16.1-16.2 - Rates And Rate Equations

16.1-16.2 - Rates And Rate Equations

Read rate equations as experimental evidence. Deduce orders from controlled comparisons, calculate rates and rate constants, and distinguish kinetic quantities from stoichiometry and yield.

Rate Is A Measured Change

A reaction rate is the change in the amount or concentration of a reactant or product per unit time. If a reactant concentration falls from 0.500moldm−30.500 \mathrm{mol} \mathrm{dm}^{-3} to 0.420moldm−30.420 \mathrm{mol} \mathrm{dm}^{-3} in 40s40 \mathrm{s}, the mean rate of disappearance of that reactant is:

mean rate=0.500−0.42040=0.080 mol dm−340 s=0.00200 mol dm−3 s−1\text{mean rate} = \frac{0.500 - 0.420}{40} = \frac{0.080\,\mathrm{mol\,dm^{-3}}}{40\,\mathrm{s}} = 0.00200\,\mathrm{mol\,dm^{-3}\,s^{-1}}

The final amount of product also depends on the starting amount of limiting reactant and the reaction yield. A greater final amount does not establish a greater rate unless the elapsed time and other relevant conditions are considered.

That unit is not decoration. It says that the concentration is changing by 0.00200moldm−30.00200 \mathrm{mol} \mathrm{dm}^{-3} each second over that interval. In other practical contexts the measured quantity could be gas volume, mass loss or colour intensity, but a rate is always a change divided by time.

TermMeaning in this lessonTrap to avoid
rate of reactionHow fast reactants are used or products form, measured as change per unit time.Do not confuse final amount made with speed.
rate equationAn experimentally determined equation linking rate to concentrations.Do not copy powers from the balanced equation unless data supports them.
half-lifeThe time taken for the amount or concentration of a reactant to halve.Calculations and graph use come later; here it is a defined kinetic term.
activation energy, EaE_{\mathrm a}The minimum energy particles need for a successful collision.It is not the energy released by a reaction.
catalystA substance that increases rate by providing an alternative route with lower activation energy and is regenerated.A catalyst changes rate, not the stoichiometric equation.

Orders Describe Concentration Dependence

The rate equation has the form:

Rate equation

rate=k[A]m[B]n\text{rate} = k[\mathrm{A}]^{m}[\mathrm{B}]^{n}

[A][\mathrm{A}], [B][\mathrm{B}] = concentrations in mol dm−3\mathrm{mol\,dm^{-3}}; mm, nn = orders (0, 1 or 2 here), found from experiment; overall order =m+n= m + n; kk = rate constant, whose units depend on the overall order.

Order with respect to a substanceMeaning when only that concentration changesExample in a rate equation
zero orderChanging that concentration has no effect on rate.rate=k[A]0\mathrm{rate} = k[\mathrm{A}]^{0}, so that factor is effectively 11.
first orderDoubling that concentration doubles the rate.rate=k[A]\mathrm{rate} = k[\mathrm{A}].
second orderDoubling that concentration increases the rate by a factor of 44.rate=k[A]2\mathrm{rate} = k[\mathrm{A}]^{2}.

The overall order is the sum of the powers in the rate equation:

rate=k[A]2[B]\text{rate} = k[\mathrm{A}]^{2}[\mathrm{B}]: overall order =2+1=3= 2 + 1 = 3

The rate constant, kk, is the proportionality constant that makes the equation work at a fixed temperature for a particular reaction and route. It does not change when concentrations change, but it does change if the temperature changes; a catalyst can also give a different value of kk because it provides a different lower-activation-energy route.

The coefficient trap

The equation A+2B→\mathrm{A} + 2\mathrm{B} \rightarrow products does not automatically mean rate=k[A][B]2\text{rate} = k[\mathrm{A}][\mathrm{B}]^{2}: orders are found from experimental rate data; balanced-equation coefficients show the overall stoichiometric ratio, not necessarily which particles are involved in the rate-controlling event.

Determining A Rate Equation From Comparisons

To determine a rate equation from simple rate data, compare two experiments where only one concentration changes. The rate change tells you the order with respect to that substance.

Use this decision rule:

Concentration changeRate changeOrder
×2\times 2no changezero order
×2\times 2×2\times 2first order
×2\times 2×4\times 4second order
×3\times 3×9\times 9second order

Worked example: determine the rate equation

For the reaction between A and B\ce{B}, the following initial-rate data is obtained at constant temperature.

Experiment[A] / mol dm−3[\mathrm{A}]\,/\,\mathrm{mol\,dm^{-3}}[B] / mol dm−3[\mathrm{B}]\,/\,\mathrm{mol\,dm^{-3}}Initial rate / moldm−3s−1\mathrm{mol} \mathrm{dm}^{-3} \mathrm{s}^{-1}
10.1000.1002.40×10−42.40\times 10^{-4}
20.2000.1009.60×10−49.60\times 10^{-4}
30.1000.2002.40×10−42.40\times 10^{-4}

Compare experiments 1 and 2. [B]\ce{[B]} is constant, [A][\mathrm{A}] doubles, and the rate increases by a factor of:

9.60×10−42.40×10−4=4\frac{9.60\times 10^{-4}}{2.40\times 10^{-4}} = 4

Doubling [A][\mathrm{A}] makes the rate four times greater, so the reaction is second order with respect to A.

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