General CHemistry
Problems: Zero, First, and Second Order Reactions
The following set of problems will help you interogate the properities of reactions that are zero, first, and second order in a particular reactant. You can answer these questions and then use what you have derived to add to a study guide, which you can find at the bottom of the page.
In lab, you study a reaction in which a green dye is depleted when reacts with a solid metal degradation catalyst. When you plot the concentration of the dye versus time, you obtain the following graph:
(a) At any given time, the concentration of the green dye will be changing, and yet your plot shows linear behavior at all points in time. What does this suggest about the reaction rate’s dependence on the concentration of the dye?
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(b) What does that in turn suggest about the order of reaction in green dye?
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(c) From the plot, estimate the half-life of the reaction. Does the length of the half-life depend on initial concentration of dye? Taking the slope of the plot to be -k, can you derive an equation for the half-life of a reaction of this order?
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(d) Write an expression for the concentration of green dye at any given time, t. This expression, when generalized, is called the integrated rate expression. How could you generalize this for all reactions of this order, using terms like [A] for concentration of a reactant, and [A]0 for the initial concentration of that reactant?
(a) Fill in the chart below about the rates you’d expect, given the concentration of H+ shown:
(b) Write the rate law for the reaction.
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(c) Using the rate law and the data in part (a), solve for the rate constant, . What are the units of for a second-order reaction like this one? To answer this generally, consider mM to simply be a concentration unit, which could have also been reported as M.
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(d) Plotting concentration versus time, you obtain the following plot, shown below. To verify the rate constant you estimated in (c), you must linearize the data. To do so, plot the five selected points indicated by black circles, which are also the points in the data from (a), versus the reciprocal concentration of H+ on the graph show. Estimate the slope, which is k. What are the units of k for this second-order reaction? Write the equation for your new linear plot in terms of [A], [A]0, t, etc.
(e) These black circles are plotted at every interval where a halving of the initial concentration of the interval occurs. From these data and the equation you wrote for the line, can you derive an equation for the half-life interval using the information given in the graph?
In lab, you are monitoring the absorbance (which is proportional to concentration) of a reaction with a highly-colored reactant. You record the time every time the absorbance halves from the time prior, obtaining the following data shown below on the left.
Solve for the half-life (t½) of this reaction. If you were to write an expression for t½, would you need any variables, such as t, [A], [A]0, etc. need to be added to this expression?
(b) What is the order of this reaction, judging from what you deduce about the half-life?
Give the rate constant for this reaction, with units, using what you know about half-life length for reactions of this order.
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(d) You make the following plot, accounting for to find the concentration of the reactant. Estimate the instantaneous rates using the plot to fill in the following table. What can be said generally about the relationship between concentration and rate for a reaction of this order?
(e) Write the rate law for this reaction.
Study Guide. Kinetic Properties by Order
Suggestions for process in filling out this chart:
The following chart is found in most General Chemistry textbooks, and collects all the information you should know about reactions that are zero, first, or second order in a given reactant, A. You should try to fill it out for yourself from your knowledge of kinetics as well as the information you generated from answering the above problems. Strive to fill out at least one property in each row and column before viewing the answers. There is one entry for each type of attribute to help you get started.
Alternate suggestions for filling out this chart:
Differential Rate Law: Start by writing the expression for the disappearance of reactant A in a reaction that is either zero, first, or second order in A with respect to time in the form , using the rate constant.
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Integrated Rate Law: Integrate over time, t, for both sides of the equation you wrote for the differential rate law. You will have to introduce a new constant, the initial concentration of A, typically given by [A]0.
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Concentration v. Time: Make a plot with Time on the x-axis and Concentration on the y-axis. Draw a line that represents the behavior of [A] according to the integrated rate law you wrote above.
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Straight-line plot to determine rate constant: Scientists like to perform mathematical operations on a data set to obtain a straight line that can be easily fit with a regression equation to obtain important data like a rate constant, . In the cases necessary, draw a new plot applying a mathematical operation to [A] on the y-axis so the data can be linearized.
