D13.2 Reaction Rate
The rate of a chemical reaction can be experimentally determined by measuring the concentration of a reactant or a product at a series of measured time intervals after the reaction has started.
Often, it is easier to measure some property related to a substance’s concentration. For example, if a reaction involves a colored reactant, light absorption can be measured at different times after the start of the reaction. Then the reactant’s concentration at a given time can be calculated from the proportionality between light absorption and concentration.
Consider the decomposition reaction of cyclobutane in the gas phase:
The rate at which cylcobutane decomposes can be expressed in terms of the rate of change of its concentration:
![\begin{array}{rcl} \text{rate of decomposition of C}_4\text{H}_8 &= & - \dfrac{\text{change in concentration of reactant}}{\text{time interval}} \\[1em] &= & - \dfrac{[\text{C}_4\text{H}_8]_{t_2}\;-\;[\text{C}_4\text{H}_8]_{t_1}}{t_2\;-\;t_1} \\[1em] &= & - \dfrac{{\Delta}[\text{C}_4\text{H}_8]}{{\Delta}t} \end{array}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-8033a9f7f32f14866e33b6a98a1c768a_l3.png "Rendered by QuickLaTeX.com")
In this equation, Δ[C4H8] represents the change in concentration of cyclobutane during the time interval Δt. (Note that even though this reaction is in the gas phase, the rate involves the concentration of cyclobutane, which is simply the amount (mol) of C4H8 divided by the volume of the reaction vessel: [C4H8] = n/V. To convert to pressure, we can use the ideal gas law, pV = nRT, where p = (n/V)RT = cRT.)
The minus sign in front of the fraction is there because reaction rate is defined to be positive. The reactant concentration decreases as the reaction proceeds, making Δ[C4H8] a negative quantity, so a negative sign is needed to make the calculated rate positive.
The table below provides an example of data collected during the decomposition of C4H8. Notice that the reaction rate varies with time, decreasing as [C4H8] decreases.
| Time (s) | [C4H8] (M) | Δ[C4H8] (M) | Δt (s) | Rate of decomposition (M/s) |
| 0.0 | 0.240 | |||
| 20.0 | 0.120 | -0.120 | 20.0 | 0.00600 |
| 40.0 | 0.060 | -0.060 | 20.0 | 0.0030 |
| 60.0 | 0.030 | -0.030 | 20.0 | 0.0015 |
| 80.0 | 0.015 | -0.015 | 20.0 | 0.00075 |
An average rate over a given time period can be calculated using the concentrations at the beginning and the end of the period. For example, the average rate for the first and last 20-second period are:
![\text{rate}_{_{\text{0s to 20s}}} = \dfrac{-{\Delta}[\text{C}_4\text{H}_8]}{{\Delta}t} = \dfrac{-(0.120\;\text{M}\;-\;0.240\;\text{M})}{(20.0\;\text{s}\;-\;0.00\;\text{s})} = 0.00600\;\dfrac{\text{M}}{\text{s}}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-b33bfd9a1630a9111f5c559f3cdaecea_l3.png "Rendered by QuickLaTeX.com")
![\text{rate}_{_{\text{60s to 80s}}} = \dfrac{-{\Delta}[\text{C}_4\text{H}_8]}{{\Delta}t} = \dfrac{-(0.015\;\text{M}\;-\;0.030\;\text{M})}{(80.0\;\text{s}\;-\;60.0\;\text{s})} = 0.00075\;\dfrac{\text{M}}{\text{s}}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-963efdd813b35ffa0c7cd6448367f4a4_l3.png "Rendered by QuickLaTeX.com")
The rate of reaction at any specific time is known as the instantaneous rate. The instantaneous rate when the reaction starts (at t0), is the initial rate. The instantaneous rate of a reaction may be determined one of two ways:
- If concentration changes can be measured at very short time intervals, then average rates over these very short time intervals provide reasonably good approximations of instantaneous rates.
- If we plot reactant concentration vs. time, the instantaneous rate at any time t is given by the negative of the slope of a straight line that is tangent to the curve at that time.
Exercise: Reaction Rates from Concentration Graph
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