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Química Geral · Chemical kinetics

Leis de velocidade integradas e meias-vidas

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O que você vai aprender

  • Use the integrated zero-, first- and second-order rate laws to find a concentration after a given time, the time to reach a given concentration, or the starting concentration.
  • Calculate half-lives for each order and use the way successive half-lives change to identify the order.
  • Determine the order and rate constant from which plot of [A], ln[A] or 1/[A] against time is a straight line.
  • Treat radioactive decay as a first-order process to find the amount or activity left, the elapsed time, or the decay constant.

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Three integrated rate laws

A rate law such as rate = k[A]ⁿ tells you how fast A is used up at one moment. Integrating it over time gives an equation that links the concentration [A]ₜ at time t to the starting concentration [A]₀. In this topic the rate is defined as −d[A]/dt, the rate of loss of A itself. Zero order (rate = k): [A]ₜ = [A]₀ − kt, with k in M/s. The concentration falls by the same amount each second until A runs out, at t = [A]₀/k. First order (rate = k[A]): ln[A]ₜ = ln[A]₀ − kt, or [A]ₜ = [A]₀e^(−kt), with k in s⁻¹. The same fraction of A disappears in each equal interval of time. The ring opening of cyclopropane to propene (drawn) is a classic first-order gas reaction. Second order (rate = k[A]²): 1/[A]ₜ = 1/[A]₀ + kt, with k in M⁻¹ s⁻¹. The rate falls quickly as A is used up, so the reaction tails off. If a rate is instead defined per mole of reaction, as rate = −(1/a)d[A]/dt for a reaction a A → products, replace k by ak in each equation.

Concentration after a time and time to reach a concentration

Choose the equation for the order, put every quantity in consistent units (k in s⁻¹ needs t in seconds), and solve for the unknown. First order: k = 4.0 × 10⁻³ s⁻¹, [A]₀ = 0.200 M, t = 250 s. Then kt = 1.00 and [A] = 0.200 × e^(−1.00) = 0.0736 M. Second order: k = 0.10 M⁻¹ s⁻¹, [A]₀ = 0.50 M, t = 20 s. Then 1/[A] = 2.0 + 2.0 = 4.0 M⁻¹ and [A] = 0.25 M. To find a time, rearrange: first order t = ln([A]₀/[A]ₜ)/k, second order t = (1/[A]ₜ − 1/[A]₀)/k, zero order t = ([A]₀ − [A]ₜ)/k. To find a starting concentration, run the equation backwards: first order [A]₀ = [A]ₜe^(kt). Use natural logarithms (ln), not log₁₀. A first-order fraction can be used directly, because only the ratio [A]ₜ/[A]₀ appears: 35% consumed means [A]ₜ/[A]₀ = 0.65.

Half-lives

The half-life t½ is the time for [A] to fall to half its current value. Set [A]ₜ = [A]₀/2 in each equation. First order: t½ = ln 2/k = 0.693/k, independent of concentration. The rate is proportional to [A], so the fraction lost per unit time never changes. Every half-life removes half of what is left, and after n half-lives the fraction remaining is (½)ⁿ. This also works when n is not a whole number: 1.5 half-lives leave (½)^1.5 = 0.354, not 0.375. Zero order: t½ = [A]₀/(2k), so each successive half-life is half as long as the one before. Second order: t½ = 1/(k[A]₀), so each successive half-life is twice as long as the one before. A table of successive half-lives therefore identifies the order: shrinking means zero order, constant means first order, doubling means second order.

Finding the order from a straight-line plot

Each integrated law has the form y = mx + b with time on the x-axis. Zero order: [A] against t is linear with slope −k. First order: ln[A] against t is linear with slope −k. Second order: 1/[A] against t is linear with slope +k and intercept 1/[A]₀. Make all three plots, and the one that is straight gives the order. k comes from the slope, and k is always positive. The decomposition of dinitrogen pentoxide (drawn) gives a straight ln[A] plot, so it is first order. Follow the reaction for at least two or three half-lives. Over the first few percent of reaction all three plots look straight, because the concentration has barely changed. A reaction with two reactants, rate = k[A][B], can be run with B in large excess. [B] then stays almost constant and the reaction is pseudo-first-order in A. The ln[A] plot gives k′ = k[B]₀, so k = k′/[B]₀. Some surface reactions are zero order. Ammonia decomposing on hot tungsten is zero order at high pressure because the surface sites are saturated, so adding more gas does not raise the rate.

Radioactive decay is first order

Each unstable nucleus has a fixed chance of decaying per unit time, so the decay rate is proportional to the number of nuclei N. The same equations apply: N = N₀e^(−kt), ln(N₀/N) = kt and t½ = 0.693/k, where k is the decay constant. Activity, the number of decays per second, equals kN, so activity, mass and number of nuclei all fall by the same factor. Nuclear half-lives are practically unaffected by temperature, pressure or chemical form, because no thermally activated step is involved. Example: iodine-131 has t½ = 8.02 days. After 20.0 days, 20.0/8.02 = 2.494 half-lives have passed, so (½)^2.494 = 0.178, or 17.8%, remains. The time for the activity to fall to 1.00% of its starting value is t = t½ × ln(100)/ln 2 = 8.02 d × 6.644 = 53.3 days.

What to carry forward

Match the order to its integrated law: [A]ₜ = [A]₀ − kt (zero), ln[A]ₜ = ln[A]₀ − kt (first), 1/[A]ₜ = 1/[A]₀ + kt (second). Keep k and t in the same time unit. First-order half-lives are constant (t½ = 0.693/k), zero-order half-lives shrink and second-order half-lives double. The straight-line plot of [A], ln[A] or 1/[A] against time reveals the order, and its slope gives k. Radioactive decay is first order, so (½)ⁿ gives the fraction left after n half-lives, whole or not.

