Question

A chemical plant accidentally discharges a pollutant A into a river. Fortunately, A degrades in time. Calculate how far from the plant the concentration of the pollutant has decreased by \(90 \%\). What is the best reactor model for this problem? What is the kinetic order for this reaction? Data Reaction rate coefficient \(\mathrm{k}=0.0008 \mathrm{~mol}^{0.5} \cdot \mathrm{m}^{1.5} \cdot \mathrm{min}^{-1}\) River flow \(\varphi_{V}=200 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) Cross section of the river \(S=300 \mathrm{~m}^{2}\) Concentration of \(\mathrm{A}\) in the plant \(\mathrm{c}_{\mathrm{A} 0}=0.02 \mathrm{~mol} \cdot \mathrm{m}^{-3}\)

Short answer

Answer: The distance from the plant where the concentration of pollutant A has decreased by 90% is 600 meters. The best reactor model for this problem is the plug flow reactor (PFR) model, and the kinetic order for this reaction is 0.5.

Step-by-step solution

Find the final concentration

Determine the concentration of the pollutant A after it has decreased by 90%. We are given the initial concentration of A, \(c_{A0}\), as \(0.02 \mathrm{~mol} \cdot \mathrm{m}^{-3}\). To find the final concentration, multiply \(c_{A0}\) by 0.1 (as 90% of the concentration has decreased):$$c_{A} = 0.1 \times c_{A0} = 0.1 \times 0.02 \mathrm{~mol} \cdot \mathrm{m}^{-3} = 0.002 \mathrm{~mol} \cdot \mathrm{m}^{-3}$$

Find the degradation time

Use the reaction rate coefficient, \(k\), to calculate the time it takes for the pollutant concentration to decrease from \(c_{A0}= 0.02 \mathrm{~mol} \cdot \mathrm{m}^{-3}\) to \(c_{A}= 0.002 \mathrm{~mol} \cdot \mathrm{m}^{-3}\).$$0.1 \times Ra = k \times c_{A}^{n}$$Here \(n\) is the kinetic order, and \(Ra = \frac{dc_{A}}{dt}\) is the reaction rate. As 90% of the concentration has decreased, we get:$$Ra = \frac{0.9 \times c_{A0}}{t}$$From this, we can find the time, \(t\):$$\frac{0.9 \times c_{A0}}{t} = k \times c_{A}^{n}$$The values of \(c_{A0}\), \(k\), and \(c_{A}\) are given, but we do not know \(n\). However, we do know the units of \(k\). We can use the units of \(k\) to find the kinetic order \(n\). The units of \(k\) are given as \(\mathrm{mol}^{0.5} \cdot \mathrm{m}^{1.5} \cdot \mathrm{min}^{-1}\). The units of \(Ra\) are \(\mathrm{mol} \cdot \mathrm{m}^{-3} \cdot \mathrm{min}^{-1}\). So, we can write:$$\mathrm{mol}^{n-0.5} \cdot \mathrm{m}^{3n-1.5} \cdot \mathrm{min}^{-1} = \mathrm{mol} \cdot \mathrm{m}^{-3} \cdot \mathrm{min}^{-1}$$From the comparison of exponents, we find \(n=0.5\). Now we can calculate time, \(t\):$$\frac{0.9 \times c_{A0}}{t} = k \times c_{A}^{0.5}$$Solving for \(t\), we get:$$t = \frac{0.9 \times c_{A0}}{k \times c_{A}^{0.5}} = \frac{0.9 \times 0.02 \text{~mol} \cdot \text{m}^{-3}}{0.0008 \text{~mol}^{0.5} \cdot \text{m}^{1.5} \cdot \text{min}^{-1} \times (0.002\text{~mol} \cdot \text{m}^{-3})^{0.5}} \approx 15\text{~min}$$

