In a pseudo first order reaction in water, the following results were obtained: | $t$/s | 0 | 30 | 60 | 90 | |---|---|---|---|---| | $[A]$/$\text{mol L}^{-1}$ | 0.55 | 0.31 | 0.17 | 0.085 | Calculate the average rate of reaction between the time interval 30 to 60 seconds.

In a pseudo first order reaction in water, the fol – Class 12 Chemistry

Exercises · 3.8

In a pseudo first order reaction in water, the following results were obtained:

$t$ /s	0	30	60	90
$[A]$ / $\text{mol L}^{-1}$	0.55	0.31	0.17	0.085

Calculate the average rate of reaction between the time interval 30 to 60 seconds.

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Concept understanding — Average Rate Of Reaction

Reaction Rate Stoichiometry – From Intuition to Precision

Imagine you are watching a simple reaction:

$2 \text{NO}_2 \rightarrow 2 \text{NO} + \text{O}_2$

As NO₂ disappears, NO appears twice as fast as O₂ appears. Why? Because the balanced equation says: for every 2 molecules of NO₂ that break apart, you get 2 molecules of NO and 1 molecule of O₂. The numbers in front of the species — the stoichiometric coefficients — tell you the relative speeds at which reactants vanish and products appear.

That is the core idea: reaction rate stoichiometry is the relationship between the rates of change of different species in a chemical reaction, dictated by their coefficients in the balanced equation.

The Intuitive Picture

Think of a factory assembly line. The balanced equation is like a recipe:

2 units of raw material A → 2 units of product B + 1 unit of byproduct C

If the line runs steadily, every time 2 units of A are consumed, 2 units of B are produced and 1 unit of C is produced. So the rate at which A disappears must be twice the rate at which C appears. The rate at which B appears equals the rate at which A disappears (both have coefficient 2).

Note

The stoichiometric coefficients are not speeds themselves — they are scaling factors that connect the speeds of different species.

The Precise Statement

For a general reaction:

aA + bB \rightarrow cC + dD

The rate of reaction (often called the rate of the process, $r$ ) is defined as:

r = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}

Here:

$\frac{d[X]}{dt}$ is the instantaneous rate of change of concentration of species X (in mol L⁻¹ s⁻¹).
The minus sign is used for reactants (their concentration decreases with time).
The plus sign is used for products (their concentration increases with time).
Dividing by the coefficient normalises the rate — so $r$ is the same number for every species.

r = -\frac{1}{a}\frac{d[A]}{dt} = \frac{1}{c}\frac{d[C]}{dt}

This single value $r$ is the intrinsic rate of the reaction, independent of which species you measure.

How to Use It (Step by Step)

Suppose you are given:

$2 \text{H}_2 + \text{O}_2 \rightarrow 2 \text{H}_2\text{O}$

And you measure that $\frac{d[\text{H}_2]}{dt} = -0.040 \text{ M/s}$ (negative because H₂ is being consumed).

Step 1: Write the rate relation:

r = -\frac{1}{2}\frac{d[\text{H}_2]}{dt} = -\frac{1}{1}\frac{d[\text{O}_2]}{dt} = \frac{1}{2}\frac{d[\text{H}_2\text{O}]}{dt}

Step 2: Plug in the known value:

r = -\frac{1}{2}(-0.040) = +0.020 \text{ M/s}

Step 3: Find the rate for O₂:

-\frac{d[\text{O}_2]}{dt} = r \quad \Rightarrow \quad \frac{d[\text{O}_2]}{dt} = -0.020 \text{ M/s}

Step 4: Find the rate for H₂O:

\frac{1}{2}\frac{d[\text{H}_2\text{O}]}{dt} = r \quad \Rightarrow \quad \frac{d[\text{H}_2\text{O}]}{dt} = 2r = 0.040 \text{ M/s}

Tip

A quick check: the coefficients tell you the relative rates. Here, H₂ disappears twice as fast as O₂, and H₂O appears at the same rate as H₂ disappears (both coefficient 2). Always verify your numbers match the coefficient ratios.

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Why this formula?

Average Rate of Reaction — Why the Formula Holds

Let’s build this from the ground up. The goal is to understand why the average rate formula looks the way it does — not just memorise it.

1. What does "rate of reaction" mean physically?

A chemical reaction changes the concentration of reactants (decreasing) and products (increasing) over time.

Rate = how fast this change happens.
If you measure the change over a finite time interval, you get the average rate.

2. The core idea: change per unit time

For any quantity $X$ that changes from $X_1$ to $X_2$ over time $t_1$ to $t_2$ :

\text{Average rate of change of } X = \frac{\Delta X}{\Delta t} = \frac{X_2 - X_1}{t_2 - t_1}

This is just the slope of the straight line connecting the two points on a concentration vs. time graph.

3. Applying this to a reaction

Consider a simple reaction:

A \rightarrow B

Reactant A is consumed: $[A]$ decreases.
Product B is formed: $[B]$ increases.

For reactant A (disappearing):

\text{Average rate} = -\frac{\Delta [A]}{\Delta t}

Why the minus sign?

Because $\Delta [A] = [A]_2 - [A]_1$ is negative (concentration drops). The rate itself must be positive (speed is never negative). So we multiply by $-1$ .

For product B (appearing):

\text{Average rate} = +\frac{\Delta [B]}{\Delta t}

Here $\Delta [B]$ is positive, so no minus sign needed.

4. The general formula for any reaction

For a balanced reaction:

aA + bB \rightarrow cC + dD

The average rate is defined per mole of reaction — so it’s the same number regardless of which species you track.

