Arrhenius Equation

9 Key excerpts on "Arrhenius Equation"

• 

  eBook - ePub

  Chemical Reactor Analysis and Applications for the Practicing Engineer

  • Louis Theodore(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  The development of the Arrhenius Equation provides an excellent introduction to the two theories to be discussed later. The reaction rate is affected not only by the concentration of species in the reacting system but also by the temperature. The Arrhenius Equation relates the reaction velocity constant with temperature. It is given by (see Chapter 3)

  (3.32)

  where	A =	frequency factor constant and is usually assumed independent of temperature
  	R =	universal gas constant
  	E a =	activation energy and is also usually assumed independent of temperature.

  The development of this equation follows. The van’t Hoff equation describes the variation of the equilibrium constant with temperature.

  (4.94)

  At equilibrium, however, it has been noted that

  (4.95)

  One may, therefore, write

  (4.96)

  Equation (4.96) may also be written

  (4.97)

  where

  and E

  a

  and E′

  a

  have yet to be defined. If one separates Equation (4.97) ,

  (4.98)

  Arrhenius noted, based on experimental data, that C = 0. He concluded

  (4.99)

  (4.100)

  where	E a =	activation energy for the forward reaction
  	E′ a =	activation energy for the reverse reaction.

  The integrated form of Equation (4.99) becomes

  (4.101)

  or

  (4.102)

  where A has been previously defined as the frequency factor. The term e

  −Ea/RT

  is simply the Boltzmann expression and can be shown to be the fraction of molecules in a system possessing energy in excess of E

  a

  . Equation (4.101) indicates that systems with large E

  a

  ’s will react slower since k is smaller. Accordingly, Equation (4.102) should yield a straight line of slope -E

  a

  /R and intercept A if ln k is plotted against (1/T) (see Figure 4.1 ). Implicit in this statement is the assumption that E

  a

  is constant over the temperature range in question.

  Figure 4.1 Activation energy information from reaction velocity constant-temperature data.

  Despite the fact that E

  a

  generally varies significantly with temperature, the Arrhenius Equation has wide applicability in industry and in the development of the theory to follow. The above method of analysis can be used to test the law, describe the variation of k with T, and/or evaluate E

  a

  . The numerical value of E

  a

  will depend on the choice and units of the reaction velocity constant (k, k

  p

  , k

  p

  *

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Physical Chemistry

  Kinetics

  • Horia Metiu(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  In its simplest and most widely used form the Arrhenius formula is a two-parameter equation, which means that if you know the values of the rate constants at two temperatures, you can use them to determine the two parameters in the equation, and then use the equation to calculate the rate constant at any other temperature. In practice it is safer to determine the rate constant at more than two temperatures and use the results in a least-squares fitting procedure to determine the constants in the Arrhenius Equation. In this chapter you will learn how to do such calculations.

  The Arrhenius Formula

  §2.  Arrhenius Formula and its Generalization.

  Arrhenius discovered empirically that the rate constant varies with temperature according to

  k = k 0 exp [ − E / R T ]

  (3.1)

  The quantity k0 is called the pre-exponential, E is the activation energy of the reaction, and R is the gas constant. The pre-exponential has the same units as k; for a first-order reaction this is s−1 .

  This equation has been remarkably successful and fits the temperature dependence of the rate constant regardless of the order of the reaction, as long as the reaction we study is direct (i.e. it is not the outcome of several reactions). More recent work has shown that sometimes the Arrhenius Equation needs to be made a little more flexible:

  k = k 0 T n exp [ − E / R T ]

  (3.2)

  This generalized Arrhenius formula has three parameters, n, k0 , and E, to be determined by fitting the data.

  The molecular theory of the rate constant is quite well developed and you can study it after you learn a bit of statistical mechanics. The theory provides a sound and interesting explanation of the Arrhenius formula. Unfortunately, for reactions of any complexity the theory does not provide an explicit formula for k, but a complex algorithm that can be implemented on a computer to calculate k. All versions of the theory, which differ in the details of their assumptions, give an exponential dependence on temperature, like that in Eqs 3.1 and 3.2. However, except for the simplest examples, the theory does not supply an explicit, general form for the temperature dependence of the pre-exponential. To make the issue harder to settle, the exponential dependence on temperature dominates the behavior of k so that the term T

  n

  in the pre-exponential does not have much of an impact; various values of n, including n

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Chemical Reactions and Chemical Reactors

