# 11 Definitions of Reaction Rate and Extent of Reactions

Learning Objectives

By the end of this section, you should be able to:

**Define **Reaction rate and reaction extent

**Calculate** the reaction rate for the reaction, the rate of formation of compounds and the reaction extent

## Reaction Rate

A reaction rate shows the rates of production of a chemical species. It can also show the rate of consumption of a species; for example, a reactant. In general though, we want an overall common reaction rate to describe changes in a chemical system.

Let’s look at an example. Say we have reaction represented as: [latex]A + 2B → 3C + D[/latex]

For this system reaction rate can be expressed as follows:

[latex]r = \frac{d[D]}{dt}[/latex]

The reaction rate is represented by the letter “r” or the Greek letter upsilon “[latex]\upsilon[/latex]”

NOTE: I will stick with r as upsilon looks like “[latex]\nu[/latex]” which we will use to represent the stoichiometric coefficient

Above we have written the reaction rate as if a substance with** a coefficient of 1** was reacting (or being produced). This is the typical form of an overall reaction rate describing a reaction.

Rearranging the above equation, we can find the rate of production/consumption for any species based on this overall reaction rate, note that stoichiometric coefficients are positive for products and negative for reactants:

[latex]r_{A}=\frac{d[A]}{dt}=-\nu_{A}×r=(-1)×r[/latex]

[latex]r_{B}=\frac{d[B]}{dt}=-\nu_{B}×r=(-2)×r[/latex]

[latex]r_{C}=\frac{d[C]}{dt}=\nu_{C}×r=3×r[/latex]

[latex]r_{D}=\frac{d[D]}{dt}=\nu_{D}×r=1×r[/latex]

The general equation for calculating the reaction rate:

General notation: J is used to denote any compound involved in the reaction.

[latex]r = \frac{1}{\nu}\frac{d[J]}{dt}[/latex] |

Reaction rate at a given time can also be found from the graph of concentration of components in a system vs. time:

Exercise: Calculating the Reaction Rate

If we have the reaction

[latex]2 NOBr_{(g)} ⇌ 2 NO_{(g)} + Br_{2(g)}[/latex]

and we measure that the rate of formation of NO is 1.6 mmol/(L·s), what are the overall reaction rate, and the rate of formation of [latex]Br_{2}[/latex] and [latex]N\!O\!Br[/latex]?

### Solution

**Step 1:** Determine the overall reaction rate from the rate of formation for NO.

\begin{align*}

r & = \frac{1}{\nu_{j}} \frac{d[NO]}{dt} \\

& = \frac{1}{2} \left( 1.6 \frac{mmol}{L·s}\right)\\

& = 0.8\frac{mmol}{L·s}

\end{align*}

**Step 2:** Use the reaction rate to determine rate of formation for the other compounds

**NOTE: **rate of formation is positive for products and negative for reactants

\begin{align*}

\frac{d[Br_{2}]}{dt}& = r \\

& = 0.8\frac{mmol}{L·s}

\end{align*}

\begin{align*}

\frac{d[NOBr]}{dt}& = -2r \\

& = -1.6\frac{mmol}{L·s}

\end{align*}

Reaction rates can be given in a variety of units over time. In this class we will just explore **molarity** and **partial pressure**, although other forms exist.

**Molarity** – molar concentration – expressed in units of [latex]\frac{mol}{volume * time}[/latex] (eg. [latex]\frac{mol}{L*s}[/latex] )

**Partial pressure** – the pressure produced by one gaseous component if it occupies the whole system volume at the same temperature, commonly used for gasses – units of [latex]\frac{pressure}{time}[/latex]( eg. [latex]\frac{Pa}{s}[/latex] )

## Extent of Reaction

We use the extent of reaction ([latex]\xi[/latex]) to describe the change in an amount of a reacting speicies J.

[latex]d n_{j} = \nu_{j} d\xi[/latex] |

where:

- [latex]dn_{j}[/latex] = change in the number of moles of a certain substance
- [latex]\nu_{j}[/latex] = the stoichiometric coefficient
- [latex]d\xi[/latex] = the extent of reaction

We can get a relationship between the reaction extent and the rate of reaction when the system volume is constant:

[latex]r = \frac{1}{V} \frac{d\xi}{dt} = \frac{1}{\nu_{j}} \frac{1}{V} \frac{dn_{j}}{dt}[/latex] |

where:

[latex]V[/latex]= volume

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