Question

A certain substance X decomposes. Fifty percent of X remains after 100 minutes. How much \(X\) remains after 200 minutes if the reaction order with respect to \(X\) is (a) zero order, (b) first order, (c) second order?

Short answer

After 200 minutes: (a) Zero-order: 0% X remains, (b) First-order: 25% X remains, (c) Second-order: More than 25% but less than 50% X remains since the rate slows down as concentration decreases.

Step-by-step solution

Understanding Zero-Order Reaction

In a zero-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactant. Therefore, the amount of substance X decreases at a constant rate over time. After 100 minutes, if 50% remains, it means that after another 100 minutes, X will be completely consumed.

Calculating Amount Remaining for Zero-Order Reaction

If 50% of substance X decomposes every 100 minutes, it will decompose another 50% in the next 100 minutes. Hence, no X will remain after 200 minutes in a zero-order reaction.

Understanding First-Order Reaction

In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant. The half-life of the substance is constant. Therefore, if 50% of substance X remains after 100 minutes, another 50% of the remaining X will decompose in the next 100 minutes.

Calculating Amount Remaining for First-Order Reaction

Since 50% of substance X remains after 100 minutes, 50% of the remaining half (25% of the original amount) will remain after another 100 minutes in a first-order reaction.

Understanding Second-Order Reaction

In a second-order reaction, the rate of reaction is proportional to the square of the concentration of the reactant. This implies that the rate of decomposition will slow down as the concentration decreases.

Calculating Amount Remaining for Second-Order Reaction

For a second-order reaction, the relationship between concentration and time is given by the equation \( \frac{1}{[X]} - \frac{1}{[X]_0} = kt \). We cannot determine the exact amount remaining without the rate constant k. However, we do know that more than 25% but less than 50% will remain because the rate slows down as the concentration decreases.

Key concepts

Understanding Zero-Order Reactions

When studying chemical kinetics, the concept of zero-order reactions is pivotal for grasping how certain reactions proceed. A zero-order reaction is one where the rate of reaction is constant and does not change as the concentration of the reactant decreases. This is an important feature because it means regardless of how much reactant you start with, the substance will decrease at a consistent rate over time.

Let's contextualize this with a simple scenario: Imagine we have 100 grams of a substance X, and this substance decomposes at a rate of 50 grams every hour. After the first hour, we'll have 50 grams left. Since the reaction is zero-order, the rate doesn't depend on how much substance X we have left. Therefore, in the next hour, the remaining 50 grams will also decompose, leaving us with 0 grams.

To make it even clearer, if we consider that 50% of substance X remains after 100 minutes as in the problem from the textbook, it's straightforward to conclude that after another 100 minutes, no substance X will remain when dealing with a zero-order reaction.

Deciphering First-Order Reactions

In contrast to zero-order reactions, first-order reactions are those where the reaction rate is directly proportional to the reactant concentration. It's like saying the more you have, the faster it goes. A good analogy might be a group of people eating a cake: The more people (reactant), the quicker the cake disappears. First-order reactions are characterized by a consistent half-life, meaning the time it takes for half the substance to decompose is always the same, regardless of the initial amount.

So, if you're told that 50% of substance X remains after 100 minutes, you can deduce that after another 100 minutes, half of what's left will also be gone. This halving continues with each successive time period equal to the half-life. In mathematical terms, if you start with 100% of substance X, after 100 minutes, you'll have 50%, and after 200 minutes, you're down to 25%.

Understanding first-order reactions is crucial because they are common in nature, especially in biological processes like radioactive decay or the metabolism of drugs in the body.

Exploring Second-Order Reactions

Moving on to second-order reactions, these involve a bit more complexity than the previous types. A second-order reaction rate is proportional to the square of the concentration of the reactant. Think of it as if two particles of the reactant need to come together for the reaction to happen, so the fewer particles you have, the less likely they are to collide and react.

In the cake-eating analogy, imagine only when two people meet can they take a bite together. As people leave the party, it gets rarer for such meetings, and so the cake eating slows down. Returning to our textbook example where 50% of substance X remains after 100 minutes, we can't simply halve it for the next 100 minutes because the rate slows down as concentration decreases.

Without the specific rate constant, we can’t calculate the exact amount remaining; nevertheless, we know it’s more than 25% because the rate of reduction in concentration decelerates over time. This decay slowdown in second-order reactions becomes very pronounced at lower concentrations, meaning they often require careful monitoring in practical applications like in chemical manufacturing or pharmaceuticals.