Finding Reaction Rate Equation

Reaction rate is important lesson in physical chemistry and it describes how reaction rate of reaction changes with reaction conditions such as concentration temperature. Also reaction rate plays a major role in chemical industries because production rate is a one of the critical thing in their economical state.

Learn Ammonia Manufacturing Process - Haber Process to understand how reaction rate effects to the production rate.

You may have seen, some reactions slowly produce products. Some are so quick and give products in rapid speed (sodium and water reaction).

Reaction Rate Definition

Rate of change of concentration of reactant or product is defined as reaction rate.

1. In a reaction, reactants decreases. So reaction rate of reactant has a negative value.
2. In a reaction products increases. So reaction rate of product has a positive value.

Reaction rate units

According to the definition, change of concentration is measured for a unit time in unit volume. So units can be written as below in different occasions or according to the given problem.

• mol dm^-3 s^-1
• kmol m^-3 hr^-1
• kmol m^-3 min^-1

Now we are going to understand more about reaction rates and their relationship between reactants.

Example 1

A + B → C

One A molecule reacts with one B molecule and produce one C molecule. So change of amount is same but difference is A and B decreases and C increases. When reaction occurs, at small time change (dt) , difference of A is dn_A, difference of B is dn_B and difference of C is dn_C. So then,

-dn_A = -dn_B = dn_C

If A,B and C contains in a same volume V.

-dn_A / V = -dn_B / V = -dn_C / V

-dc_A = -dc_B = dc_C

Now change of concentrations are denoted by above equations.

Next, we can write change of concentrations per dt, small time range.

-dc_A / dt = -dc_B / dt = dc_C / dt

Now you can see, to write the reaction rate, a change of concentration of any reactant or product can be used.

Example 2

P + 2Q → 3S

One P molecule reacts with two Q molecule and produce three S molecule. When reaction occurs, at small time change (dt) , difference of P is dn_P, difference of Q is dn_Q and difference of S is dn_S. So then,

2dn_P = dn_Q

-dn_P = -dn_Q / 2 = dn_S / 3

If P,Q, and S contains in a same volume V.

-dn_A / V = -dn_B / V = -dn_C / V

If p,Q ans S contain in same volume V, change of amount in dt time period, we can write reaction rates rates for reactants and products as below.

dP, dQ and dS denotes concentration changes of P,Q and S respectively at dt time period.

Next, we can write change of concentrations per dt, small time range.

2dc_P / dt = dc_Q / dt

-3dc_P / dt = dc_S / dt

-dc_P / dt = -(1/2)dc_Q / dt = (1/3)dc_S /dt

Now you can see, to write the reaction rate, a change of concentration of any reactant or product can be used.

It's clear now that reaction rates of reactants and products are different due to different stoichiometric coefficients. Therefore we cannot express one reaction rate for the reaction from this way. Thus we introduce a general expression for reaction rate.

aA + bB → cC +dD

Reaction rate equation

Lets take an example reaction to explain this.

2NO + O₂ → 2NO₂

Stoichiometry of balanced equation is important in reaction rate equation. Two NO molecules react with one O₂ molecule and produce two NO₂ molecules. So we can say, reaction rate of NO is twice as O₂.

Also NO₂ forming rate is twice as reaction rate of O₂.

If we take reaction rate of O₂ as -R and then write reaction rate for other constituents.

• Reaction rate of NO = -2R
• Reaction rate (forming) of NO₂ = +2R

Express reaction rate equation using concentration

aA + bB → cC + dD

We can write a general equation for reaction rate like this.

Reaction rate ∝ C_A^m C_Bⁿ

Reaction rate = K C_A^m C_Bⁿ

K is a proportional constant and it is a function of temperature. C_A and C_B are concentrations of A and B respectively. m and n are defined as order of A and order of B.

Find reaction rate constant - Arrhenius equation

Reaction rate constant(K) is given by Arrhenius equation.