  • George W. Roberts(Author)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Chapter 2 Reaction Rates—Some Generalizations LEARNING OBJECTIVES After completing this chapter, you should be able to 1. use the Arrhenius relationship to calculate how reaction rate depends on temperature; 2. use the concept of reaction order to express the dependence of reaction rate on the individual species concentrations; 3. calculate the frequency of bimolecular and trimolecular collisions; 4. determine whether the rate equations for the forward and reverse rates of a reversible reaction are thermodynamically consistent; 5. calculate heats of reaction and equilibrium constants at various temperatures (review of thermodynamics). In order to design a new reactor, or analyze the behavior of an existing one, we need to know the rates of all the reactions that take place. In particular, we must know how the rates vary with temperature, and how they depend on the concentrations of the various species in the reactor. This is the field of chemical kinetics. This chapter presents an overview of chemical kinetics and introduces some of the molecular phenomena that provide a foundation for the field. The relationship between kinetics and chemical thermodynamics is also treated. The information in this chapter is sufficient to allow us to solve some problems in reactor design and analysis, which is the subject of Chapters 3 and 4. In Chapter 5, we will return to the subject of chemical kinetics and treat it more fundamentally and in greater depth. 2.1 RATE EQUATIONS A ‘‘rate equation’’ is used to describe the rate of a reaction quantitatively, and to express the functional dependence of the rate on temperature and on the species concentrations. In symbolic form, r A ¼ r A ðT , all C i Þ where T is the temperature. The term ‘‘all C i ’’ is present to remind us that the reaction rate can be affected by the concentrations of the reactant(s), the product(s), and any other compounds that are present, even if they do not participate in the reaction.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Engineering Chemistry

  Fundamentals and Applications

  • Shikha Agarwal(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  13.13 Variation of Reaction Rates with Temperature – Arrhenius Equation The rate of a chemical reaction increases with the rise in temperature and it has been found that it becomes almost double for every 10 °C rise in temperature. This is known as the temperature coefficient, which may be defined as the ratio of rate constants of the reaction at two temperatures differing by 10 °C. Thus, Temperature coefficient = Rate constant ( 10 C) Rate constant at C T T + ° ° Figure 13.8 Variation of reaction rate with temperature The curve shows that at higher temperature the curve is shifted towards the right indicating that at higher temperatures the molecules have higher energies. Since the rate of reaction depends on effective collisions, that is, collisions with sufficient energy and proper orientation (to be discussed in collision theory of reaction rates); hence, as seen from the graph the number of effective collision doubles on increasing the temperature by 10 °C; therefore, the rate of reaction also doubles. Arrhenius Equation and calculation of activation energy Swedish chemist Arrhenius in 1889 gave a method for expressing the influence of temperature on reaction velocity. He proposed a quantitative relationship between rate constant and temperature: / Ea RT k Ae − = (1) where k is the rate constant, A is the pre-exponential factor which is related to the frequency of collision, E a is the activation energy or the energy barrier which the reactants must cross to form products, R is the gas constant and T is the temperature in K. Chemical Kinetics 723 Taking logarithm of both sides, ln ln a E k A RT = − (2) Converting to the base 10 ( 10 ln 2.303 log x x = ), we get 2.303 log 2.303 log a E k A RT = − log log 2.303 a E k A RT = − (3) The above equation shows that the value of k decreases as activation energy increases; hence, the rate of reaction decreases with the rise in activation energy.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Kinetic Modeling of Reactions In Foods

  • Martinus A.J.S. van Boekel(Author)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  In other words, the Arrhenius Equation becomes k ( t ) ¼ A exp E a RT ( t ) (5 : 43) Integration of Equation 5.42, after some rearrangement, now results in ð c c 0 1 c n d c ¼ A ð t 0 exp E a RT ( t ) d t (5 : 44) One has to know the dependence of T on t , T ( t ), in order to be able to solve this equation. This dependency could be a temperature varying linearly with time T ( t ) ¼ T 0 þ at (5 : 45) where the coef fi cient a gives the rate of temperature change, or a quadratic change T ( t ) ¼ T 0 þ at þ bT 2 (5 : 46) or a sinusoidal change T ( t ) ¼ T m þ a 2 sin 2 p t L þ 3 p 2 (5 : 47) or an increase described by an exponential T ( t ) ¼ T h þ ( T 0 T h ) exp ( Jt ) (5 : 48) In these equations T ( t ) represents temperature as a function of time, T 0 the starting temperature, T m the average temperature, and T h is the fi nal temperature. The parameter J in Equation 5.48 accounts for the heat transfer coef fi cient and speci fi c heat and mass of the product fl owing through a heat exchanger. To be sure, other equations than the ones given here are possible, as long as they describe the T -t pro fi le correctly. It is important to realize that the change in concentration is quite different for isothermal kinetics and nonisothermal kinetics. For a fi rst-order isothermal reaction it looks as depicted in Figure 5.17A, but for a linearly changing temperature the same fi rst-order reaction would look like the curve in Figure 5.17B. In the latter case, the reaction is slow at fi rst because the temperature is low but as the temperature rises the reaction rate increases, until the end when it decreases because the reactant becomes depleted (Figure 5.17C). Temperature and Pressure Effects 5 -23 On the assumption that E a and A do not depend on temperature and that one knows how temperature varies with time, one can derive the activation parameters directly from the concentration curve obtained from a nonisothermal experiment.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Statistical Design and Analysis of Stability Studies

