As described in the previous quanta, the rate of a reaction is affected by the concentrations of reactants. Rate laws or rate equations are mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentration of its reactants. Show Consider a general reaction like this: a A + b B → c C + d D for which the rate law is: rate = k[A]m[B]n in which [A] and [B] represent the molar concentrations of reactants (although it is also possible products and catalysts may be represented in the rate law as well), and k is the rate constant, which is specific for a particular reaction at a particular temperature. The exponents, m and n, are usually positive integers (although it is possible for them to be fractions or negative numbers). These exponents are not necessarily the coefficients from the reaction and should not be assumed to be so. We will cover this more in depth later. The rate constant, k, and the order exponents must be determined experimentally by observing how the rate of a reaction changes as the concentrations of the reactants are changed. The exponents in a rate law describe the effects of the reactant concentrations on the reaction rate and define the reaction order. If the exponent m is 1, the reaction is first order with respect to A. If m is 2, the reaction is second order with respect to A. If n is 1, the reaction is first order in B. If n is 2, the reaction is second order in B. If m or n is zero, the reaction is zero order in A or B, respectively, and the rate of the reaction is not affected by the concentration of that reactant. The overall reaction order is the sum of the orders with respect to each reactant. If m = 1 and n = 1, the overall order of the reaction is second order (m + n = 1 + 1 = 2). Example 1Writing Rate Laws from Reaction Orders NO2(g) + CO(g) → NO(g) + CO2(g) is second order in NO2 and zero order in CO at 100 °C. What is the rate law for the reaction? Solution rate = k[NO2]m[CO]n The reaction is second order in NO2; thus m = 2. The reaction is zero order in CO; thus n = 0. The rate law is: rate = k[NO2]2[CO]0 = rate = k[NO2]2 Remember that a number raised to the zero power is equal to 1, thus [CO]0 = 1, which is why we can drop the concentration of CO from the rate equation: the rate of reaction is solely dependent on the concentration of NO2. Check Your Learning H2(g) + 2 NO(g) → N2O(g) + H2O(g) has been determined to be rate = k[NO]2[H2]. What are the orders with respect to each reactant, and what is the overall order of the reaction? Answer: order in NO = 2; order in H2 = 1; overall order = 3 Check Your Learning CH3OH + CH3CH2OCOCH3 → CH3OCOCH3 + CH3CH2OH The rate law for the reaction between methanol and ethyl acetate is, under certain conditions, determined to be: rate = k[CH3OH] What is the order of reaction with respect to methanol and ethyl acetate, and what is the overall order of reaction? Answer: order in CH3OH = 1; order in CH3CH2OCOCH3 = 0; overall order = 1 Determining the Rate Law Using the Method of Initial RatesIt is sometimes helpful to use a more explicit algebraic method, often referred to as the method of initial rates, to determine the orders in rate laws. To use this method, we select two sets of rate data that differ in the concentration of only one reactant and set up a ratio of the two rates and the two rate laws. After canceling terms that are equal, we are left with an equation that contains only one unknown, the coefficient of the concentration that varies. We then solve this equation for the coefficient. Example 2Determining a Rate Law from Initial Rates NO(g) + O3(g) → NO2(g) + O2(g) This reaction has been studied in the laboratory, and the following rate data were determined at 25 °C. Trial[NO] (M)[O3] (M)Initial Rate (M/s)11.00 × 10−63.00 × 10−66.60 × 10−521.00 × 10−66.00 × 10−61.32 × 10−431.00 × 10−69.00 × 10−61.98 × 10−442.00 × 10−69.00 × 10−63.96 × 10−453.00 × 10−69.00 × 10−65.94 × 10−4Determine the rate law and the rate constant for the reaction at 25 °C. Solution rate = k[NO]m[O3]n We can determine the values of m, n, and k from the experimental data using the following three-part process:
The large value of k tells us that this is a fast reaction that could play an important role in ozone depletion if [NO] is large enough. Check Your Learning CH3CHO(g) → CH4(g) + CO(g) Determine the rate law and the rate constant for the reaction from the following experimental data: Trial[CH3CHO] (M)Initial Rate (M/s)11.75 × 10−32.06 × 10−1123.50 × 10−38.24 × 10−1137.00 × 10−33.30 × 10−10Answer: rate = k[CH3CHO]2 with k = 6.73 × 10−6 M-1s-1 Example 3Determining Rate Laws from Initial Rates 2 NO(g) + Cl2(g) → 2 NOCl(g) Trial[NO] (M)[Cl2] (M)Initial Rate (M/s)10.100.100.0030020.100.150.0045030.150.100.00675Solution rate = k[NO]m[Cl2]n As in Example 2, we can approach this problem in a stepwise fashion, determining the values of m and n from the experimental data and then using these values to determine the value of k. In this example, however, this time we will use a different approach to determine the values of m and n :
Check Your Learning OCl–(aq) + I–(aq) → OI–(aq) + Cl–(aq) Trial[OCl−] (M)[I−] (M)Initial Rate (M/s)10.00400.00200.0018420.00200.00400.0009230.00200.00200.00046Determine the rate law expression and the value of the rate constant k with appropriate units for this reaction. Answer: rate = k[OCl–]2[I–] k = 5.75 x 104 M-2 s-1 Reaction Order and Rate Constant UnitsReaction orders also play a role in determining the units for the rate constant k. In Example 2, a second-order reaction, we found the units for k to be M-1s-1, whereas in Example 3, a third order reaction, we found the units for k to be M−2s-1. The following summarizes the rate constant units for common reaction orders: Reaction OrderUnits of kzeroM s-1firsts−1secondM-1 s-1thirdM-2 s−1m+nM1-(m+n) s-1Note that the units in the table can also be expressed in terms of mol/L instead of molarity (M). Also, units of atoms, molecules, atm, or units of time other than the second (such as minutes, hours, days) may be used, depending on the situation. Key Concepts and SummaryAn extremely useful way to determine the rate of reaction is by using a rate law, which contains a rate constant and the concentration of all reactants to a reaction order. The reaction order must be determined experimentally, with one way being the method of initial rates. The method of initial rates looks at how the initial rate of a reaction changes when the initial concentration of the reactants are also changed, one reactant at a time. Once the orders of the reaction are known, the rate constant can also be calculated. Key EquationsGlossarymethod of initial ratesan algebraic method that can be used to determine the rate law of a reactionrate constantdefined as k, which is specific for a particular reaction at a particular temperaturerate law (or rate equation)a mathematical expression that describes the relationship between the rate of a chemical reaction and the concentration of its reactantsreaction orderthe exponents in a rate law that describe the effect of a reactant concentration on the reaction rate. Adding together all reaction orders gives the overall reaction order.
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