature dependence of the pre-exponential factor. The modified equation is usually of the form The original Arrhenius expression above corresponds to n = 0. Fitted rate constants typically lie in the range . Theoretical analyses yield various predictions for n. It has been pointed out that "it is not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted T1/2 dependence of the pre-exponential factor is observed experimentally". However, if additional evidence is available, from theory and/or from experiment (such as density dependence), there is no obstacle to incisive tests of the Arrhenius law. Another common modification is the stretched exponential form where β is a dimensionless number of order