Q.In a reaction, $2A \rightarrow$ Products, the concentration of A decreases from $0.5\ \text{mol L}^{-1}$ to $0.4\ \text{mol L}^{-1}$ in 10 minutes. Calculate the rate during this interval?

The key idea is the average rate of reaction, defined as the change in concentration per unit time, adjusted for the stoichiometric coefficient.

For the reaction $2A \rightarrow \text{Products}$, the average rate is:

Step 1: Find the change in concentration of A.

Step 2: Time interval is $\Delta t = 10\ \text{minutes}$.

Step 3: Substitute into the formula: …

The key idea here is that the rate of reaction is defined as the change in concentration of a reactant or product per unit time, adjusted for stoichiometry. For a reactant like A, its concentration decreases over time, so the rate is expressed as a positive number by taking the negative of the change.

Why do we divide by the stoichiometric coefficient? Because the rate of reaction should be the same regardless of which species we measure. In $2A \rightarrow$ Products, two molecules of A disappear for every reaction event, so the rate at which A disappears is twice the rate of the reaction itself. Dividing by 2 corrects for this.

Let’s work through the calculation step by step.

1. Identify the given data

   Initial concentration of A: $[A]_0 = 0.5\ \text{mol L}^{-1}$

   Final concentration of A: $[A]_t = 0.4\ \text{mol L}^{-1}$

   Time interval: $\Delta t = 10\ \text{minutes}$

2. Find the change in concentration of A

   Since A is a reactant, its concentration decreases:

   $\Delta [A] = [A]_t - [A]_0 = 0.4 - 0.5 = -0.1\ \text{mol L}^{-1}$

   The negative sign indicates a decrease.

3. Apply the formula for average rate of reaction

   For a general reaction $aA \rightarrow$ products, the average rate over an interval is:

Here $a = 2$, so:

4. Simplify the expression …

Method: Average Rate of Reaction (Using Change in Concentration Over Time)

This method calculates the average rate over a given time interval, not the instantaneous rate.

Steps:

1. Write the formula for average rate For a reactant $A$ in the reaction $2A \rightarrow \text{Products}$, the average rate is:

The factor $\frac{1}{2}$ comes from the stoichiometric coefficient of $A$. The negative sign accounts for the decrease in concentration of the reactant.

2. Calculate the change in concentration

3. Calculate the time interval

4. Plug into the formula …

Here are the common mistakes students make when solving this exact problem, along with how to avoid each one.

Mistake 1: Forgetting the Stoichiometric Coefficient

The error:

Students often write the rate as simply:

and plug in numbers directly. This ignores the coefficient 2 in front of A in the reaction $2A \rightarrow \text{Products}$.

Why it’s wrong:

The average rate of reaction is defined in terms of the stoichiometry. For the general reaction $aA \rightarrow \text{Products}$, the rate is:

Here $a = 2$, so you must divide by 2.

How to avoid:

Always write the rate formula with the coefficient before substituting. Memorise:

Mistake 2: Using the Wrong Sign for $\Delta[A]$

The error:

Students compute $\Delta[A] = [A]_{\text{final}} - [A]_{\text{initial}} = 0.4 - 0.5 = -0.1$, then plug this into the formula without the negative sign in front, getting a negative rate.

Why it’s wrong:

The rate of reaction is always positive (it measures how fast the reaction proceeds). The negative sign in $-\frac{\Delta[A]}{\Delta t}$ is there to cancel the negative $\Delta[A]$ (since concentration decreases).

How to avoid:

• First find $\Delta[A] = [A]_{\text{final}} - [A]_{\text{initial}}$ (which will be negative for a reactant).
• Then apply the formula with the explicit negative sign in front.
• Alternatively, use the magnitude: $\text{Rate} = \frac{1}{a} \cdot \frac{\text{decrease in concentration}}{\Delta t}$.

Mistake 3: Confusing Units or Time

The error:

Using time in seconds instead of minutes (or vice versa) without checking what the question expects. Here, time is given in minutes, so the rate will be in $\text{mol L}^{-1} \text{min}^{-1}$.

Why it’s wrong:

The unit of rate depends on the time unit used. If the question doesn’t specify, stick to the unit given. Changing units without converting correctly leads to wrong numerical answers.

How to avoid:

• Note the time unit in the question.
• If you must convert (e.g., to seconds for a standard unit), do it after calculating the rate in the given unit, or convert carefully. …