  • Shein-Chung Chow(Author)
  • 2007(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  The relationship between the rate constant (or reaction rate) and absolute temperature can be expressed by the following Arrhenius Equation (see, e.g., Bohidar and Peace, 1988; Davies and Hudson, 1981; Carstensen, 1990): d ln K dT = E RT 2 , (2.13) where T is the absolute temperature, E is the activation energy, and R is the gas constant. Integrating both sides of Equation 2.13 gives ln K = − E R · 1 T + ln A , or K = A exp − E RT , (2.14) Binod April 12, 2007 10:56 C9055 Chapter 2 30 Accelerated Testing where A is a frequency factor. Substituting Equation 2.14 into Equations 2.11 and 2.12 yields the following equations: Y ( t ) = 100 − A exp − E RT t , (2.15) ln[ Y ( t )] = ln(100) − A exp − E RT t . (2.16) Rearranging the terms in Equations 2.15 and 2.16 gives Y ( t ) − 100 t = − A exp − E RT , (2.17) and ln[ Y ( t ) / 100] t = − A exp − E RT . (2.18) The quantities on the left-hand side of Equations 2.17 and 2.18 are the degradation per time unit based on either the original scale of the strength for the zero-order reaction or the log scale of the strength for the first-order reaction. [ Y ( t ) − 100] / t or ln[ Y ( t ) / 100] / t can be interpreted as the observed reaction rates that can be used for estimation of the unknown parameters A and − E / R in Equation 2.14. For the purpose of estimating the parameters A and − E / R , it is preferable to use the following negative observed reaction rates: 100 − Y ( t ) t = A exp − E RT , (2.19) ln[100 / Y ( t )] t = A exp − E RT . (2.20) The estimates obtained from Equations 2.19 and 2.20 are the same as those from Equations 2.17 and 2.18. 2.2 Statistical Analysis and Prediction Let Y i j be the strength of a sample stored after time t j at temperature T i , where i = 1 , 2 ,..., I and j = 1 ,..., J .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Physical Chemistry for Engineering and Applied Sciences

  • Frank R. Foulkes(Author)
  • 2012(Publication Date)
  • CRC Press

    (Publisher)

  CHEMICAL REACTION KINETICS 26-45 k = A exp G RT a u < › • – ³ — ˜ µ 6 where A is a constant––called the pre-exponential factor ––that is essentially independent of temperature . It can be seen that k increases as the temperature increases because the term exp G RT a < › ( ) 6 increases with increasing temperature. Thus, the reaction goes faster at higher temperatures. Basically this is because at higher temperatures, owing to the Maxwell-Boltzmann Distribution of kinetic energy among the molecules, a larger fraction of the molecules will possess enough kinetic energy to overcome the activation energy “barrier,” and therefore react to form the activated complex. Taking logarithms of both sides: lnk = ln A G RT a < › 6 A plot of ln k vs . 1 T should give a straight line with slope < › 6 G R a and intercept ln A . This type of plot is used experimentally to determine 6 G a › and A . ln k 0 1 T ln A slope G R a < › 6 We have seen this type of behavior before when we studied the Clausius-Clapeyron equation for phase equilibrium and the van’t Hoff equation for equilibrium constants. If we know the rate constant at T 1 , then, knowing 6 G a › for the reaction, we can evaluate the rate constant at T 2 from ln k k T T 2 1 £ ¤ ² ¥ ¦ ´ = < › < • – ³ — ˜ µ 6 G R 1 T 1 T a 2 1 15. Rates of ionic reactions : Often the rates of reaction between ions in solution are dependent on the ionic strength of the solution. This dependency is known as the primary kinetic salt effect . Consider the following elementary reaction between two ions in solution: A B z z A B + A Products where z A and z B are the valences (with signs) of the two ions. Since this is an elementary reaction it will be first order in each reactant, and the reaction rate will be = 26-46 CHEMICAL REACTION KINETICS r = < d[A ] dt z A = < d[B ] dt z B = k [A ][B ] z z A B .

