Arrhenius Equation
The Arrhenius Equation is a formula that describes the temperature dependence of reaction rates in chemical reactions. It states that the rate constant of a reaction increases exponentially with temperature. The equation is expressed as k = A * e^(-Ea/RT), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin.
5 Key excerpts on "Arrhenius Equation"
eBook - ePub- Jeffrey Gaffney, Nancy Marley(Authors)
- 2017(Publication Date)
- Elsevier(Publisher)
...The quantitative basis of the relationship between these factors and the reaction rate is described by the Arrhenius Equation ; k = Ae − E a / RT (15) where “k” is the reaction rate constant, “E a” is the activation energy, “R” is the ideal gas constant (8.314 × 10 − 3 kJ • K − 1 • mol − 1), and “T” is the temperature in Kelvin. The parameter “A” is called the frequency factor. It is related to the number of collisions with the correct orientation for reaction. The factor e − E a /RT is the fraction of molecules with the minimum amount of energy required for reaction. The frequency factor (molecular orientation) has a very small temperature dependence and can be considered to be independent of temperature. The exponential factor is strongly temperature-dependent. Taking the natural logarithm of Eq. (15) gives: ln k = ln A – E a / RT ln k = ln A – E a / R 1 / T (16) Eq. (16) is in the form of a straight line with a slope of − E a / R and an intercept of ln A. If the reaction rate constants are measured experimentally at different temperatures and plotted as ln k versus 1/ T, the graph is known as an Arrhenius plot. The activation energy of the reaction can be obtained from the slope of the Arrhenius plot as; Slope = – E a / R E a = – R × slope. An Arrhenius plot for the reaction of an oxygen atom with the organic molecule acrolein (C 3 H 4 O) in the gas phase is shown in Fig. 9.8. This reaction has an activation energy of 2.4 kcal • mol − 1 and and frequency factor of 1.4 × 10 10 mol − 1 • s − 1. This very high frequency factor shows that the reaction has a high frequency of effective collisions. Fig...
eBook - ePub- Mark E. Davis, Robert J. Davis(Authors)
- 2013(Publication Date)
- Dover Publications(Publisher)
...The Arrhenius form of the reaction rate constant is an empirical relationship. However, transition-state theory provides a justification for the Arrhenius formulation, as will be shown below. Note that the Arrhenius law (Equation 2.2.1) gives a linear relationship between ln k and T −1. EXAMPLE 2.2.1 The decomposition reaction: can proceed at temperatures below 100°C and the temperature dependence of the first-order rate constant has been measured. The data are: Suggest an experimental approach to obtain these rate constant data and calculate the activation energy and pre-exponential factor. (Adapted from C. G. Hill, An Introduction to Chemical Engineering Kinetics & Reactor Design, Wiley, New York, 1977.) Answer Note that the rate constants are for a first-order reaction. The material balance for a closed system at constant temperature is: where is the number of moles of N 2 O 5. If the system is at constant volume (a closed vessel), then as the reaction proceeds the pressure will rise because there is a positive mole change with reaction. That is to say that the pressure will increase as N 2 O 5 is reacted because the molar expansion factor is equal to 0.5. An expression for the total moles in the closed system can be written as: where n is the total number of moles in the system. The material balance on the closed system can be formulated in terms of the fractional conversion and integrated (see Example 1.5.2) to give: Since the closed system is at constant T and V (PV = nR g T): and the pressure can therefore be written as: If the pressure rise in the closed system is monitored as a function of time, it is clear from the above expression how the rate constant can be obtained at each temperature. In order to determine the pre-exponential factor and the activation energy, the ln k is plotted against T −1 as shown below: From a linear regression analysis of the data, the slope and intercept can be obtained, and they are 1.21 × 10 4 and 30.4, respectively...
- Kevin Reel, Derrick C. Wood, Scott A. Best(Authors)
- 2014(Publication Date)
- Research & Education Association(Publisher)
...time will result in a straight line whose slope is the rate constant. DID YOU KNOW? Nuclear decay processes can occur in fractions of a second but others, such as those that occur with nuclear waste, can take thousands of years! Effect of Temperature on Rate Temperature is a measure of the average kinetic energy of molecules. In general, raising the temperature will cause the reaction rate to increase. This is due to the fact that reactant molecules will be traveling at increased speed and will collide more frequently with greater kinetic energy. This results in an increased number of effective collisions that result in the formation of products. The Arrhenius Equation represents the relationship between temperature and the rate constant of a reaction: Note that since the Arrhenius Equation deals with the energy of molecules, the value of R should be 8.31 J mol –1 K –1. If a plot of the natural log of k (ln k) versus the inverse of temperature (1/T) is created, it will produce a straight line whose slope allows for the calculation of the energy of activation, E a for the reaction. Reaction Mechanisms A reaction mechanism represents the pathway by which reactants form products. Reaction mechanisms are classified by the number of steps involved, whether it is one (unimolecular), two (bimolecular), or three (termolecular). There are two criteria that must be met for a plausible mechanism of a chemical reaction. 1. The steps of the reaction mechanism must add up to and equal the overall reaction. 2. The experimental rate law must match the rate law derived from the mechanism. Some reactions occur in one step and are known as concerted reactions. These types of reactions most often involve two reactants as shown in the following equation. When determining the rate law from a mechanism, the coefficients of the reactants represent the orders of the reactants...
