Problem 23 - Chemistry and Chemical Reactivity

Question

The decomposition of ammonia on a metal surface to form \(\mathrm{N}_{2}\) and \(\mathrm{H}_{2}\) is a zero-order reaction (Figure \(14.7 \mathrm{c}) .\) At \(873^{\circ} \mathrm{C},\) the value of the rate constant is \(1.5 \times 10^{-3} \mathrm{mol} / \mathrm{L} \cdot \mathrm{s}\). How long it will take to completely decompose 0.16 g of \(\mathrm{NH}_{3}\) in a \(1.0-\mathrm{L}\) flask?

Step-by-Step Solution

Answer

It will take approximately 6,260 seconds to completely decompose 0.16 g of NH3.

Step 1: Convert grams to moles

First, we need to convert the mass of ammonia (NH_3 m) into moles. The molar mass of ammonia is calculated using the atomic weights: 1 nitrogen (N) = 14.01 g/mol + 3 hydrogen (H) = 3 x 1.008 g/mol = 17.034 g/mol. Given 0.16 g of NH_3 m, the number of moles of NH_3 m is calculated as follows: \[ \text{moles of } \mathrm{NH}_{3} = \frac{0.16 \text{ g}}{17.034 \text{ g/mol}} \approx 0.00939 \text{ moles} \].

Step 2: Use the zero-order rate equation

For a zero-order reaction, the rate of reaction is constant and given by: \[ \text{Rate} = k \].Where \( k \) is the rate constant, and the concentration changes with time are expressed by: \[ \text{[NH}_3\text{]}_t = \text{[NH}_3\text{]}_0 - kt \].Here,\( \text{[NH}_3\text{]}_t \) is the concentration at time \( t \), \( \text{[NH}_3\text{]}_0 \) is the initial concentration.In this case, we decompose the whole amount, so \( \text{[NH}_3\text{]}_t \ = 0 \).

Step 3: Calculate initial concentration

The initial concentration \( \text{[NH}_3\text{]}_0 \) is determined as follows: \[ \text{[NH}_3\text{]}_0 = \frac{\text{moles of } \mathrm{NH}_{3}}{\text{volume of flask}} = \frac{0.00939 \text{ moles}}{1.0 \text{ L}} = 0.00939 \text{ M} \].

Step 4: Rearrange rate equation and solve for time

We rearrange the equation\[ 0 = \text{[NH}_3\text{]}_0 - kt \]to solve for time \( t \):\[ t = \frac{\text{[NH}_3\text{]}_0}{k} \].Substitute the known values \( \text{[NH}_3\text{]}_0 = 0.00939 \text{ M} \) and \( k = 1.5 \times 10^{-3} \text{ mol/L} \cdot \text{s} \):\[ t = \frac{0.00939 \text{ M}}{1.5 \times 10^{-3} \text{ mol/L} \cdot \text{s}} \approx 6.26 \times 10^{3} \text{ s} \].

Key Concepts

Ammonia Decomposition

The decomposition of ammonia ( H_{3} ) on a metal surface is an important zero-order reaction. This type of reaction typically occurs on a catalyst surface, where the rate is independent of the concentration of the reactant. As ammonia decomposes, it breaks down into nitrogen ( _{2} ) and hydrogen ( _{2} ). This transformation is achieved at a higher temperature, such as 873°C in the given example. The process is crucial in various industrial applications, including the production of hydrogen gas. Understanding the decomposition of ammonia helps us understand complex reaction pathways and energy transformations.

Rate Constant

In chemical kinetics, the rate constant () plays a pivotal role. It is a measure of how quickly a reaction proceeds. For the zero-order reaction described, the rate constant is given as \(1.5 \times 10^{-3} \text{ mol/L} \cdot \text{s}\). Here are some key points about the rate constant:

• It is a fixed value at a specific temperature.
• For zero-order reactions, it translates directly to the rate because changes in concentration do not affect the rate.
• Knowing the rate constant allows us to determine the time needed for a reaction to reach completion, especially when the initial concentration is known.

This constant is critical for calculating how fast a certain amount of ammonia decomposes under specified conditions.

Reaction Kinetics

Reaction kinetics explores how and why reactions occur at specific rates. This field of study looks at various factors influencing the reaction rate, including temperature, concentration, and the presence of catalysts. In zero-order reactions:

• The rate is constant over time.
• The concentration of the reactant decreases linearly with time.
• The half-life depends on the initial concentration and decreases as the reaction progresses.

By studying kinetics, chemists can predict reaction behavior, improve processes, and design better industrial applications.

Chemical Concentration Calculations

Calculating chemical concentrations involves converting amounts from one unit to another (e.g., grams to moles) and using those values to determine molarity. Here's how it applies to our example:

• First, convert the mass of ammonia to moles using its molar mass. For NH_3, this is approximately 17.034 g/mol. Thus, 0.16 g of NH_3 equals about 0.00939 moles.
• The initial concentration is found by dividing the moles by the volume of the container (1.0 L), giving 0.00939 M.
• Using the concentration and rate constant, the time to fully decompose the ammonia can be calculated. Using \(t = \frac{\text{[NH}_3\text{]}_0}{k}\), we find the time to be approximately 6260 seconds.

These calculations form the backbone of understanding reaction kinetics and their real-world applications.