Problemas de prática (40)

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Half-lives

  • Predict a half-life at a new concentrationDeduce the order from how the half-life depends on starting concentration and predict the half-life at another concentration.

Integrated rate laws

  • A zero-order reaction after 40 minutesRecognize that a zero-order reactant is used up at t = [A]₀/k, after which the integrated law no longer applies.

Classificar 4

Integrated rate laws

  • Which equation is the first-order integrated rate law?Recognize the integrated rate law for a first-order reaction among the zero-, first- and second-order forms.

Half-lives

  • Which half-life does not depend on concentration?Identify the reaction order whose half-life is independent of the starting concentration.

Order from plots and data

  • A straight line for ln[A] against timeAssign the reaction order when a plot of ln[A] against time is a straight line.

Radioactive and first-order decay

  • The kinetics of radioactive decayClassify radioactive decay as a first-order process and state what that implies for its half-life.

Comparar 3

Half-lives

  • Rank half-lives across ordersCompare the first half-lives of zero-, first- and second-order reactions using the expression for each order.

Radioactive and first-order decay

  • Which isotope lasts longest over a month?Rank radioisotopes by the fraction remaining after a fixed time from their half-lives.

Integrated rate laws

  • Same start, different ordersCompare how much reactant remains for zero-, first- and second-order reactions that start at the same concentration and rate.

Calcular 18

Integrated rate laws

  • Butadiene left after one hourUse the second-order integrated rate law, with time converted to the units of k, to find a concentration.
  • Cyclopropane remaining after 1500 sUse the first-order integrated rate law to find a concentration after a given time.
  • Time for HI to fall to 0.100 MRearrange the second-order integrated rate law to find the time needed to reach a given concentration.
  • Second-order decay with a stoichiometric coefficientInclude the stoichiometric coefficient when a second-order rate constant is defined per mole of reaction.
  • Ammonia left on a hot tungsten surfaceUse the zero-order integrated rate law to find a concentration after a given time.

Radioactive and first-order decay

  • Cobalt-60 left in a sourceCalculate the mass of a radioisotope remaining after a time that is not a whole number of half-lives.
  • Waiting for fluorine-18 to fadeCalculate the time for a radioisotope's activity to fall to a given percent of its starting value.
  • Decay constant of phosphorus-32Calculate a radioactive decay constant from the half-life.
  • Technetium-99m activity after a dayFind the activity of a radioactive sample after a whole number of half-lives.
  • Ratio of two isotopes after 40 daysCompare the first-order decay of two radioisotopes that start with equal numbers of nuclei.

Half-lives

  • Percent left after 60 minutesFind the percent of reactant remaining after a whole number of first-order half-lives.
  • Half-life from percent decomposedFind a first-order half-life from the fraction of reactant consumed in a measured time.
  • Time to reach a quarter in a second-order reactionUse the doubling second-order half-life to find the time for a reactant to fall to a quarter of its starting value.
  • Half-life of sulfuryl chlorideCalculate a first-order half-life from the rate constant and express it in hours.
  • Two half-lives of a zero-order reactionUse the shrinking zero-order half-life to find the time for a reactant to fall to a quarter of its starting value.

Order from plots and data

  • Half-life from the slope of an ln[A] plotConvert the slope of a linear ln[A]-against-time plot into a rate constant and a half-life.
  • Second-order k from a pseudo-first-order runExtract a second-order rate constant from a pseudo-first-order ln[A] plot run with one reactant in large excess.
  • Rate constant for N₂O₅ from two pointsCalculate a first-order rate constant from two points on a linear ln[A]-against-time plot.

Encontre o erro 3

Radioactive and first-order decay

  • Find the error with a fractional half-lifeDiagnose the error of treating radioactive decay as a fixed amount lost per half-life when the time is not a whole number of half-lives.

Order from plots and data

  • Find the error in a half-life from a 1/[A] plotRecognize that the half-life of a reaction with a linear 1/[A] plot depends on the starting concentration.

Integrated rate laws

  • Find the error in a first-order calculationIdentify a mismatch between the time unit of k and the time used in an integrated rate law.

Interprete as evidências 4

Order from plots and data

  • Order and k from a concentration tableDetermine the order and rate constant by testing which transformed concentration changes linearly with time.
  • The linear plot for a second-order reactionName the linear plot for a second-order reaction and relate its slope to k.
  • Reading time from a 1/[A] plotUse the slope and intercept of a linear 1/[A] plot to find the time at which a given concentration is reached.

Half-lives

  • What order do these half-lives show?Identify the reaction order from the way successive half-lives change in a concentration–time table.

Trabalhe de trás para frente 2

Integrated rate laws

  • Starting concentration of hydrogen peroxideWork backwards with the first-order integrated rate law to find a starting concentration.

Radioactive and first-order decay

  • How long ago was the radon sealed?Work backwards from the fraction of a radioisotope remaining to the time elapsed.

Planeje uma síntese 1

  • Plan an experiment to tell first from second orderChoose a concentration–time experiment that distinguishes first-order from second-order kinetics.

Explique por quê 3

Order from plots and data

  • Why ammonia on tungsten gives a straight [A] plotExplain zero-order kinetics in a surface-catalysed decomposition by saturation of the catalyst surface.

Radioactive and first-order decay

  • Why heating does not speed up radioactive decayExplain why a nuclear decay constant, unlike a chemical first-order rate constant, does not depend on temperature.

Half-lives

  • Why the first-order half-life never changesExplain the constant first-order half-life from the proportionality between rate and concentration.

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