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  PPI FE Chemical Review Manual eText - 1 Year

  • Michael R. Lindeburg(Author)
  • 2016(Publication Date)
  • PPI, a Kaplan Company

    (Publisher)

  A  < 0). The rate of a reaction depends primarily on temperature and reactant concentration, although pressure and other factors may also affect the reaction rate. The units of rate are moles per second (i.e., mol/s).

  Equation 52.11

  through

  Eq. 52.13

  : Isothermal, Constant Volume Rate of Reaction

  52.11

  52.12

  52.13

  Variation

  Description

  Equation 52.11

  through

  Eq. 52.13

  describe the rate of consumption of reactant A with time.14  

  Eq. 52.12

  , along with C A = N A /V , is used when volume, V , is constant.15  

  Eq. 52.13

  and its variation are generalized rate equations in which the exponents of each individual component, C x , give the reaction order for that component. The reaction rate constant (or just rate constant ), k , accounts for temperature effects and is described by the Arrhenius Equation. (See

  Sec. 52.5.

  .) The variation equation is applicable to reactions involving reactants that combine in multiple molecules. No exponents are needed when one reactant molecule forms one product molecule in the absence of other molecules.

  Figure 52.3

   shows the reaction rate influence factors such as temperature, concentration, and size of the pieces.

  Figure 52.3

  Reaction Rate Influence Factors

  Example

  A reaction occurs as shown. What is the most logical form of the rate equation?

  (A)

  r = k

  (B)

  r = k [S ]

  (C)

  r = k [P ]

  (D)

  r = k [S ][P ]

  Solution

  Rate equations do not usually include the concentrations of the products. A logical first assumption would be that the reaction is first order, in which case, the reaction rate would be proportional to the reactant concentration.

  The answer is (B).

  5. Arrhenius Equation

  Equation 52.14

  and

  Eq. 52.15

  : Arrhenius Equation16

  52.14

  52.15

  Description

  The Arrhenius Equation (see

  Eq. 52.14

  ) provides a method of calculating the rate constant, k , from relates the activation energy , E a , and the temperature, T . If the pre-exponential factor 17 (collision frequency factor ), A , is unknown, the activation energy can be found from the rate constants at two different temperatures, as shown in

  Eq. 52.15

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Physical Chemistry

  Understanding our Chemical World

  • Paul M. S. Monk(Author)
  • 2005(Publication Date)
  • Wiley

    (Publisher)

  (2) As an intercept, 1 ÷ T = 0, so the only sensible temperature to include as T in the intercept term would be T = ∞, which means that S ‡ = −∞. Again, this is not realistic. (3) More importantly, however, is a physicochemical concept behind the equa- tions: if both equations are written as ln(k) (as ‘y ’) as a function of 1 ÷ T (as ‘x ’), then it is dishonest to suppose that the gradients of the respective graphs can be different, one a function of E a and the other a function of H ‡ . As a further implication, H ‡ cannot be the same as E a . In fact, from the Eyring theory, we can show readily that H ‡ = E a + RT (8.57) This equation explains why values of H ‡ are not Eyring theory says that H ‡ = E a + RT , explaining why values of H ‡ are not con- stant, but depend on temperature. constant, but depend on temperature. Conversely, E a is a true constant. We will employ the form of the Eyring equation writ- ten as Equation (8.56). SAQ 8.26 The following table contains the rate constant k for the demethylation reaction of N-methyl pyridinium bromide by aqueous sod- ium hydroxide as a function of temperature: T /K 298 313 333 353 k/10 2 dm 3 mol −1 s −1 8.39 21.0 77.2 238 THERMODYNAMIC CONSIDERATIONS 419 (1) Calculate the activation energy E a and pre-exponential factor A by plotting an Arrhenius graph. (2) Calculate G ‡ , H ‡ and S ‡ for the reaction at 298 K by plotting an Eyring graph. (3) What is the relationship between H ‡ and E a ? Justification Box 8.6 From Eyring, the rate constant of reaction k depends on a pseudo equilibrium constant K ‡ , relating to the formation of a transition-state complex, TS. Clearly, K ‡ will always be virtually infinitesimal. The values of k and K ‡ are related as Care: there are three different types of ‘k’ in Equation (8.58), so we must be careful about the choice of big or small characters, and subscripts. k = k B T h K ‡ (8.58) where T is the thermodynamic temperature.

  Sign up to read

  Learn more about book