K(T) = A.e^(-E/RT)

• A - frequency factor
• E - Activation Energy(J/mol)
• R - Gas constant = 8.314 J/mol.K = 1.987 cal/mol
• T - absolute temperature(K)

We want to make a relation between absolute temperature (T) and reaction rate constant(K).

Take ln for both sides.

ln( K(T) ) = ln( Ae^(-E/RT) )

ln( K(T) ) = ln(A) + ( -E/RT )

Plotting reaction rate constant against 1/(absolute temperature)

Units of reaction rate constant

Units of reaction rate constant depends on the units of function of concentrations.

Reaction types

There are two types of reactions according to the mechanism.

1. Elementary reactions
2. Non-elementary reactions

Elementary reaction

A one step reaction which give products directly. No intermediate states or products. Elementary reaction are rare in the world. Reaction of methyl bromide and sodium ethoxide is an example to elementary reaction.

CH₃I + CH₃CH₂O^- → CH₃OCH₂CH₃ + I^-

Non-elementary reactions

When a reaction occurs through multiple reactions to give products, we call that reaction is a non-elementary reaction.

Write reaction rate equation for elementary reactions

In this tutorials, we are going to discuss only about elementary reactions. Reaction rate for elementary reaction is written by as below.

For elementary reaction

Reaction rate is proportional to the product of the molar concentrations of reactants with each concentration term raises to the power equals to the stoichiometry number of respective reactants

Lets take our reaction to write the reaction by assuming it is a elementary reaction.

aA + bB → cC + dD

Reaction rate = K C_A^a C_B^b

Here m and n are replaced by a and b. a and b represent Stoichiometric coefficients of A and B reactants respectively.

Write reaction rate equation for the following elementary reaction

CH₃I + CH₃CH₂O^- → CH₃OCH₂CH₃ + I^-

Reaction rate = K [ CH₃I ] [ CH₃CH₂O^- ]

[ ] are used to denote concentrations.

Write reaction rate equation for following irreversible elementary reactions

Question 1

A + 2B → C

In this reaction one A molecule is reacted with two B molecules and produce one C molecule. Therefore concentration of both A and B is reduced (negative reaction rate) and C is increased (positive reaction rate).

• rate = -K [A] [B]²

In previous lesson, we have obtained molar balance equations for different reactors.

All of those equations contains r_A (reaction rate). Two solve obtained equations, we have to find r_A as a function of concentrations.

Write reaction rate equation for reversible reactions

Write reaction rate of A of following reversible reaction.
A + 2B ⇌ C
In this reversible reaction, one A molecule reacts with two B molecules and produce one C molecule (forward reaction). Also one C molecule produce one A molecule and two B molecules (backward reaction).
When we write reaction rate of A for forward reaction, A is reduced (disappearing). Therefore rection rate of A of forward reaction is negative. But A is formed by backward reaction from C. Therefore rection rate of A of backward reaction is positive.

r_A|_forward = -K₁ * C_A * C_B²
r_A|_backward = K₂ * C_C
r_A|_net = -K₁ * C_A * C_B²   +   K₂ * C_C



when we write reaction rate equation, we should be careful about stoichiometry number of reactants. In the above reaction we can one hydrogen molecule and one iodine molecule react and produce two hydrogen iodide molecules.

Forward reaction: Reaction rate of producing HI is twice as reaction rate of consuming H₂
Backward reaction: Reaction rate of producing H₂ is half as reaction rate of consuming HI

Why reaction rate equation is important in chemical reaction engineering?

In chemical engineering, new reactors are designed to do reactions. So how do we know, how much time is required to complete the reaction. If we don't know the reaction completing time, how do we know the, at what time we should enter new raw materials to the reactor. So we have so many more questions when we talk about reactors. To find answers them, we use reaction rate equation.

Questions

Is reaction rate constant throughout the reaction?

When we study how reaction rate depends, it is very obivious that reaction rate is not constant during the reaction. When concnetration of reactants decreases, reaction rate decreases.

If temperature increases, reaction rate can be increased. (if concentrations of reactants do not decrease much).