# 4.5: Differential Rate Law Resources

- Page ID
- 362218

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Experimental Design Considerations

Consideration \(\PageIndex{1a}\): Rate Dependence on Fe^{+3}

Describe the overall design of how you could design an experiment that would allow you to measure the affect of ferric ion concentration on the rate? Explain what would be varied and what would be held constant, do not explain how you would actually make the measurements. Describe the function relating the rate dependence on ferrous ion concentration under these conditions.

**Answer**-
You would vary [Fe

^{+3}] at constant T and [I^{-}] and measure the initial rate at the different concentrations. Since T and [I^{-}] are constant the complete rate law becomes\[R=k'[Fe^{+3}]^m \\ \text{where} \\ k'=Ae^{-\frac{E_A}{RT}}[I^-]^n\nonumber \]

Consideration \(\PageIndex{1b}\)

For the above consideration, what would you plot to get a straight line relationship, and how would you determine the order of reaction with respect to the iron concentration

**Answer**-
Since the above equation is a power function would would make a log/log plot and the slope of the line would be the order of reaction with respect to ferrous ion concentration. The equation would be

\[logR=mlog[Fe^{+3}] + logk' \nonumber \]

Consideration \(\PageIndex{2a}\): Rate Dependence on I^{-}

Describe the overall design of how you could design an experiment that would allow you to measure the affect of iodide ion concentration on the rate? Explain what would be varied and what would be held constant, do not explain how you would actually make the measurements. Write the equation and show what is held constant

**Answer**-
You would vary [I

^{-}] at constant T and [Fe^{+3}] and measure the initial rate at the different concentrations. Since T and [Fe^{+}] are constant the complete rate law becomes\[R=k''[I^-]^n \\ \text{where} \\ k''=Ae^{-\frac{E_A}{RT}}[Fe^{+3}]^m\nonumber \]

Consideration \(\PageIndex{2b}\)

For the above consideration, what would you plot to get a straight line relationship, and how would you determine the order of reaction with respect to the iodide concentration

**Answer**-
Since the above equation is a power function would would make a log/log plot and the slope of the line would be the order of reaction with respect to ferrous ion concentration. The equation would be

\[logR=mlog[I^{-}] + logk'' \nonumber \]

Consideration \(\PageIndex{3a}\): Rate Dependence on T

Describe the overall design of how you could design an experiment that would allow you to measure the affect of temperature (T) on the rate? Explain what would be varied and what would be held constant, do not explain how you would actually make the measurements. Describe what the function relating the rate dependence on temperature under these conditions.

**Answer**-
You would vary the temperature while keeping [Fe

^{+3}] and [I^{-}] constant. The rate law becomes:\[R=k'''e^{-\frac{E_A}{RT}} where K''' = A[Fe^{+3}]^m [I^-]^n \nonumber \]

Consideration \(\PageIndex{3b}\)

For the above consideration, what would you plot to get a straight line relationship, and how would you determine the Energy of Activation (E_{a})?

**Answer**-
The above equation is an exponential function you would think we would plot the natural log of R as a function of the reciprocal temperature, and we could

\[lnR=-\frac{E_a}{RT}+ lnk''' \nonumber \]

but that is not what we normally do. Instead, we solve the value of k using the known values of [Fe

^{+3}] and [I^{-}] ,\[R = k [Fe^{+3}]^m [I^-]^n \\ to \\ \frac{R}{[Fe^{+3}]^m[I^-]^n} = \nonumber\]

noting

\[k=e^-\frac{E_a}{RT} \nonumber\]

and then using the Arrhenius equation we plot

\[lnk=\frac{-E_a}{R}\left ( \frac{1}{T} \right ) + lnA \nonumber \]

The slope of the line is -\(\frac{E_a}{R}\), where R=8.314\(\frac{J}{mol \cdot K}\)

Consideration \(\PageIndex{1}\)

Why can't you just use the ratio method of the two state technique (section 14.3.3.2) for this lab report?

**Answer**-
There will be a lot error because "seeing" the solution turn blue is subjective. If your data is exact, you can choose any two points.

Consideration \(\PageIndex{1}\)

Discuss the challenges of measuring the temperature for the hot and cold runs (experiments 6 & 7)

**Answer**-
The room is the final heat sink and the building's heating and ventilation system is maintaining a constant temperature, so the hot run will cool to room temperature and the cold will warm. The problem is the rate of heat flow (and thus the rate in change in temperature ) is an exponential function of the temperature difference between the system and the room. If the room is 25

^{o}C a hot object at 99^{o}C will cool to 98^{o}C much faster than an object at 27^{o}C will cool to 26^{o}C. If the solution cools while you are waiting for it to turn blue you will not have an accurate measure of the temperature, as the temperature changed while you were making your measurement.

Consideration \(\PageIndex{1}\)

Why do we not want to run the experiment at real hot temperatures.

**Answer**-
There are essentially three reasons.

- The hotter it is the easier it is to get burned, and so it is safer.
- At high temperatures it cools real fast and so the temperature will change quickly and it will be harder to run the experiment at constant temperature.
- The reaction speeds up and could be so fast that you may have difficulties measuring it. (If it changes in a second or two, the relative error will be very great)

There is an additional reason, and that is that at high temperatures you may also start seeing competing reactions occurring.

Consideration \(\PageIndex{1}\)

What could we do to prevent the temperature from changing during the experiment?

**Answer**-
We could use a calorimeter to hold the solutions so that no heat flows between the solution and the room.

Consideration \(\PageIndex{1}\)

You may come up with a non-integer order of reaction and we want you to round off to an integer. Please look at the reactions in equations 4.2.3 and 4.2.4. In the first, iodide reduces ferric iron to ferrous iron. In the second reaction thisulfate reduces iodine back to iodide. Can you come up with a competing reaction for the consumption of ferric iron that may interfere with our estimate of its order of reaction.

**Answer**-
Both iodide and thiosulfate are reducing agents, and so the thiosulfate can also reduce ferric iron to ferrous iron

\[2Fe^{+3}(aq) + 2S_2O_3^{-2}(aq) \rightarrow 2Fe^{+2}(aq) + S_4O_6^{-2}(aq) \nonumber \]

This is a competing reaction, fortunately it is slower than the reaction we are investigating, but when you have competing reactions that you are not taking into account, you may end up with non-integer orders of reaction.