eBook - ePubPharmaceutical Dosage Forms and Drug Delivery
Revised and Expanded
- Ram I. Mahato, Ajit S. Narang(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...Plot of the variation of the rate constant, k, versus reciprocal of the absolute temperature, T. The Arrhenius expression can also be written as (Figure 7.4): (7.53) ln k = ln A − E a R T Or (7.54) log k = log A − E A 2.303 R T This equation is of the form y = mx + c for a straight-line plot. Thus, an Arrhenius plot of log k on the y -axis against reciprocal of the absolute temperature (1 /T) on the x -axis yields E a from the slope of the straight line (Figure 7.4). This equation is not amenable for direct application for the measurement of reaction rates, since A and E a are unknown. Nevertheless, activation energy, E a, can be calculated by comparing reaction rates at two different temperatures. Thus, for. temperatures T 1 and T 2, (7.55) k 1 = A e − E a / R T 1 (7.56) k 2 = A e − E A / R T 2 Thus, (7.57) k 2 k 1 = A e − E a / R T 2 A e − E a / R T 1 = e E a / R T 1 − E a / R T 2 =[--=. PLGO-SEPARATOR=--]e E a / R (1 / T 1 − 1 / T 2) = e E a / R ((T 2 − T 1) / T 1 T 2) Which is same as: (7.58) ln k 2 k 1 = ln E a R (T 2 − T 1 T 1 T 2) Or, as in Figure. 7.4, (7.59) log k 2 k 1 = log E a 2.303 R (T 2 − T 1 T 1 T 2) Thus, measurement of the reaction rate constant, k, at two different temperatures allows the calculation of the activation energy, E a, for a given reaction. 7.3.1.2 Shelf life The Arrhenius plot can be used to determine the shelf life of the drug. The half-life (t 1/2) and shelf life (t 0.90) expressions from the reaction order can be substituted for the reaction rate constants, k, in the above equations to directly infer product’s shelf life at a given temperature. These calculations allow the calculation of temperature of optimum drug stability over its shelf life. If a drug is stable at room temperature (25°C), it is usually labeled for storage at controlled room temperature (range 15°C–30°C)...
eBook - ePub- Athel Cornish-Bowden(Author)
- 2013(Publication Date)
- Wiley-Blackwell(Publisher)
...The equilibrium constant for denaturation, K, varies with temperature according to the van’t Hoff equation (Section 1.8.1) : § 1.8.1, pages 15–17 where R is the gas constant, T is the absolute temperature and Δ G 0 ′, Δ H 0 ′ and Δ S 0 ′ are the standard Gibbs energy, enthalpy and entropy of reaction, respectively. This relationship can be rearranged to provide an expression for K : § 1.8.1, pages 15–17 The rate equation k for the catalytic reaction may be governed by the integrated Arrhenius Equation: where A is a constant and E a is the Arrhenius activation energy. The rate of the catalytic reaction is given by v = k [E] [A], but to use this equation the concentration [E] of active enzyme has to be expressed in terms of the total concentration e 0 = [E] + [E′],and so The observed rate constant, K obs, may be defined as k /(1 + K), and varies with the temperature according to the following equation: (11.1) At low temperatures, when Δ S 0 ′/ R is small compared with Δ H 0 ′/ RT, the exponential term in the denominator is insignificant, and so k obs varies with temperature in the ordinary way according to the Arrhenius Equation. At temperatures above Δ H 0′ /Δ S 0′, however, the denominator increases steeply with temperature and the rate of reaction decreases. This behavior is illustrated in Figure 11.3. Figure 11.3. The temperature dependence of an enzyme rate is typically the result of two effects in opposite directions, as given by equation 11.1. The numerator increases steeply with temperature over the whole range; the denominator is negligible when Δ S 0 ′/ R < Δ H 0 ′ /RT but overwhelms the increasing numerator when the temperature is higher. 11.2 Irreversible denaturation For the sake of simplicity the previous section treated thermal denaturation as a reversible equilibrium (Figure 11.1), but in reality it is likely to be irreversible, at least in part, and Figure 11.2 shows a more realistic model...
