What will the initial rate be if [a] is halved and [b] is tripled?

When you ask, “What will the initial rate be if [a] is halved and [b] is tripled?” You’re diving into the fascinating world of chemical reactions. This question relates to the relationship between reactant concentrations and the reaction rate, a critical part of chemical kinetics. Let’s break down, step by step, what happens when reactant concentrations change and how those changes affect the reaction rate.

Understanding the Rate Law

The rate of a chemical reaction isn’t just a random occurrence—it’s a well-defined relationship between the reactants’ concentrations and the reaction’s speed. This relationship is expressed through something called the rate law. The general form of the rate law is:

Rate=k[A]m[B]n\text{Rate} = k[A]^m[B]^nRate=k[A]m[B]n

Where:

• The rate represents the speed of the reaction.
• k is the rate constant, a proportionality constant that remains constant for a particular response under continuous conditions.
• [A] and [B] are the reactants A and B concentrations, respectively.
• m and n are the reaction orders for A and B. These orders can vary depending on the specifics of the reaction and are typically determined experimentally.

The Impact of Changing Concentrations

Let’s look at the situation where A is halved and B is tripled. This change will directly impact the initial rate, and it’s crucial to understand how the reaction orders (m and n) influence this impact.

What Happens When A is Halved?

When A is halved, its concentration decreases. The exact impact of this change depends on the value of m, the reaction order concerning A:

• If m = 1 (a first-order reaction concerning A), halving [A] will also halve the reaction rate. This is because the rate is directly proportional to the concentration of A.
• If m = 2 (a second-order reaction concerning A), halving [A] will reduce the rate by a factor of 4. This is because the rate is proportional to the square of the concentration of A.

The general effect of halving [A] will be:

New Rate due to A=(12)m\text{New Rate due to A} = \left( \frac{1}{2} \right)^mNew Rate due to A=(21​)m

What Happens When B is Tripled?

Next, consider what happens when B is tripled. The reaction order concerning B, denoted by n, determines how much the rate will change:

• If n = 1 (a first-order reaction relating to B), tripling [B] will increase the rate by 3.
• If n = 2 (a second-order reaction relating to B), tripling [B] will increase the rate by 9.

In general, the effect of tripling [B] can be written as:

New Rate due to B=3n\text{New Rate due to B} = 3^nNew Rate due to B=3n

Combining the Effects

Now that we know how the changes in concentrations affect the rate individually let’s combine them. We multiply the changes to find the overall effect on the initial rate when both A is halved, and B is tripled. This means that the new initial rate will be:

New Initial Rate=Initial Rate×(12)m×3n\text{New Initial Rate} = \text{Initial Rate} \times \left( \frac{1}{2} \right)^m \times 3^nNew Initial Rate=Initial Rate×(21​)m×3n

Example Calculation

Let’s put this into practice with an example. Suppose the rate law is:

Rate=k[A]2[B]\text{Rate} = k[A]^2[B]Rate=k[A]2[B]

This means m = 2 and n = 1.

Now, if A is halved and B is tripled, the new initial rate would be:

New Initial Rate=Initial Rate×(12)2×31\text{New Initial Rate} = \text{Initial Rate} \times \left( \frac{1}{2} \right)^2 \times 3^1New Initial Rate=Initial Rate×(21​)2×31New Initial Rate=Initial Rate×14×3\text{New Initial Rate} = \text{Initial Rate} \times \frac{1}{4} \times 3New Initial Rate=Initial Rate×41​×3New Initial Rate=34×Initial Rate\text{New Initial Rate} = \frac{3}{4} \times \text{Initial Rate}New Initial Rate=43​×Initial Rate

In this case, the new initial rate is 3/4 of the original rate, meaning it has decreased because A was halved, but the increase from tripling B only partially compensates for that decrease.

Why Does This Matter?

Understanding how changes in concentration affect reaction rates is essential in many fields, including chemical engineering, environmental science, and pharmacology. For example, in industrial chemical processes, optimizing concentrations can help improve the efficiency of the reaction. Similarly, in environmental studies, understanding how pollutants react at different concentrations can provide insights into how to control or mitigate harmful effects.

Key Takeaways

• The rate law defines how reactant concentrations affect a chemical reaction’s speed.
• The reaction orders m and n determine the magnitude of the effect of concentration changes on the rate.
• When A is halved and B is tripled, the new initial rate can be calculated by multiplying the impact of these changes together: (12)m×3n\left( \frac{1}{2} \right)^m \times 3^n(21​)m×3n.
• This calculation allows us to predict how the reaction rate will change under different conditions, which is critical for understanding and controlling chemical reactions.

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Last Reflections

So, when you’re asked, “What will the initial rate be if [a] is halved and [b] is tripled?” you’re looking at how these changes in concentration, influenced by the reaction orders m and n, will impact the reaction rate. Understanding these relationships is key, whether you’re dealing with complex industrial processes or just studying chemical kinetics for fun.