Square brackets with concentrations

The rate of a process is expressed by the derivative of a concentration (square brackets) with respect to time, d[ ]/dt. If the concentration of a reaction product is used, this quantity is positive if a reactant is used, it is negative and a minus sign must be included. Also, each derivative d[ ]/dt should be divided by the coefficient of that component in the chemical equation which describes the reaction so that a single rate is described, whichever component in the reaction is used to monitor it. A rate law describes the rate of a reaction as the product of a constant k, called the rate constant, and various concentrations, each raised to specific powers. The power of an individual concentration term in a rate law is called the order with respect to that component, and the sum of the exponents of all concentration terms gives the overall order of the reaction. Thus in the rate law Rate = k[X] [Y], the reaction is first order in X, second order in Y, and third order overall. 

The proportionality constant k, called the rate constant, has a constant value for a given reaction at a given temperature. The terms in square brackets are concentration terms (compare Chap. 14), and x and y are exponents which are often integral. The exponent x is called the order with respect to A, and y is called the order with respect to B. The sum x +y is called the overall order of the reaction. The values for x and y can be 0, 1, 2, 3 or 0.5, 1.5, or 2.5, but never more than 3. These values must be determined by experiment, and do not necessarily equal the values of a and b in the chemical equation. 

Quantities in square brackets represent concentrations and the proportionality constant k is called the rate constant. If this mechanism is right, then the rate of the reaction will be simply and linearly proportional to both [n-BuBr] and to [HO ]. And it is. Ingold measured the rates of reactions like these and found that they were second-order (proportional to two concentrations) and he called this mechanism Substitution, Nucleophilic, 2nd Order or 8 2 for short. The rate equation is usually given like this, with 2 representing the second-order rate constant. 

The letter k with the indices (-1, +1, or +2) symbolizes the velocity constants of the reactions defined by the rates of changes in concentrations. Square brackets symbolize concentrations of the respective substances. The velocity of the first reaction step is so fast that it is impossible to determine the constants k, and k+1 separately, and the equilibrium constant Kd is more useful ... 

Fig. 11. Active transport using the simple carrier. Formal kinetic schemes for (a) countertransport and (b) co-transport. A and B are the two substrates of the carrier E either of which (in countertramsport) or both which (in co-transport) combine with E to form EA, EB or EAB. Subscripts 1 or 2 refers to substrate at side 1 or 2 of the membrane. The rate constants b, d, f, g, k are defined in the figure. AT" is the equilibrium constant of the chemical reaction /I =4.42 in primary active transport (and is equal to unity in the case of secondary active transport). Af=A /A 2 in co-transport. The square brackets denote concentrations and terms such as /4, =[/l,]/Arj, where is the relevant dissociation constant, here = dj/gi. J is the net flux in the 1 ->2 direction. (Figure taken, with permission, from [30].)...

Square brackets indicate concentrations (as they do in an equilibrium expression - see Chapter 7), the exponents a and b are the individual orders of the reaction with respect to the reactants A and B, and k is the proportionality constant, known as the rate constant. The values ofa,b and k have to be determined experimentally. The sum of the individual orders, a + b, is known as the overall order of the reaction. The order of reaction with respect to a particular reactant indicates precisely what happens to the rate of reaction if that concentration is changed. 

Square brackets are commonly used for two purposes to denote concentrations and also to include the whole of a complex ion for the latter purpose curly brackets (braces) are sometimes used. With careful scrutiny there should be no confusion regarding the sense in which the square brackets are used with complexes there will be no charge signs inside the brackets. 

A note on good practice In chemical kinetics, the square brackets denote molar concentration, with the units mol-L 1 retained. 

When dealing with the kinetic or thermodynamic behaviour of transition-metal systems, square brackets are used to denote concentrations of solution species. In the interests of simplicity, solvent molecules are frequently omitted (as are the square brackets around complex species). The reaction (1.1) is frequently written as equation (1.2). 

The first two expressions involving the reactants are negative, because their concentrations will decrease with time. The square brackets represent moles per liter concentration (molarity). 

The term in the square bracket is an effective diffusion coefficient DAB. In principle, this may be used together with a material balance to predict changes in concentration within a pellet. Algebraic solutions are more easily obtained when the effective diffusivity is constant. The conservation of counter-ions diffusing into a sphere may be expressed in terms of resin phase concentration Csr, which is a function of radius and time. 

When a reaction occurs between gaseous species or in solution, chemists usually express the reaction rate as a change in the concentration of the reactant or product per unit time. Recall, from your previous chemistry course, that the concentration of a compound (in mol/L) is symbolized by placing square brackets, [ ], around the chemical formula. The equation below is the equation you will work with most often in this section. 

The rate, or velocity, of a reaction is usually defined as the change with time t of the concentration (denoted by square brackets) of one of the reactants or of one of the products of the reaction that is. 

As discussed earlier, the dependence of pe on the concentrations of reductants and oxidants is often small in comparison with its dependence on pH. The term in the square brackets in Equation (4.25) can therefore be replaced by pe terms giving for the approximate standard free energy change ... 

A mole (mol) is Avogadro s number of particles (atoms, molecules, ions, or anything else). Molarity (M) is the number of moles of a substance per liter of solution. A liter (L) is the volume of a cube that is 10 cm on each edge. Because 10 cm = 0.1 m, 1 L = (0.1 m)3 = 10-3 m3. Chemical concentrations, denoted with square brackets, are usually expressed in moles per liter (M). Thus means the concentration of H+ ... 

According to Nernst s equation, there should be linear relationship between the equilibrium potential of the metal/metal-ion electrode (M/M2+) and the logarithm of the concentration of 1VF+ ions [Eq. (5.13)]. This linear relationship was experimentally observed for low concentration of the solute MA, for instance, 0.01 mol/L and lower. For higher concentrations a deviation from linearity was observed, see, for example, Figure 5.12. The deviation from linearity is due to ion-ion interactions. In the example in Figure 5.12, the ion-ion interactions include interaction of the hydrated Ag+ ions with one another and with NO 3 ions. The linear relationship between the equilibrium potential E and the log of concentration is obtained if the square brackets in Eq. (5.13) signify the activity of species within those brackets. The activity of the species i is defined by the equation... 

The basic strategy in the application of electroanalytical methods in studies of the kinetics and mechanisms of reactions of radicals and radical ions is the comparison of experimental results with predictions based on a mechanistic hypothesis. Thus, equations such as 6.28 and 6.29 have to be combined with the expressions describing the transport. Again, we restrict ourselves to considering transport governed only by linear semi-infinite diffusion, in which case the combination of Equations 6.28 and 6.29 with Fick s second law, Equation 6.18, leads to Equations 6.31 and 6.32 (note that we have now replaced the notation for concentration introduced in Equation 6.18 earlier by the more usual square brackets). Also, it is assumed here that the diffusion coefficients of A and A - are the same, i.e. DA = DA.- = D. 

Since we are dealing in all these equations with dimensionless ratios of mole fractions, we may replace mole fractions by molarities (designated by square brackets) or, indeed, any convenient molecular concentration units. The distribution of H and D among isotopically mixed species is then generally given by a Poisson distribution, i.e. we have for the fractional abundance F of a species XDJ)Hni i, (Cadogan et al., 1955 Kresge, 1964) ... 

The terms concentration and molarity are synonymous terms. Concentration of a reactant or product, c, often represented with square brackets [ ], is measured by the amount of substance, n divided by the volume of the solution, mol dm 3 are the usual units used for molarity or concentration ... 

Do not use square brackets to indicate concentration with a spelled-out name. 

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