How can you determine the rate law of the following reaction? $2NO(g) + O_2(g) \rightarrow 2NO_2(g)$

How can you determine the rate law of the following reaction? 2NO(g) + O₂(g) arrow 2NO₂(g)

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2NO(g) + O_2(g) \rightarrow 2NO_2(g)

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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.

--- …

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 rate law is determined experimentally from initial-rate data, not from the stoichiometric coefficients. For the reaction $2NO + O_2 \rightarrow 2NO_2$ , the experimentally observed rate law is $\text{Rate} = k[NO]^2[O_2]$ , making it third-order overall.

The biggest mistake students make here is looking at the balanced equation and writing $\text{Rate} = k[NO]^2[O_2]$ because "the coefficients say so." That is wrong. The rate law is an experimental fact, not a prediction from the balanced equation. The coefficients in the balanced equation tell you the stoichiometric relationship — how much of each reactant is consumed or product formed — but they do not tell you how the rate depends on concentration.

For example, the reaction $2NO + O_2 \rightarrow 2NO_2$ could, in principle, have a rate law like $\text{Rate} = k[NO][O_2]$ or $\text{Rate} = k[NO]^2$ or even $\text{Rate} = k[O_2]$ . Only experiment can decide.

\text{Rate} = k[\text{NO}]^m[\text{O}_2]^n

Here is how you actually determine the rate law, step by step.

Collect initial-rate data. You run the reaction several times, each time changing the initial concentration of one reactant while keeping the other constant. You measure the initial rate (the slope of concentration vs. time at $t=0$ ) for each trial. A typical data set for this reaction looks like:

Trial	$[NO]_0$ (M)	$[O_2]_0$ (M)	Initial Rate (M/s)
1	0.10	0.10	$2.5 \times 10^{-3}$
2	0.20	0.10	$1.0 \times 10^{-2}$
3	0.10	0.20	$5.0 \times 10^{-3}$

Find the order with respect to NO. Compare trials where $[O_2]$ is constant and $[NO]$ changes. Here, trials 1 and 2: $[O_2]$ is fixed at 0.10 M, and $[NO]$ doubles from 0.10 to 0.20 M. The rate goes from $2.5 \times 10^{-3}$ to $1.0 \times 10^{-2}$ M/s — that is a factor of 4 increase. Since $2^m = 4$ , we get $m = 2$ . So the reaction is second order in NO. …

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Method: Initial Rates Method

This is the standard experimental approach to determine the rate law when you have concentration vs. time data from multiple trials.

Steps

Step 1: Write the general rate law form

For the reaction:

2NO(g) + O_2(g) \rightarrow 2NO_2(g)

The rate law is:

\text{Rate} = k[\text{NO}]^m[\text{O}_2]^n

Here, $m$ and $n$ are the orders with respect to NO and O₂, and $k$ is the rate constant.

Step 2: Collect data from at least two trials

You need initial rates at different initial concentrations. A typical data table looks like:

Trial	[NO] (M)	[O₂] (M)	Initial Rate (M/s)
1	0.10	0.10	$R_1$
2	0.20	0.10	$R_2$
3	0.10	0.20	$R_3$

Step 3: Find the order with respect to NO ( $m$ )

Compare trials where [O₂] is constant (e.g., Trial 1 and Trial 2):

\frac{R_2}{R_1} = \frac{k[0.20]^m[0.10]^n}{k[0.10]^m[0.10]^n} = \left(\frac{0.20}{0.10}\right)^m = 2^m

If $R_2/R_1 = 2$ , then $m = 1$ (first order in NO).

If $R_2/R_1 = 4$ , then $m = 2$ (second order in NO).

Step 4: Find the order with respect to O₂ ( $n$ )

Compare trials where [NO] is constant (e.g., Trial 1 and Trial 3):

\frac{R_3}{R_1} = \frac{k[0.10]^m[0.20]^n}{k[0.10]^m[0.10]^n} = \left(\frac{0.20}{0.10}\right)^n = 2^n

If $R_3/R_1 = 2$ , then $n = 1$ (first order in O₂). …

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✗ Mistake 1: Assuming the rate law from the balanced equation

What students do:

They see $2NO + O_2 \rightarrow 2NO_2$ and write:

\text{Rate} = k[NO]^2[O_2]

Why it’s wrong:

Rate law is experimentally determined, not derived from stoichiometric coefficients. The coefficients only tell you the order with respect to each reactant if the reaction is elementary — but most reactions are multi-step.

How to avoid:

Always remember: Rate law is an experimental fact, not a prediction from the balanced equation. Unless the problem explicitly says “elementary reaction,” never use coefficients as exponents.

✗ Mistake 2: Confusing average rate with instantaneous rate

What students do:

They calculate the average rate over a time interval and treat it as the rate law constant.

Why it’s wrong:

Average rate changes with time. The rate law uses instantaneous rate (the slope of concentration vs. time at a specific moment).

How to avoid:

For rate law determination, always use initial rates (rate at $t \approx 0$ ).
Average rate is useful for monitoring progress, not for finding the rate law.

✗ Mistake 3: Forgetting to account for stoichiometric coefficients in rate definition

What students do:

They write:

\text{Rate} = -\frac{\Delta[NO]}{\Delta t}

without dividing by the coefficient.

Why it’s wrong:

By IUPAC definition, for $aA + bB \rightarrow cC$ :

\text{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}

For this reaction:

\text{Rate} = -\frac{1}{2}\frac{\Delta[NO]}{\Delta t} = -\frac{\Delta[O_2]}{\Delta t} = \frac{1}{2}\frac{\Delta[NO_2]}{\Delta t}

How to avoid:

Always write the rate definition with the reciprocal of the coefficient in front. This ensures consistency when comparing rates from different species.

✗ Mistake 4: Using wrong units or forgetting to check units

What students do:

They calculate $k$ without verifying that the units match the overall order.

Why it’s wrong: …

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Q.How can you determine the rate law of the following reaction? 2NO(g)+O2(g)→2NO2(g)2NO(g) + O_2(g) \rightarrow 2NO_2(g)2NO(g)+O2​(g)→2NO2​(g)

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Q.How can you determine the rate law of the following reaction?
$2NO(g) + O_2(g) \rightarrow 2NO_2(g)$

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