Q.In a reaction if the concentration of reactant A is tripled, the rate of reaction becomes twenty seven times. What is the order of the reaction?

The key idea is that the rate of a reaction depends on the concentration of reactants raised to a power, called the order.

Step 1: For a reaction of order $n$ with respect to A, the rate law is:

$\text{Rate} = k[A]^n$ …

Why this approach works

The rate law for a reaction involving a single reactant A is:

where $n$ is the order of the reaction with respect to A. When we change the concentration of A, the rate changes according to that exponent $n$. If we triple $[A]$, the new rate becomes:

The problem tells us that $\text{Rate}_\text{new} = 27 \times \text{Rate}_\text{old}$. So we simply need to find $n$ such that $3^n = 27$.

Step-by-step solution

1. Write the general rate law For a reaction where only the concentration of A affects the rate:

2. Express the new rate after tripling $[A]$ New concentration: $[A]_\text{new} = 3[A]_\text{old}$

3. Use the given ratio of rates The problem states: $\text{Rate}_\text{new} = 27 \times \text{Rate}_\text{old}$ Therefore:

4. Solve for $n$ Since $27 = 3^3$, we have: …

Method: Initial Rate Method (Order from Concentration-Rate Dependence)

This method uses the direct proportionality between rate and concentration raised to the power of the order.

Steps

1. Write the general rate law For a reaction:

where $n$ is the order with respect to A.

2. Set up the ratio of two conditions

   Let initial rate $= r_1$ when $[A] = C$

   New rate $= r_2$ when $[A] = 3C$

   Given: $r_2 = 27 \times r_1$

3. Write the rate expressions

4. Divide the equations

5. Substitute the given ratio

6. Solve for $n$ …

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

Mistake 1: Confusing the factor change with the order

What students do wrong:

They see "tripled" and "twenty seven times" and guess the order is 3 because $3 \times 9 = 27$ or because $3^3 = 27$. That part is correct, but they often skip writing the rate law and jump to the answer without checking the logic.

How to avoid:

Always write the general rate law first:

Then apply the change:

• Initial: $\text{Rate}_1 = k [A]^n$
• After tripling: $\text{Rate}_2 = k (3[A])^n = k \cdot 3^n [A]^n$

Given $\text{Rate}_2 = 27 \times \text{Rate}_1$, we have:

Cancel $k[A]^n$:

Since $27 = 3^3$, we get $n = 3$.

Key takeaway: Always set up the ratio $\frac{\text{Rate}_2}{\text{Rate}_1} = (\text{factor})^n$ and solve for $n$.

Mistake 2: Forgetting that concentration change applies to the entire reactant

What students do wrong:

They treat "tripled" as adding 3 to the concentration instead of multiplying by 3. For example, they might write $[A] + 3$ instead of $3[A]$.

How to avoid:

Remember: "tripled" means multiply by 3, not add 3. The rate law uses powers, so:

Never write $k ([A] + 3)^n$ — that is incorrect.

Mistake 3: Misidentifying the order when the factor is not a perfect power

What students do wrong:

If the rate increases by a factor that is not a simple power (e.g., 8 times for doubling concentration), they might guess the order incorrectly. Here, 27 is $3^3$, so it's clean — but students sometimes think $27 = 3^2 \times 3$ and get confused.

How to avoid:

Use logarithms if needed:

For non-integer orders, take $\log$ of both sides:

Mistake 4: Forgetting that the rate constant $k$ cancels out

What students do wrong:

They try to solve for $k$ or think they need its value. This wastes time and leads to errors.

How to avoid:

Remember: $k$ is constant at a fixed temperature. When you take the ratio of two rates for the same reaction at the same temperature, $k$ cancels. So you only need the concentration factor and the rate factor.

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