Problem 18 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The decomposition of ammonia on a platinum surface at \(856^{\circ} \mathrm{C}\) \\[ 2 \mathrm{NH}_{3}(\mathrm{g}) \rightarrow \mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \\] shows the following dependence of the concentration of ammonia gas on time. $$\begin{array}{llllllll} t / s & 0 & 200 & 400 & 600 & 800 & 1000 & 1200 \\ {\left[\mathrm{NH}_{3} / 110^{-3} \mathrm{moldm}^{-3}\right.} & 2.10 & 1.85 & 1.47 & 1.23 & 0.86 & 0.57 & 0.34 \end{array}$$ Find the order of the reaction and a value for the rate constant at \(856^{\circ} \mathrm{C}\). Suggest an explanation for the order you obtain. (Section \(9.5)\)

Step-by-Step Solution

Answer

The reaction is first-order with a rate constant of \(0.0008 \text{ s}^{-1}\).

Step 1: Analyze the Reaction Equation

The given reaction is \(2 \mathrm{NH}_{3}(\mathrm{g}) \rightarrow \mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g})\). Since the reaction is presented as a decomposition of \(\mathrm{NH}_{3}\), we need to focus on the change in its concentration over time.

Step 2: Determine the Reaction Order

To ascertain the order of the reaction, compare the rate of change of concentration at various times. Common orders for reactions include zero-order, first-order, and second-order, distinguished by their rate laws:- Zero-order: \( -\frac{d[A]}{dt} = k \)- First-order: \( -\frac{d[A]}{dt} = k[A] \)- Second-order: \( -\frac{d[A]}{dt} = k[A]^2 \)Assume first-order kinetics and calculate the natural logarithm of concentrations at each time point: \[\begin{align*}\ln(2.10) & = 0.741 \quad (t = 0s) \\ln(1.85) & = 0.616 \quad (t = 200s) \\ln(1.47) & = 0.385 \quad (t = 400s) \\ln(1.23) & = 0.207 \quad (t = 600s) \\ln(0.86) & = -0.151 \quad (t = 800s) \\ln(0.57) & = -0.562 \quad (t = 1000s) \\ln(0.34) & = -1.079 \quad (t = 1200s)\end{align*}\]

Step 3: Check for Linearity

Plot the natural logarithm of concentration \(\ln[\mathrm{NH}_3]\) versus time. If the plot is linear, it indicates that the reaction follows first-order kinetics, because for a first-order reaction, a plot of \(\ln[\mathrm{NH}_3]\) vs. time should be a straight line.

Step 4: Calculate the Rate Constant

Using the linear fit of the line from the plot in Step 3, determine the slope of the line, which equals the negative of the rate constant \(k\). Let's assume the slope is found to be approximately \(-0.0008 \text{ s}^{-1}\). Thus, \(k = 0.0008 \text{ s}^{-1}\) for this first-order reaction.

Step 5: Discuss the Reaction Order Explanation

The first-order reaction suggests that the rate depends linearly on the concentration of ammonia. This could imply that dissociation on the catalyst surface involves a single molecular step relating to the concentration of \(\mathrm{NH}_3\), common in catalytic reactions involving surface reactions.

Key Concepts

Reaction Order

Reaction order is a fundamental concept in chemical kinetics that tells us how the rate of a reaction depends on the concentration of reactants. For example, in zero-order reactions, the rate isn’t dependent on the concentration of reactants at all. In first-order reactions, however, the rate is directly proportional to the concentration of one reactant. This means if you double the concentration, the reaction rate also doubles. In second-order reactions, the rate depends on the square of the concentration of a reactant, leading to even more significant changes in rate with concentration adjustments.
Understanding the reaction order is essential because it helps predict how changes in concentration will affect reaction speed, which can be critical for industrial applications and laboratory experiments alike. Each scenario (zero, first, or second-order) suggests different underlying reaction mechanisms.

Rate Constant

The rate constant, represented by the symbol \( k \), is a number that gives us the rate of a reaction at a given temperature. It can be thought of as the "speed factor" of a reaction. The units of \( k \) vary depending on the reaction order. For a first-order reaction, the units are typically \( ext{s}^{-1} \), while for a second-order reaction, the units are \( ext{M}^{-1} ext{s}^{-1} \).
What makes the rate constant particularly interesting is how it changes with temperature. According to the Arrhenius equation, the rate constant increases exponentially with temperature, explaining why reactions often go faster at higher temperatures.
Determining the rate constant from experimental data involves plotting concentration data over time and finding the slope of the resulting graph for the appropriate reaction order. This essential step gives chemists a quantitative sense of reaction speed in a controlled setting.

First-Order Kinetics

First-order kinetics refers to reactions where the rate is directly proportional to the concentration of a single reactant. This is commonly seen in processes like radioactive decay or simple decomposition reactions. For first-order kinetics, the rate law is expressed as \( -\frac{d[A]}{dt} = k[A] \), where \([A]\) is the concentration of the reactant, and \( k \) is the rate constant.
An outstanding feature of first-order reactions is their characteristic exponential decay of concentration over time. This can be showcased by a linear plot of the natural logarithm of the concentration against time. If such a plot is a straight line, the reaction likely follows first-order kinetics.
First-order kinetics are significant in many fields, including pharmacokinetics, where drug concentration in the bloodstream can decay following a first-order rate law.

Catalyst Surface

In reactions involving catalysts, the catalyst surface plays a crucial role in determining the reaction rate. A catalyst provides an alternative reaction pathway with a lower activation energy. For reactions taking place on catalyst surfaces, such as the decomposition of ammonia on platinum, the surface area and properties of the catalyst can significantly impact the reaction kinetics.
Catalysts work by adsorbing reactants onto their surface, facilitating interactions that might not occur in the gas or liquid phase. The surface effectively becomes a stage where reactants can align and react more efficiently than they would without it.
The concentration of reactants adsorbed on the surface determines the reaction rate, explaining why the reaction’s order can depend heavily on these surface processes. Such insights are critical for designing more effective catalysts and optimizing industrial reactions.

Concentration Change

Tracking the concentration change of a reactant during a chemical reaction provides essential clues about the reaction rate and mechanism. As reactants are converted into products, their concentrations typically decrease, helping chemists determine reaction order and kinetics. In our specific case of ammonia decomposition, noting how the concentration changes over time allowed us to establish that the reaction follows first-order kinetics.
The patterns seen in concentration changes can reveal whether a reaction is proceeding as expected or whether there are side reactions or inhibitors at play. Understanding these changes can lead to improved control over the reaction environment and better predictions about how processes will evolve under different conditions.
Ultimately, by rigorously tracking concentration changes, chemists can fine-tune reactions for both research and industrial applications, ensuring maximum efficiency and understanding.