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Relative rate vs. concentration: Fill out the chart for rates given changes in the initial concentration of A. Applying your differential rate law may help you.
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Half life: The half life is defined as the amount of time that it takes for one-half of A to be depleted by the reaction. Plugging in for [A] in the integrated rate law to represent this one-half depletion, solve for t. You may have to look up rules of exponents and logarithms if you don’t remember them.
Units of k, rate constant: Fill out the units for the rate constant in terms of concentration, M, and time, s. It may be helpful to look at your differential rate law, knowing that the rate is given in units of M/s and [A] is given in units of M.
Offshoot: Beer's Law Deep-Dive
Now what if we change the species inside the cuvettes, but held the concentration the same? Now order the cuvettes in order of increasing molar absorptivity, which is a constant characteristic of a given compound, can range many orders of magnitude, and can be thought of as characterizing how much light a compound absorbs at a particular wavelength and thus how saturated it is in solution.
Lab Application:
Scientists (including those in CHEM 136L) often apply Beer’s law by first making a concentration curve of around five standard solutions of a compound at known concentrations, measuring the absorbance of these solutions and then plotting them against concentration. Then, they will measure the absorbance of a solution with an unknown concentration of the analyte of choice, and use the concentration curve to find the unknown concentration. A visual schematic of this process is shown below:
Want to practice your hand at this process? Use the following data (absorption spectra of KMnO4, samples measured are the samples shown in the picture in the first question above) to determine the molar extinction coefficient of KMnO4 for the wavelength it absorbs maximally. The cuvette used in this experiment was a standard 1 cm path-length cell. Also indicate what this wavelength is (also known as , pronounced “lambda max”). The concentration curve chart is provided for you, so you can plot the data from the absorption curves, and then estimate a best linear fit and the slope of that line. Note as these are real data, they may not fit perfectly onto the line you sketch in.
Potassium permanganate (KMnO4) is a dark purple solid that makes beautiful pinky-purpley solutions shown in the first question in this section. A chemist wants to find the concentration of KMnO4 in a sample. She creates a standard curve of concentration versus absorbance for a series of solutions of known concentration of KMnO4 and then finds the molar absorptivity to be 2.33 x 104 M-1cm-1. She then measures the absorbance of her unknown sample to find an absorbance of 0.867. Her cuvette has a pathlength of 1 cm, as is standard. What is the concentration?
Medicine Application, Enrichment/Challenge Question:
A patient is suspected to be anemic (low hemoglobin/iron in blood). After her blood is drawn and sent to the lab, a technician makes a solution from a sample of her blood, K3Fe(CN)6 (called “potassium ferricyanide”, an oxidant) and KCN (potassium cyanide, a salt that will bind to oxidized hemoglobin to make it highly colored) and employs the Beer-Lambert law to find the patient’s iron levels. The technician knows that the molar absorptivity of the highly colored, oxidized hemoglobin-cyanide adduct (nicknamed “HiCN”) is 1.106 x 103 M-1cm-1 from their previous work in the hospital lab. When they measure the sample prepared from the patient’s blood using a cuvette with a 1 cm pathlength, they find the absorbance is 0.174. What is the concentration of hemoglobin in the patient’s blood (assume every molecule of hemoglobin is converted directly to HiCN)? The range of healthy hemoglobin levels for an adult female is considered to be 11.5-15.5 g/L. Is this patient anemic? (The formula weight of hemoglobin is 65,000 g/mol-- proteins are huge!)
Hint: Note that the majority of the information in this question sets the stage for the application, rather than being the heart of the question about Beer’s Law. This question is equivalent, with different numbers, to the basic application question above. You will also have to convert the concentration of HiCN from M to g/L of hemoglobin from the blood sample. Assume for every unit of hemoglobin in the blood sample, one unit of HiCN is produced.