We divide each $\frac{\Delta [\text{species}]}{\Delta t}$ by its stoichiometric coefficient:

\text{Average rate} = -\frac{1}{a} \frac{\Delta [A]}{\Delta t} = -\frac{1}{b} \frac{\Delta [B]}{\Delta t} = +\frac{1}{c} \frac{\Delta [C]}{\Delta t} = +\frac{1}{d} \frac{\Delta [D]}{\Delta t}

Why divide by the coefficient?

If 2 moles of A disappear for every 1 mole of C formed, then $\frac{\Delta [A]}{\Delta t}$ is twice as large as $\frac{\Delta [C]}{\Delta t}$ . Dividing by the coefficient normalises them to the same "per mole of reaction" rate.

5. Key exam point: the formula in one line

For any species $X$ with stoichiometric coefficient $\nu_X$ (negative for reactants, positive for products):

$\text{Average rate} = \frac{1}{\nu_X} \frac{\Delta [X]}{\Delta t}$ …

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The average rate of reaction is the change in concentration divided by the time interval. For the interval 30–60 s, the rate is $4.67 \times 10^{-3} \, \text{mol L}^{-1} \text{s}^{-1}$ .

The average rate of reaction tells us how fast the concentration of a reactant (or product) changes over a specific time interval. Unlike the instantaneous rate (which is the slope of the tangent at a single point), the average rate is simply the total change in concentration divided by the total time elapsed. It’s like calculating the average speed of a car on a trip — you don’t care about every bump and turn, just the total distance covered over the total time.

For a reaction $\text{A} \to \text{products}$ , the average rate over a time interval $\Delta t = t_2 - t_1$ is:

\text{Average rate} = -\frac{\Delta [\text{A}]}{\Delta t} = -\frac{[\text{A}]_{t_2} - [\text{A}]_{t_1}}{t_2 - t_1}

The negative sign is crucial: since $[\text{A}]$ decreases with time, $\Delta [\text{A}]$ is negative, so the negative sign makes the rate positive. Always remember — rate is a positive quantity.

Now let’s apply this to the given data.

Identify the time interval and concentrations.

We need the average rate between $t = 30$ s and $t = 60$ s.

From the table:

$[\text{A}]_{30} = 0.31 \, \text{mol L}^{-1}$

$[\text{A}]_{60} = 0.17 \, \text{mol L}^{-1}$
Calculate the change in concentration.

\Delta [\text{A}] = [\text{A}]_{60} - [\text{A}]_{30} = 0.17 - 0.31 = -0.14 \, \text{mol L}^{-1}

Calculate the time interval.

\Delta t = 60 - 30 = 30 \, \text{s}

Plug into the formula. $\text{Average rate} = -\frac{\Delta [\text{A}]}{\Delta t} = -\frac{(-0.14)}{30} = \frac{0.14}{30}$ …

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Common Mistakes & How to Avoid Them

Mistake 1: Using the wrong formula for average rate

The error: Students often confuse average rate with instantaneous rate, or use the formula for rate of disappearance of reactant incorrectly.

Correct approach: Average rate of reaction over a time interval $\Delta t$ is:

\text{Average rate} = -\frac{\Delta[A]}{\Delta t} = -\frac{[A]_{t_2} - [A]_{t_1}}{t_2 - t_1}

The negative sign is crucial — it converts the negative change in reactant concentration into a positive rate.

How to avoid: Always write the formula first, then substitute values. Remember: rate is always positive.

Mistake 2: Forgetting the negative sign

The error: Students compute $\frac{0.17 - 0.31}{60 - 30} = \frac{-0.14}{30} = -0.00467$ and leave it negative.

Correct calculation:

\text{Average rate} = -\frac{0.17 - 0.31}{60 - 30} = -\frac{-0.14}{30} = \frac{0.14}{30} = 0.00467 \text{ mol L}^{-1} \text{s}^{-1}

How to avoid: Train yourself to always include the negative sign in the formula. The final answer must be positive.

Mistake 3: Swapping the time interval order

The error: Writing $\frac{0.31 - 0.17}{30 - 60}$ instead of $\frac{0.17 - 0.31}{60 - 30}$ .

Both give the same numerical value, but the first is mathematically incorrect and can cause sign errors in more complex problems.

How to avoid: Always use: $\frac{[\text{final}] - [\text{initial}]}{\text{final time} - \text{initial time}}$

Mistake 4: Confusing average rate with rate constant

The error: Students try to use the integrated rate law for a pseudo first-order reaction:

k = \frac{2.303}{t} \log\frac{[A]_0}{[A]_t}

This gives the rate constant, not the average rate.

How to avoid: Read the question carefully. "Average rate of reaction" means $\frac{-\Delta[A]}{\Delta t}$ , not $k$ .

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Concept understanding — Average Rate Of Reaction

Reaction Rate Stoichiometry – From Intuition to Precision

The Intuitive Picture

The Precise Statement

How to Use It (Step by Step)

Why this formula?

Average Rate of Reaction — Why the Formula Holds

1. What does "rate of reaction" mean physically?

2. The core idea: change per unit time

3. Applying this to a reaction

For reactant A (disappearing):

For product B (appearing):

4. The general formula for any reaction

5. Key exam point: the formula in one line

Method: Average Rate of Reaction from Concentration–Time Data

Steps

Common Mistakes & How to Avoid Them

Mistake 1: Using the wrong formula for average rate

Mistake 2: Forgetting the negative sign

Mistake 3: Swapping the time interval order

Mistake 4: Confusing average rate with rate constant

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