Calculate the distance from the plant

Using the time it takes for the pollutant concentration to decrease, find the distance from the plant. The river flow is given as \(\varphi_{V} = 200 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) and the cross-section of the river is given as \(S = 300 \mathrm{~m}^{2}\). We can use these values to calculate the average velocity of the river:$$v_{\text{ave}} = \frac{\varphi_{V}}{S} = \frac{200 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}}{300 \mathrm{~m}^{2}} = \frac{2}{3} \mathrm{~m} \cdot \mathrm{s}^{-1}$$Now, we can find the distance from the plant where the concentration has decreased by 90%:$$x = v_{\text{ave}} \times t$$To match the units of time, we must convert 15 minutes to seconds$$t = 15 \text{~min} \times \frac{60 \text{~s}}{1 \text{~min}} = 900 \text{~s}$$Now, calculate \(x\):$$x = \left(\frac{2}{3} \mathrm{~m} \cdot \mathrm{s}^{-1}\right) \times 900 \mathrm{~s} = 600 \mathrm{~m}$$

Determine best reactor model

Given the details of the problem, the plug flow reactor (PFR) model best suits this situation. A PFR model assumes a continuous flow of fluid with distinct inlet and outlet points, which is a good representation of the river with a chemical plant discharging pollutant A into it. Moreover, the PFR model considers fluid elements flowing along the length of the reactor, which represents the degradation of pollutant A along the river.

State the kinetic order for this reaction

As we found the kinetic order \(n\) in Step 2, the kinetic order for this reaction is \(0.5\). In conclusion, the distance from the plant where the concentration of pollutant A has decreased by 90% is 600 meters. The best reactor model for this problem is the plug flow reactor (PFR) model, and the kinetic order for this reaction is 0.5.

Key concepts

Reaction Rate Coefficient

Understanding the reaction rate coefficient is essential when delving into chemical kinetics. Essentially, it quantifies the rate at which a chemical reaction proceeds under specific conditions. It integrates factors such as temperature, pressure, and concentration, and is represented by the symbol 'k'. The rate law for a reaction generally follows the form \( rate = k \times [A]^{m} \times [B]^{n} \), where \( [A] \) and \( [B] \) are the concentrations of reactants, and \( m \) and \( n \) are their respective orders of reaction.

In our textbook exercise, we find that the reaction rate coefficient \( k \) is \( 0.0008 \) with units of \( \mathrm{mol}^{0.5} \cdot \mathrm{m}^{1.5} \cdot \mathrm{min}^{-1} \). The units indicate the specific reaction order and the necessary conditions for which the coefficient is valid. Knowing 'k' allows us to predict how rapidly the concentration of a pollutant decreases, which is crucial for addressing environmental issues like an accidental discharge into a river.

To improve comprehension, focus on the practical applications of 'k' in predicting reaction progress and relate it to real-world scenarios like environmental clean-ups and industrial processes.

Kinetic Order

The kinetic order of a reaction reveals the relationship between the concentration of reactants and the reaction rate. It is deduced by determining how the rate varies as concentrations change. If the rate doesn’t change with varying concentrations, the reaction is zero-order in the respective reactant. A first-order reaction implies that the rate is directly proportional to the concentration.

How do we determine this? We look at the units of the reaction rate coefficient, as they provide vital clues to the kinetic order. In this case, the exercise demonstrates that given the units of \( k \), we can deduce the kinetic order by examining the dependencies of the rate on concentration, ultimately finding that the order is 0.5. This is also called a half-order reaction, which is not as common as zero or first-order reactions but occurs under certain rate-determining mechanisms.

Recognizing the kinetic order is more than an academic exercise; it's directly applicable to designing chemical processes, ensuring safety, and advancing our ability to respect and restore natural water sources.

Plug Flow Reactor Model

The plug flow reactor (PFR) model is applied when looking at reactors where reactants flow through a pipe or channel. It is a common model used in engineering because of its simplicity and ability to model flow systems like pipes, rivers, and other continuous flow channels. In a PFR, the reactants enter one end and flow through the system, reacting over time without any mixing or backflow.

For a PFR, we assume that the concentration of reactants and the reaction rate remain constant across any cross-section perpendicular to the flow. This theory was employed in our exercise to determine the distance downstream where the concentration of pollution has reduced by 90%. The model is valuable in environmental engineering when assessing the dispersal and degradation of contaminants in rivers and waterways.

As an exercise improvement tip, envision the PFR not just as a theoretical construct but as a simplified representation of real-world systems, which helps us comprehend and tackle environmental issues that can impact ecosystems and human health.