Question

For the zero-order decomposition of ammonia on tungsten $$ \mathrm{NH}_{3}(g) \stackrel{\mathrm{W}}{\longrightarrow} \frac{1}{2} \mathrm{~N}_{2}(g)+\frac{3}{2} \mathrm{H}_{2}(g) $$ the rate constant is \(2.08 \times 10^{-4} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\). (a) What is the half-life of a \(0.250 \mathrm{M}\) solution of ammonia? (b) How long will it take for the concentration of ammonia to drop from \(1.25 \mathrm{M}\) to \(0.388 \mathrm{M}\) ?

Short answer

Answer: The half-life of a 0.250 M solution of ammonia is 600 seconds. It takes about 4144.23 seconds for the concentration of ammonia to drop from 1.25 M to 0.388 M.

Step-by-step solution

Calculate the Initial Concentration

In this case, the initial concentration of ammonia is given as 0.250 M.

Calculate Half-life for Zero-order Reaction

For a zero-order reaction, we need to find the half-life, which is the time it takes for the concentration to be reduced by half. We can rewrite our rate equation for a zero-order reaction as follows: $$ \frac{1}{2}[\mathrm{A}]_{0} - [\mathrm{A}]_{0} = -k \times t_{1/2} $$ Substitute the given values: $$ \frac{-1}{2}(0.250\ \mathrm{M}) = -(2.08\times10^{-4}\ \mathrm{M/s}) \times t_{1/2} $$

Solve for the Half-life Time

Now we can solve the equation for the half-life, \(t_{1/2}\): $$ t_{1/2}=\frac{\frac{1}{2}(0.250\ \mathrm{M})}{ 2.08 \times10^{-4}\ \mathrm{M/s}} $$ $$ t_{1/2}= 600\ \mathrm{s} $$ The half-life of a 0.250 M solution of ammonia is 600 seconds. (b) Time for the concentration of ammonia to drop from 1.25 M to 0.388 M

Calculate the Change in Concentration

We are given the initial concentration ([A]₀) as 1.25 M and the final concentration ([A]₁) as 0.388 M. We need to determine the change in concentration: $$ \Delta[\mathrm{A}] = [\mathrm{A}]_{1} - [\mathrm{A}]_{0} = 0.388\ \mathrm{M} - 1.25\ \mathrm{M} = -0.862\ \mathrm{M} $$

Calculate the Time Required for the Concentration Change

Using the rate equation for a zero-order reaction, we can find the time it takes for the concentration to change: $$ \mathrm{rate} = k = \frac{\Delta[\mathrm{A}]}{\Delta t} $$ Now, we can solve for the time (\(\Delta t\)): $$ \Delta t = \frac{-0.862\ \mathrm{M}}{2.08\times 10^{-4}\ \mathrm{M/s}} $$ $$ \Delta t = 4144.23\ \mathrm{s} $$ It will take about 4144.23 seconds for the concentration of ammonia to drop from 1.25 M to 0.388 M.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the rates of chemical reactions and the factors that affect these rates. Understanding kinetics is crucial for predicting how quickly a reactant will be consumed or how fast a product will form in a chemical process. In the case of a zero-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactants. This is counterintuitive to many students used to first-order kinetics, where the rate of the reaction is directly proportional to the concentration of the reactant. An everyday example to illustrate zero-order kinetics might be a catalyzed reaction on a surface when the surface is fully covered by the reactant, leading to a reaction rate that is independent of how much more reactant you add to the system.

Reaction Rate Constant

The reaction rate constant, symbolized as 'k', is a proportionality factor that provides a connection between the reaction rate and the concentrations of the reactants according to the rate law. For a zero-order reaction, the rate constant has units of concentration per time, typically M/s or mol/(L·s). This constant is a measure of how quickly a reaction proceeds. It's important to note that the rate constant is determined experimentally and varies with temperature, which follows the Arrhenius equation. A higher rate constant indicates a faster reaction rate. In the ammonia decomposition exercise, knowing the rate constant allowed us to calculate how long it will take for the ammonia concentration to decrease to a certain level.

Concentration Change

Concentration change refers to the difference in the amount of a substance in a certain volume over time during a chemical reaction. Recognizing that for a zero-order reaction the concentration change occurs at a constant rate is key to mastering this concept. In our ammonia decomposition example, this means that regardless of the starting concentration, the ammonia will be used up at the same rate. By calculating the change in concentration, students can determine both the half-life, which is a specific instance of concentration change, and the total time required for a reaction to reach a particular endpoint. This concept is not only pivotal for understanding kinetics but also has practical implications in fields like pharmaceuticals, where dosing is based on how quickly a drug concentration decreases in the body.