Problem 90 - A Complete Resource Book in Chemistry for JEE Main

Question

In a zero-order reaction, \(47.5 \%\) of the reactant remains at the end of \(2.5\) hours. The amount of reactant consumed in one hour is (a) \(11.0 \%\) (b) \(33.0 \%\) (c) \(42.0 \%\) (d) \(21.0 \%\)

Step-by-Step Solution

Answer

The amount of reactant consumed in one hour is \(21.0\%\).

Step 1: 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. The rate equation is given by:\[[A] = [A]_0 - k t\]where \([A]\) is the concentration of the reactant at time \(t\), \([A]_0\) is the initial concentration of the reactant, and \(k\) is the rate constant.

Step 2: Define Known Values

Let's denote initial concentration \([A]_0\) as 100% since we are dealing with percentages. At \(t = 2.5 \text{ hours}\), \([A] = 47.5\%\). We need to find the amount consumed in 1 hour.

Step 3: Calculate Rate Constant

Using the zero-order equation:\[[A] = [A]_0 - k t\]Substitute to find \(k\):\[47.5 = 100 - k \times 2.5\]which simplifies to\[k = \frac{52.5}{2.5} = 21 \text{ percent per hour}.\]

Step 4: Amount Consumed in One Hour

The amount consumed in one hour can now be determined because the rate is constant (21% per hour). Therefore, in one hour, 21% of the reactant is consumed.

Key Concepts

Rate of Reaction

The rate of reaction refers to the speed at which reactants are converted into products over time. It’s an important concept in chemistry, revealing how fast a chemical process occurs. In a zero-order reaction, like the one described above, the rate of reaction is constant. This means that it does not change even if the concentration of the reactant decreases. This differs from first and second-order reactions, where the rate depends on the concentration of the reactants.

The constancy of the rate in a zero-order reaction can be expressed by the equation: \[ ext{Rate} = - \frac{d[A]}{dt} = k \] Here, \(k\) is the rate constant and \(d[A]/dt\) represents the change in concentration of \(A\) over time \(t\). In this context, a constant rate signifies that the concentration of the reactant decreases linearly over time. This behaviour is particularly observed in reactions carried out under certain conditions, like those that involve catalysts on surfaces.

Rate Constant

The rate constant \(k\) is a crucial part of understanding reaction rates and kinetics. In the context of a zero-order reaction, the rate constant reflects a fixed decomposition rate of the reactant. Unlike other reactions, where the rate depends on reactant concentrations, here it only relies on the nature of the reaction and conditions like temperature, the presence of a catalyst, or surface area.

To determine the rate constant for a given zero-order reaction, you can use the formula: \[ k = \frac{{[A]_0 - [A]}}{t} \] By substituting known values into this equation, you can find \(k\). In our example, with an initial reactant concentration of 100% that falls to 47.5% after 2.5 hours, the calculation yielded a rate constant of 21% per hour. This constant rate means that each hour, 21% of the reactant is converted to product, showcasing the predictable nature of zero-order kinetics.

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical processes occur and the factors affecting these rates. It is essential for understanding how reactions progress over time.

In the case of a zero-order reaction, reaction kinetics simplifies greatly since the rate does not depend on reactant concentration. The zero-order reaction can be graphically represented by a straight line when you plot the concentration of a reactant versus time. The slope of this line is the negative rate constant \( -k \). This distinctive feature helps chemists identify zero-order kinetics in experimental data.

Kinetics studies also help in understanding the mechanism of the reaction, identifying whether it’s driven by a surface process, or if it's under saturation. This informs how process conditions might need adjustments to optimize the reaction yield, aimed at a consistent production rate by managing parameters such as temperature or catalyst use.