Solution reactions enthalpy profile

In this chapter, we describe how time-resolved photoacoustic calorimetry (PAC) can be used to obtain both the energetics and kinetics of carbenes in solution.7-9 PAC measures the magnitude and temporal profile of volume changes in solution following deposition of energy. These time-resolved volume changes can be directly related to carbene reaction enthalpies. We will first discuss the principles of this photoacoustic technique and then how it has been... 

Figure 3.2 An enthalpy profile for a unimolecular reaction in solution, involving the formation of a radical pair inside a solvent cage. Adapted from [61],...

An implicit assumption of the foregoing treatment is that A// remains independent of temperature over the range investigated. This is very nearly correct in general, and is particularly the case given that studies of reactions in solution are usually conducted over a temperature interval of only some 30-50°. In certain circumstances the temperature profiles show curvature outside the experimental error. Such cases have, or appear to have, temperature-dependent activation enthalpies. Here we explore one of the reasons for that another is given in Section 7.3. 

In order to develop an intuition for the theory of flames it is helpful to be able to obtain analytical solutions to the flame equations. With such solutions, it is possible to show trends in the behavior of flame velocity and the profiles when activation energy, flame temperature, diffusion coefficients, or other parameters are varied. This is possible if one simplifies the kinetics so that an exact solution of the equation is obtained or if an approximate solution to the complete equations is determined. In recent years Boys and Corner (B4), Adams (Al), Wilde (W5), von K rman and Penner (V3), Spalding (S4), Hirschfelder (H2), de Sendagorta (Dl), and Rosen (Rl) have developed methods for approximating the solution to a single reaction flame. The approximations are usually based on the simplification of the set of two equations [(4) and (5)] into one equation by setting all of the diffusion coefficients equal to X/cpp. In this model, Xi becomes a linear function of temperature (the constant enthalpy case), and the following equation is obtained ... 

When a fast reaction is highly exothermic or endothermic and, additionally, the effective thermal conductivity of the catalyst is poor, then significant temperature gradients across the pellet are likely to occur. In this case the mass balance (eq 32) and the enthalpy balance (eq 33) must be simultaneously solved using the corresponding boundary conditions (eqs 34-37), to obtain the concentration profile of the reactant and the temperature profile inside the catalyst pellet. The exponential dependence of the reaction rate on the temperature thereby imposes a nonlinear character on the differential equations which rules out an exact analytical treatment. Approximate analytical solutions [83, 99] as well as numerical solutions [13, 100, 110] of eqs 32-37 have been reported by various authors. 

In kinetic and thermodynamic analyses of chemical processes in solution, there are basically two physical parameters that can be varied experimentally, viz, temperature and pressure. The temperature dependence is used to obtain reaction and activation free energy, enthalpy, and entropy, which are used to construct an energy profile for the reaction under study. The thermodynamic activation parameters reveal information regarding the energetics of the reaction and the nature of the transition state, especially in terms of structural order as obtained from the activation entropy. The pressure dependence is used to obtain reaction and activation volumes that are used to construct a volume profile for the reaction under study (2, 3). 

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