Problem 35 - Chemistry in 30 days

Question

Two colourless substances \(X\) and \(Y\) react to give a coloured substance \(Z\). The time \((t)\) taken for various initial concentrations of \(X\) and \(Y\) to produce a certain colour intensity are recorded in the table \begin{tabular}{|c|c|c|} \hline\([X] / m o l \mathrm{~L}^{-1}\) & {\([Y] / m o l \mathrm{~L}^{-1}\)} & time/s \\ \hline \(0.05\) & \(0.05\) & 44 \\ \hline \(0.05\) & \(0.10\) & 22 \\ \hline \(0.10\) & \(0.05\) & 44 \\ \hline \end{tabular} Which rate equation is consistent with these results? (a) rate \(=k[Y]^{1 / 2}\) (b) rate \(=k[X]^{1 / 2}[Y]^{1 / 2}\) (c) rate \(=k[Y]\) (d) rate \(=k[X][Y]\)

Step-by-Step Solution

Answer

The correct rate equation is (c) rate \( = k[Y] \).

Step 1: Understand Reaction Rates

The rate of reaction is typically expressed as \( \text{rate} = \frac{k}{t} \), where \( k \) is the rate constant and \( t \) is the time taken for the reaction to reach a certain state.

Step 2: Analyze Reaction Order

We need to determine how changing the concentrations of \([X]\) and \([Y]\) affect the reaction time \( t \). This will help us derive the order of reaction with respect to each substance.

Step 3: Compare Trials to Determine Effects of Concentration

Compare the given experimental data:1. When \([X] = 0.05\) and \([Y] = 0.05\), \( t = 44 \).2. When \([X] = 0.05\), doubling \([Y]\) to 0.10 halves \( t \) to 22.3. Doubling \([X]\) to 0.10 with \([Y] = 0.05\) keeps \( t = 44 \).

Step 4: Determine Order of Reaction for Each Reactant

- For \([Y]\): Doubling \([Y]\) decreases time \( t \) by half, suggesting first-order dependence (\( t \propto \frac{1}{[Y]} \)).- For \([X]\): Doubling \([X]\) does not affect \( t \), suggesting zero-order dependence (\( t \) is independent of \([X]\)).

Step 5: Identify Matching Rate Law

Considering that the reaction is first-order with respect to \([Y]\) and zero-order with respect to \([X]\), the rate equation should reflect this dependence. Therefore, the correct rate equation is \( \text{rate} = k[Y] \).

Key Concepts

Rate Equation

The rate equation is an essential concept for understanding how chemical reactions progress over time. It shows the relation between the rate of a reaction and the concentration of the reactants. Typically, the rate equation is expressed as:

• rate = k[X]^m[Y]^n

In this formula,

• k is the rate constant, a factor that allows us to calculate the rate of reaction and varies with temperature.
• [X] and [Y] represent the molar concentrations of the reactants, X and Y, respectively.
• m and n are the orders of reaction concerning X and Y, dictating how changes in concentrations affect the rate.

Using this equation, we can explore how different concentrations of reactants affect the reaction's speed. In our exercise, the determined rate equation is rate = k[Y], indicating a first-order dependence on Y and zero-order on X.

Reaction Order

Reaction order is crucial to understanding how a reaction's rate depends on the concentration of reactants. It tells us the power to which the concentration of a reactant is raised to in the rate equation. It doesn't have to match the stoichiometric coefficients from the chemical equation, as it is determined empirically.

• Zero Order: The rate is independent of the reactant’s concentration. For example, if [X] changes but rate remains the same, X is zero-order.
• First Order: The rate is directly proportional to reactant’s concentration. Doubling [Y] initially led to halving the time, indicating first-order dependence.
• Second Order: The rate is proportional to the square of the reactant’s concentration.

In our specific exercise, we deduced that Y affects the time inversely, confirming that it's first-order concerning Y. Meanwhile, X doesn't affect the reaction rate, signifying a zero-order relationship.

Rate Constant

The rate constant, denoted by the symbol k, is a key factor in a rate equation. It links the concentration of the reactants to the rate of the reaction. It's unique for each reaction and constant at a given temperature. However, it will change with temperature shifts or under the influence of catalysts.

• The rate constant is derived from experimental data.
• It helps calculate the reaction rate quickly, once the reactant concentrations and reaction order are known.
• A higher k indicates a faster reaction at the same concentrations.

In our context, since we are evaluating a first-order reaction with respect to Y, knowing k allows predictions about how fast the product Z will form under different conditions.

First-Order Reaction

A first-order reaction is one where the reaction rate is directly proportional to the concentration of a single reactant, i.e., changing the concentration of this reactant will change the rate of the reaction proportionately. The formula often used is:

• rate = k[A]

Here, the rate depends linearly on the concentration of reactant A.

• In our exercise, when the concentration of [Y] doubled, the time required to complete the reaction was halved, confirming the first-order nature with respect to Y.
• This relationship means that if [Y] increases, the rate of producing Z through the reaction will also increase, cutting the time needed to reach a certain extent of the reaction.

First-order reactions often imply a simple, single-step process where only one reactant participates in the rate-determining step. This simplicity makes them easy targets for analysis and understanding in chemical kinetics.